\begin{equation}\label{mujkL1la}
\|\M_{j,k,(\iota_1,0)}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}.
\end{equation}

Again, we can sum in $\iota_1\lesssim j,$ and combining all our estimates we see that the contribution $\M_{j,k,|Y_1|\gg 1}^\la$
of the region where $|Y_1|\gg 1$ to  $\M_{j,k}^\la$ can be estimated by

\begin{equation}\label{Mjk>1}
\|\M_{j,k,|Y_1|\gg 1}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-j}+ \la^{1+\ve}2^{-j-k},
\end{equation}

for every $\ve>0.$

What remains is the contribution $\M_{j,k,|Y_1|\lesssim 1}^\la$ of the region where $|Y_1|\lesssim 1$ to  $\M_{j,k}^\la.$
A look at the phase (<ref>)  shows that we are now in the position to apply Lemma <ref> to the oscillatory integral $F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)$ in the variables $(s_1,z)$ in place of $(\eta,x),$ with $N:=\la 2^{-j}\ge 1.$
Moreover, observe that $s_1$ and $z$   result  from our original coordinates $s_1$ and $z_1=x_1$ by means of  a smooth change of coordinates.  Passing back to our original coordinate, according to Remarks <ref>  we therefore must evaluate the original phase $\breve\phi_1(x_1, s_2)+s_1x_1-s_1y_1$ in (<ref>) at $x_1=y_1,$ and thus get

\begin{equation}\label{applemma5.2}
F_{j,k,|Y_1|\lesssim1}(\la,s_2,s_3,y_1)   = (\la2^{-j})^{-1} 2^{-j}e^{-i\la s_3(\breve\phi_1(y_1, s_2)}    \eta(y_1,2^{-k}\si_2).
\end{equation}

This implies that

\mu_{j,k,|Y_1|\lesssim1}^\la(y+\Ga)=\la^{\frac32}2^{-k}
\int  e^{-i\la s_3\big(\breve\phi_1(y_1, 2^{-k}\si_2)+2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3\big)} \chi_1(\si_2) \chi_1(s_3) \,\eta d\si_2ds_3.

By (<ref>), the complete phase is here given by

2^{-2k}\big[\si_2^2 B(2^{-k}\si_2)+y_1\si_2^2 q(y_1, 2^{-k}\si_2) -2^{k}\si_2 (y_2 -y_1^m\omega(y_1))\big]   +y_1^n\beta(y_1)-y_3.

In order to estimate the result of the integration in $\si_2,$ we can argue now in a similar way as before and decompose the region where $|Y_1|\lesssim 1$ into the subregions

\begin{eqnarray*}
\Delta_{\iota_2}&:=& \{y:|Y_1|\lesssim 1,2^k |y_2 -y_1^m\omega(y_1)|\sim 2^{\iota_2}\}, 1\ll \iota_2\le k,\\
\Delta_{0}&:=& \{y:|Y_1|\lesssim 1, 2^k|y_2 -y_1^m\omega(y_1)|\lesssim 1\}.
\end{eqnarray*}

|\Delta_{\iota_2}|\lesssim 2^{-j} 2^{\iota_2-k}.

If $\iota_2\gg 1,$ then we may integrate by parts in $\si_2$ and obtain

|\mu_{j,k,|Y_1|\lesssim1,\iota_2}^\la(y+\Ga)|\lesssim \la^{\frac32}2^{-k}(\la 2^{-2k+\iota_2})^{-1}=\la^{\frac 12}2^{k-\iota_2},

\|\M_{j,k,|Y_1|\lesssim1,\iota_2}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12}2^{-j}.

And, if $\iota_2=0,$ then we apply van der Corput's estimate of order 2 in $\si_2,$ which leads to

|\mu_{j,k,|Y_1|\lesssim1,\iota_2}^\la(y+\Ga)|\lesssim \la^{\frac32}2^{-k}(\la 2^{-2k})^{-\frac 12}=\la,

\|\M_{j,k,|Y_1|\lesssim1,\iota_2}^\la\|_{1+\ve\to 1+\ve}\lesssim \la 2^{-k} 2^{-j}.

These estimates match with (<ref>), and thus altogether we find that, for every $\ve>0,$

\begin{equation}\label{Mjk}
\|\M_{j,k}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}+  \la^{\frac 12+\ve}2^{-j}.
\end{equation}

§.§ $L^p$ - estimates of $\M_{j,k}^\la$  for the contribution by $E_l$ when $\la\ge 2^j$

Interpolation between the $L^2$ - estimate (<ref>) and the $L^{1+\ve}$ - estimate (<ref>) leads to

\begin{eqnarray}\nonumber
\|\M_{j,k}^\la\|_{p\to p}&\lesssim&
(\la^{1+\ve}2^{-k})^{\frac 2p -1}(1+\la d_{j,k})^{-\frac 12 (2-\frac 2p)} 2^{-j}
+ (\la^{\frac 12+\ve})^{\frac 2p -1}(1+\la d_{j,k})^{-\frac 12 (2-\frac 2p)} 2^{-j}\\ \nonumber
\la^{\ve'} (\la d_{j,k})^{\frac 2p -1} (1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 2p-1}\, 2^{-k(\frac 2p -1)}\,2^{-j} \\
&&\quad +\, \la^{\ve'}
(\la d_{j,k})^{\frac 1p -\frac 12}(1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 1p-\frac 12}\, 2^{-j},\label{Mlajkp}
\end{eqnarray}

if $1<p< 2.$

* Since $1/p-1/2>0,$ we can first  sum both terms over all dyadic $\la$s such that $\la d_{j,k}\le 1,$ provided we choose $\ve'>0$ sufficiently small.

* If $\la d_{j,k}> 1,$ then the total exponent of $\la d_{j,k}$ in the first summand is

\frac 2p-1+\frac 1p-1=\frac 3p-2<0,

since we are assuming that $p>3/2.$ Similarly, the  total exponent of $\la d_{j,k}$ in the second summand is

\frac 1p-\frac 12+\frac 1p-1=\frac 2p-\frac 32<\frac 43-\frac 32<0.

Thus, again both terms can be summed over all dyadic  $\la$s such that $\la d_{j,k}>1,$ provided we choose $\ve'>0$ sufficiently small.

This easily shows that, for every $\ve'>0,$

\begin{equation}\label{suminla1}
\sum\limits_{\la\gg 1}\|\M_{j,k}^\la\|_{p\to p}\lesssim (d_{j,k}^{-1})^{\frac 2p-1+\ve'}\, 2^{-k(\frac 2p -1)}\,2^{-j}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}.
\end{equation}

In order to also sum over the $j$s and $k$s associated to $E_l,$  it turns out that we shall have to make use of the condition
(<ref>). To this end, we shall first derive a little lemma which  will help to relate any edge of $\N(\Phi)$ having the
vertex $(B,A):=(B_l,A_l)$ associated to the transition domain $E_l$ as an endpoint.

§.§ A geometric lemma and $L^p$-estimation of $\M^{\tau_l}, l=1,\dots,L$

Given  a weight $\ka=(\ka_1,\ka_2)$ such that $\ka_1>0, \ka_2\ge 0,$ and the associated  line

L_\ka:=\{(t_1,t_2): \ka_1t_1+\ka_2 t_2=1\},

assume that $(B,A)$ is any point on $L_\ka.$ We define the number $n_\ka-2$ as the $t_1$-coordinate of the intersection of the line $L_\ka$ with the horizontal line $t_2=1$ (compare Figure <ref>). Then

where $a:=\ka_1/\ka_2$ is the modulus of the slope of the line $L_\ka.$

The condition
$\ka_1B+\ka_2 A=1$ is equivalent to $A/a+B=1/\ka_1,$ hence to

\frac {A-1}a+B+2=\frac 1{\ka_1}-\frac {\ka_2}{\ka_1}+2=\frac {1-\ka_2}{\ka_1}+2=(n_\ka-2)+2=n_\ka.

Meaning of $n_\ka$

The next lemma will allow us to show that the estimates in (<ref>) can be summed over the relevant $j$s and $k$s. Since in Step 2 and higher we shall have to deal with functions $\Phi(z,s_2)$ which are analytic in the first variable $z,$ but only fractionally analytic in the second variable, we shall state it more generally for Newton-Puiseux polyhedra.

So, assume that $\Phi$ is such a function, with Newton-Puiseux polyhedron $\N(\Phi),$ whose vertices are the points
$(B_l,A_l), l=0,\dots L,$ where $(B_0,A_0)=(n-2,0),$ somewhat   as in Subsection <ref> (compare Figure <ref>). Note, however, that in contrast to the case of Newton polyhedra, where necessarily  $A_1\ge 1,$ we may possibly have $A_1<1.$ Otherwise, we adapt the same notation as in Subsection <ref>.

Assume that $A_1\ge 1,$ so that $A_l\ge 1$ for every $l\ge 1.$ Then

\begin{equation*}%\label{keyest}
\frac{A_l-1}{a_l}+B_l+2\le \frac{A_1-1}{a_1}+B_1+2=\frac{A_0-1}{a_1}+B_0+2
\end{equation*}

for $l=1,\dots,L.$

The proof is geometrically evident in view of Lemma <ref> (note that apparently the sequence $\{n_{\ka^l}\}_l$ is decreasing) and can easily by carried out by means of an  induction over $l\ge0.$ It will therefore be omitted.

We are now in the position to show that the estimates in (<ref>) can be summed over all relevant $j$s and $k$s associated to a given transition domain $E_l, l=1,\dots, L,$ provided $p>\max\{3/2, h\}.$  Actually, we shall only need condition (<ref>) here, i.e.,
$ j\ge k a_l+M/2.$   Indeed, we shall prove that under these assumptions

\begin{equation}\label{suminjk1}
\sum\limits_{j,k\gg1:  j\ge k a_l}\Big((d_{j,k}^{-1}2^{-k})^{\frac 2p-1+\ve'}\, 2^{-j}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\Big)<\infty,
\end{equation}

provided we choose $\ve'>0$ sufficiently small.

In combination with estimate (<ref>) this will clearly imply that for if $l=1,\dots,L,$

\begin{equation}\label{Lpest1}
\|\M^{\tau_l}\|_{p\to p} <\infty,
\end{equation}

provided $p>\max\{3/2, h\}.$

Now, from (<ref>) and (<ref>), we find that the  first two vertices of $\N(\Phi)$ are given by $(B_0,A_0)=(n-2,0)$ and  $(B_1,A_1)=(m-2,1),$  and that $\ka_1^1=1/(n-2)>0$  and

\begin{equation*}%\label{a1}
a_1=\frac {\ka_1^1}{\ka^1_2}=\frac 1{n-m}.
\end{equation*}

(compare also Figure <ref>). Thus, Lemma <ref> implies that

\begin{equation}\label{leqm}
\frac{A_l-1}{a_l}+B_l+2\le m
\end{equation}

for $l=1,\dots, L.$  Fix such an $l,$ and put $A:=A_l, B:= B_l$ and $a=a_l>a_1=1/(n-m)>0.$ Then
d_{j,k}^{-1}= 2^{kA+j(B+2)}
and  $k\le j/a$ in (<ref>). Thus, we can re-write  (<ref>) as

\begin{equation}\label{crucsum}
\sum\limits_{j,k\gg1:  k\le j/a}\Big(( 2^{k(A-1)+j(B+2)})^{\frac 2p-1+\ve'}\, 2^{-j}
+  ( 2^{kA+j(B+2)})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\Big)<\infty.
\end{equation}

Since the exponents $\frac 2p-1+\ve'$ and $\frac 1p-\frac 12+\ve'$ are strictly positive, we can first sum over all
$k\le j/a$ and are left with showing that

\begin{equation*}%\label{crucsum3}
\sum\limits_{j\gg1}\Big( 2^{j\big[(\frac{A-1}a+B+2)(\frac 2p-1+\ve')-1\big]}
+  2^{j\big[\frac Aa+B+2)(\frac 1p-\frac 12+\ve')-1\big]}\Big)<\infty.
\end{equation*}

But, by (<ref>),

\frac{A-1}a+B+2\le m \quad \text{and} \quad \frac 2p-1<\frac 2n \le \frac 1m,

since $1/p<1/h=1/2+1/n$ and $n\ge 2m,$ so that we may assume that the exponent of the first summand in (<ref>) is strictly negative and we can sum in $j.$

\frac{A}a+B+2\le m +\frac 1a\le m+(n-m)=n\quad \text{and} \quad \frac 1p-\frac 12<\frac 1n,

so that we may assume that the exponent of the second summand in (<ref>) is strictly negative  too and we can again  sum in $j.$
This completes the proof of (<ref>).

§.§ $L^p$-estimation of $\M^{\tau_0}$

Let us finally look at the case $l=0.$ Recall that here only the condition (<ref>), i.e., $j\le ka_1-M/2,$ is available, and that now

d_{j,k}^{-1}= 2^{kA_0+j(B_0+2)}=2^{jn}.

Thus, here we would like  to show that

\begin{equation}\label{suminjk2}
\sum\limits_{j,k\gg1:  k\ge j/a_1}\Big(( 2^{jn-k})^{\frac 2p-1+\ve'}\, 2^{-j}
+  ( 2^{jn})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\Big)<\infty.
\end{equation}

In the first summand, we can first sum over all $k\ge j/a_1$ and arrive at the sum

\sum\limits_{j\gg 1} 2^{j\big[(n-\frac 1{a_1})(\frac 2p-1+\ve')-1\big]}.

n-\frac 1{a_1}=m, \quad \text{and} \quad \frac 2p-1<\frac 2n \le \frac 1m,

so that we may assume that the exponent of the first summand in (<ref>) is strictly negative by choosing $\ve'$ sufficiently small,  and  thus we can sum in $j.$

The second summand is insufficient for a summation in $k,$ unless, say,  $k\lesssim j.$  In the latter case, we can sum in $k\lesssim j$ and  arrive at the sum

\begin{equation}\label{suminj1}
\sum\limits_{j\gg 1}2^{jn(\frac 1p-\frac 12+\ve')}\, 2^{-j},
\end{equation}

with any slightly bigger $\ve'>0$ than before. This series is  convergent for $\ve'$ sufficiently small, since we are assuming that  $p>h.$

So, let us assume henceforth that, say, $k> jn.$
The crucial observation  is that  the second term in (<ref>)  can be improved when  $\la2^{-k}\le 1,$ and this will be needed for the case $l=0.$

To be more precise, assume first that  $\la2^{-k}>1,$ i.e., $k\lesssim\log \la.$ Then we can keep the second term in (<ref>), which is here given by

\la^{\ve'} (\la 2^{-jn})^{\frac 1p -\frac 12}(1+\la 2^{-jn})^{\frac 1p-1} \, (2^{jn})^{\frac 1p-\frac 12}\, 2^{-j}.

Summing now first in $k\lesssim\log \la,$ we can estimate by the same kind of expression, but with a slightly bigger exponent $\ve'>0.$
After summing in $\la,$ we are again left with the series in (<ref>) and are done.

Assume next that  $\la2^{-k}\le 1.$ Then the exponential factor of the oscillatory integral $\mu_{j,k}^\la(y+\Gamma)$ which collects all terms depending  only on $\si_2$  (compare (<ref>)) is essentially non-oscillatory, and so we  shall rather estimate for $y\in \Delta_{\iota_1}$ by

|\mu_{j,k,\iota_1}^\la(y+\Ga)|\lesssim \la^{\frac 52} 2^{-j-k} (\la 2^{-j} 2^{\iota_1})^{-1}=\la^{\frac 32} 2^{-k-\iota_1}.

Since $|\Delta_{\iota_1}|\lesssim 2^{\iota_1-j},$ we obtain the improved estimate

\|\M_{j,k,\iota_1}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 32}2^{-j-k}.

This shows that we can replace the second term in (<ref>) by $\la^{\frac 32+\ve}2^{-j-k},$ and finally, after interpolation with our $L^2$-estimate (<ref>), we are led to summing

M(\la,j,k):=(\la^{\frac 32+\ve}2^{-k})^{\frac 2p -1}(1+\la 2^{-jn})^{\frac 1p-1}2^{-j}

over our $\la$s, $j$s and $k$s.

Assume first that $2^k\ge \la>2^{jn}.$ Then
M(\la,j,k)\sim(\la^{\frac 32+\ve}2^{-k})^{\frac 2p -1}(\la 2^{-jn})^{\frac 1p-1}2^{-j},

\begin{eqnarray*}
\sum\limits_{k:2^k\ge \la} M(\la,j,k)&\lesssim&  \la^{\frac 1p -\frac 12+\ve'}(\la 2^{-jn})^{\frac 1p-1}2^{-j}=(\la 2^{-jn})^{\frac 1p-\frac 12+\frac 1p-1+\ve'} 2^{jn(\frac 1p-\frac 12+\ve')} 2^{-j}\\
&=&(\la 2^{-jn})^{\frac 2p-\frac 32+\ve'} 2^{j\big[n(\frac 1p-\frac 12+\ve')-1\big]}.
\end{eqnarray*}

But, since we are assuming that $p>\max\{3/2, h\},$ we have $ 2/p- 3/2=-1/6<0,$ and again $n(1/p- 1/2)-1<0,$
so that we can first  sum in $\la>2^{jn},$ and then in $j\gg1,$ provided $\ve'>0$ is sufficiently small.

We are thus left with the case where $ \la\le 2^{jn}$ and $k>jn.$ Then
M(\la,j,k)\sim(\la^{\frac 32+\ve}2^{-k})^{\frac 2p -1}2^{-j},
and thus

\sum\limits_{\la:\la\le 2^{jn}} M(\la,j,k)\lesssim  (2^{jn(\frac 32+\ve)}2^{-k})^{\frac 2p -1}2^{-j}.

This implies

\sum\limits_{k: k\ge jn}\sum\limits_{\la:\la\le 2^{jn}} M(\la,j,k)\lesssim  (2^{jn(\frac 12+\ve)})^{\frac 2p -1}2^{-j}.

and again we can finally also sum in $j,$ since $p>h.$

Consequently, estimate (<ref>) holds true even for $l=0.$

§.§ On the contributions by the homogeneous domains $D_l, l=1,\dots,L$

Let us fix $l_1\in \{1,\dots,L\}.$ We recall that

D_{l_1}:=\{(z_1,s_2):   2^{-M} s_2^{a_{l_1}}\le |z_1|\le 2^{M} s_2^{a_{l_1}}\}, $$

where we may assume that $\rho_{l_1}$ is supported in $D_{l_1}.$ We also recall from Subsection <ref> that we may assume that

J^{\rho_{l_1}}(\la,s)=\sum\limits_{j,k: |j-k a_{l_1}|\le M/2}J_{j,k}(\la,s).

We shall decompose  $D_{l_1}$ into a finite number of very narrow subregions. To this end, fix any point $c$  in the compact interval  $I:=[2^{-M}, 2^M],$ and put

D_{l_1}^c:=\{(z_1,s_2):   |z_1-cs_2^{a_{l_1}}|<  \ve s_2^{a_{l_1}}\},

where $\ve>0$ is supposed to be sufficiently small. More precisely, recall from Subsection <ref> that every root $r(s_2)$ in the cluster
$\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]$
associated to the compact edge $\ga_{l_1}$ of $\N(\Phi)$ admits a Puiseux series expansion

c_{l_1\cdots l_p}^{\al_1\cdots \al_p}s_2^{a_{l_1\cdots
l_p}^{\al_1\cdots \al_{p-1}}}+\cdots,

with leading term $c_{l_1}^{\al_1}s_2^{a_{l_1}},$ where the leading coefficient $c_{l_1}^{\al_1}$ may be real, or complex.
By having chosen $M$ to be sufficiently large, we are of course assuming that  all real coefficients  $c_{l_1}^{\al_1}$ will be contained in the interval $I.$ Denote by  $\mathcal C_{l_1}$ the set of those real coefficients $c_{l_1}^{\al_1}.$

If $c\notin \mathcal C_{l_1},$ we choose $\ve>0$ at least so small that $\ve\ll\dist(c,\mathcal C_{l_1}).$

Comparison between $\N(\Phi)$ and $\N(\tilde\Phi)$

Now, for any $c\in I,$ let us introduce the new variable $\tilde z:=z_1-cs_2^{a_{l_1}},$ and accordingly define

\begin{equation*}%\label{tildePhi}
\tilde\Phi(\tilde z,s_2):= \Phi(\tilde z+cs_2^{a_{l_1}}, s_2).
\end{equation*}

Note that $\tilde \Phi$ is still analytic in $\tilde z,$ but may be only fractionally analytic in $s_2.$ The region $D_{l_1}^c$ then corresponds to

\tilde D_{l_1}^c:=\{(\tilde z, s_2):   |\tilde z|<  \ve s_2^{a_{l_1}}\}.

In Step 2 of our resolution algorithm,  which will be discussed in the next section, we shall base our analysis on the Newton polyhedron of $\tilde\Phi$  in place of $\N(\Phi).$ The corresponding  vertices and compact edges of  $\N(\tilde\Phi)$ will  be denoted by $(\tilde B_l, \tilde A_l), l=0,\dots, \tilde L$ and  $\tilde \ga_l:=(\tilde B_{l},\tilde A_{l}),(\tilde B_{l-1},\tilde A_{l-1})], l=1,\dots, \tilde L.$
The  line supporting the edge $\tilde \ga_l$ will be denoted by $\tilde L_l,$ the associated weight by
$\tilde \ka^l,$ and the corresponding modulus of slope  by $\tilde a_l= \tilde \ka_1^l/\tilde \ka_2^l.$ The corresponding homogeneous domains  and transition domains into which we shall decompose the upper half-plane where $\tilde z\in \RR, s_2>0,$ will be denoted by $\tilde D_l, l=1,\dots, \tilde L,$  and $\tilde E_l, l=0,\dots, \tilde L.$

Coming back to $\tilde D_{l_1}^c,$  note that  by (<ref>),

\begin{equation}\label{tildePhifac}
\tilde\Phi(\tilde z,s_2)=\tilde U_0(\tilde z,s_2)(\tilde z+cs_2^{a_{l_1}})^{\nu_1}  \prod_{l=1}^L
\Phi\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ](\tilde z+cs_2^{a_{l_1}},s_2),
\end{equation}

\Phi\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ](\tilde z+cs_2^{a_{l_1}},s_2)
=\prod_{r\in \left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]}(\tilde z+cs_2^{a_{l_1}}-r(s_2))=\prod_{r\in \left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]}(\tilde z-\tilde r(s_2)),

if we define adjusted  roots $\tilde r$  by

\tilde r(s_2):=r(s_2)-cs_2^{a_{l_1}}.

If now $l<l_1$  and
$r\in \left[ \begin{matrix}
\cdot\\
\end{matrix}\right ], $
then $a_l<a_{l_1},$ so the roots $\tilde r(s_2)$ and $r(s_2)$ do have the same leading term and the same multiplicity.

Thus the identities (<ref>) and (<ref>) show that we indeed have

\begin{equation*}%\label{}
(\tilde B_l, \tilde A_l)=( B_l, A_l), \quad 0\le l<l_1.
\end{equation*}

In particular,

\tilde \ga_l=\ga_l \quad \text{and}  \quad \tilde a_l=a_l, \  l< l_1

(compare Figure <ref>).

Case 1: $c\notin \mathcal C_{l_1}.$ Then, if $l=l_1$ and
$r\in \left[ \begin{matrix}
\cdot\\
\end{matrix}\right ], $
we can factor

\tilde z-\tilde r(s_2)=s_2^{a_{l_1}}\Big(c-c_{l_1}^{\al_1}+\frac {\tilde z}{s_2^{a_{l_1}}}\Big)+\text{higher degree terms},

where $|{\tilde z}/{s_2^{a_{l_1}}}|<\ve.$ Moreover, if $c_{l_1}^{\al_1}$ is real, then $|c-c_{l_1}^{\al_1}|\gg\ve,$ and otherwise
$|c-c_{l_1}^{\al_1}|\ge |\Im  c_{l_1}^{\al_1}|>0,$ so that, by choosing $\ve $ small enough, we can assume that in both cases
$|c-c_{l_1}^{\al_1}|\gg\ve.$ Thus we may assume that
|c-c_{l_1}^{\al_1}+ {\tilde z}/{s_2^{a_{l_1}}}|\sim 1.

Arguing in a similar way also for $l\ne l_1$ (compare our discussion in Subsection <ref>), we find that the edge
$\tilde\ga_{l_1}$  has the same slope  and right endpoint as $\ga_{l_1},$  but as left endpoint a point of the form
$(0,\tilde A_{l_1}),$ and that we can factor

\begin{equation*}%\label{Dreso1}
\tilde\Phi(\tilde z,s_2)=V(\tilde z,s_2)\, s_2^{\tilde A_{l_1}},\qquad (\tilde z,s_2)\in \tilde D_{l_1}^c,
\end{equation*}

where  the function $V(\tilde z,s_2)$ behaves in a similar way is in Subsection <ref>. In particular, here $\tilde L=l_1.$

This shows that, by changing from $z_1$ to $\tilde z,$ we can regard the  region $\tilde D_{l_1}^c$  now like a transition domain $\tilde E_{\tilde L},$ in the coordinates $(\tilde z, s_2).$  With view to Lemma <ref>, we also remark that here $\tilde A_{l_1}\ge A_{l_1}\ge A_1=1.$
These kind of  domains will be  handled in Step 2 of our resolution algorithm.

Case 2: $c\in \mathcal C_{l_1}.$
Then $c$ is of the form $c=c_{l_1}^{\al_1}\in\RR.$  This will fix $\al_1$ in the sequel. We then recall from (<ref>) that, up to a constant factor,  the $\ka^{l_1}$-principal part of $\Phi$ is given by

\Phi_{\ka^{l_1} }(z_1,s_2)= s_2^{A_{l_1-1}} z_1^{B_{l_1}}\prod_\al \Big(z_1-c^\al_{l_1} s_2^{a_{l_1}}\Big)^{N\left[ \begin{matrix}
\al\\
\end{matrix}\right ] }.

Consequently, the   $\ka^{l_1}$-principal part of $\tilde \Phi$ is given by

\tilde\Phi_{\ka^{l_1} }(\tilde z,s_2)= s_2^{A_{l_1-1}} (\tilde z+cs_2^{a_{l_1}})^{B_{l_1}}\prod_\al \Big(\tilde z-(c^\al_{l_1}-c_{l_1}^{\al_1})\ s_2^{a_{l_1}}\Big)^{N\left[ \begin{matrix}
\al\\
\end{matrix}\right ] }.

We must therefore distinguish two sub-cases:

Subcase (i):
$B_{l_1}>0, $ or
$\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]
\supsetneqq\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]. $
Then $\tilde\Phi_{\ka^{l_1}}$ is not a monomial, and thus the edge $\tilde \ga_{l_1}$  lies on the same line as $ \ga_{l_1}.$ It thus will have the same slope  $\tilde a_{l_1}= a_{l_1}$ and right endpoint $(B_{l_1-1},A_{l_1-1})$ as $\ga_{l_1},$ but the left endpoints of these two edges may be different.

\tilde E_{l_1}=\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l_1+1}}<|\tilde z|<2^{-M} s_2^{a_{l_1}}\}.

Thus, if we choose $\ve<2^{-M},$ and decompose

\tilde D_{l_1}^c=\tilde D_{l_1,1}^c \cup \tilde D_{l_1,2}^c,

\begin{eqnarray*}
\tilde D_{l_1,1}^c&:=&\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l_1+1}}\le |\tilde z|<\ve s_2^{a_{l_1}}\},\\
\tilde D_{l_1,2}^c&:=&\{(\tilde z,s_2):  |\tilde z|<2^Ms_2^{a_{l_1+1}}\},
\end{eqnarray*}

then $\tilde D_{l_1,1}^c\subset \tilde E_{l_1}.$ Note that for  $l_1=\tilde L,$ we even have $\tilde D_{l_1}^c\subset \tilde E_{l_1}.$

Moreover, clearly $\tilde D_{l_1,2}^c$ is contained in the union of the $\tilde E_l$s  and $\tilde D_l$s  with $l\ge l_1+1,$ and therefore

\begin{equation*}%\label{coverDl12}
\tilde D_{l_1}^c\subset \bigcup\limits_{l=l_1}^{\tilde L} \tilde E_l\, \cup \bigcup\limits_{l=l_1+1}^{\tilde L}\tilde D_l.
\end{equation*}

(a) We thus see that in Subcase (i) the contribution by $D_{l_1}^c$ to our maximal operator can be covered by the contributions by the
$\tilde E_l$s  with $l\ge l_1$ and the $\tilde D_l$s  with $l\ge l_1+1.$ Those by the $ \tilde E_l$s will be completely covered  by  Step 2  of our resolution algorithm in the next section, and to the $\tilde D_l$s we shall apply a similar kind of reasoning as in the present section,  which will lead to Step 3 of our resolution algorithm, and so forth. Note also that $l_1+1\ge 2,$ so that the first edge with right endpoint $(n-2,0)$  will be the same for all steps $\ge 2,$ namely $\tilde \ga_1.$  Thus, if $\tilde A_1\ge 1,$ then we can apply Lemma <ref> in Step 2, and as well in any higher step.

(b) Note also that  if $\tilde a_l$  with $l\ge l_1+1$ is the leading exponent of a root $\tilde r(s_2)$ of  $\tilde\Phi,$  then $\tilde a_l$ is of the form $a_{l_1l_2}^{\al_1},$ and the leading term of $\tilde r(s_2)$ is of the form

Subcase (ii):  $B_{l_1}=0$ and
$\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]
=\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ], $
i.e., all roots in $\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]$  have the same leading term $cs_2^{a_{l_1}}.$  Then
$\tilde\Phi_{\ka^{l_1} }(\tilde z,s_2)= s_2^{A_{l_1-1}}
\tilde z^{N\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ] }
=s_2^{A_{l_1-1}} \tilde z^{B_{l_1-1}}
is a monomial, so that the edge $\tilde \ga_{l_1}$ has “shrunk”  to the vertex $(B_{l_1-1},A_{l_1-1})$ of $\N(\tilde\Phi)$ by the change of coordinates from $z_1$ to $\tilde z.$ Here,  the edge $\tilde \ga_{l_1}$ will thus  have a slope $\tilde a_{l_1}> a_{l_1}.$

\tilde E_{l_1-1}=\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l_1}}<|\tilde z|<2^{-M} s_2^{a_{l_1-1}}\},

since $\tilde a_{l_1-1}=a_{l_1-1}.$ And, we claim that we may assume that

2^M s_2^{\tilde a_{l_1}}< \ve s_2^{a_{l_1}}\le 2^{-M} s_2^{a_{l_1-1}}.

Indeed, the first inequality is equivalent to $ s_2^{\tilde a_{l_1}-a_{l_1}}<\ve 2^{-M}.$ But, $\tilde a_{l_1}-a_{l_1}>0,$ so that the latter inequality  can be assumed to be true  because  we may assume  that  $s_2\ll1 $ is  sufficiently small.  The second inequality  follows in a similar way. Let us here decompose
\tilde D_{l_1}^c=\tilde D_{l_1,1}^c \cup \tilde D_{l_1,2}^c,
where now

\begin{eqnarray*}
\tilde D_{l_1,1}^c&:=&\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l_1}}\le |\tilde z|<\ve s_2^{a_{l_1}}\},\\
\tilde D_{l_1,2}^c&:=&\{(\tilde z,s_2):  |\tilde z|<2^Ms_2^{\tilde a_{l_1}}\}.
\end{eqnarray*}

Then $\tilde D_{l_1,1}^c\subset \tilde E_{l_1-1},$  in a similar way as in the previous subcase we see that here

\begin{equation}\label{coverDl122}
\tilde D_{l_1}^c\subset \bigcup\limits_{l=l_1-1}^{\tilde L} \tilde E_l\, \cup \bigcup\limits_{l=l_1}^{\tilde L}\tilde D_l.
\end{equation}

(a) If $l_1=1,$ then for our phase $\Phi$ in (<ref>), by (<ref>) we have

\begin{eqnarray}\label{l=1}
\begin{split}
\ka^1&=\big(\frac 1{n-2},\frac {n-m}{n-2}\big), \quad a_1=\frac 1{n-m},\\
\Phi_{\ka^{1} }(z_1,s_2)&= z_1^{n-2} n(n-1)\beta(0)+s_2 z_1^{m-2} m(m-1)\omega(0).
\end{split}
\end{eqnarray}

Note that here $B_{l_1}=m-2.$  Moreover, since we are assuming that $n\ge 2m,$ we have $n-m\ge m\ge 2,$ so that the polynomial $z_1^{n-m} n(n-1)\beta(0)+s_2  m(m-1)\omega(0)$ has at least 2 different (real or complex) roots, each of multiplicity 1, hence
$\left[ \begin{matrix}
\cdot\\
\end{matrix}\right ]
\supsetneqq\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]. $
This shows that here  only Subcase (i) can arise.

In particular, in Step 2, Subcase (ii) can only arise when $l_1\ge 2,$ and the same applies then to all higher steps, so that $l_1-1\ge 1$ in (<ref>) – thus  no transition domains $\tilde E_0$ will appear in (<ref>).

(b)  By (<ref>) we see that  in Subcase (ii) the contribution by $D_{l_1}^c$ to our maximal operator can be covered by the contributions by the
$\tilde E_l$s  with $l\ge l_1-1\ge 1$ and the $\tilde D_l$s  with $l\ge l_1\ge 2.$ Those by the $ \tilde E_l$s will be completely covered  by  Step 2  of our resolution algorithm in the next section, and to the $\tilde D_l$s we shall apply a similar kind of reasoning as in the present section,  which will lead to Step 3 of our resolution algorithm, and so forth. Note again that since here $l_1\ge 2,$ the first edge with right endpoint $(n-2,0)$  will be the same for all steps $\ge 2,$ namely $\tilde \ga_1.$  Thus, if $\tilde A_1\ge 1,$ then we can apply Lemma <ref> in Step 2, and as well in any higher step.

(c)  Note also that  if $\tilde a_l$  with $l\ge l_1$ is the leading exponent of a root $\tilde r(s_2)$ of  $\tilde\Phi,$  then $\tilde a_l$ is of the form $a_{l_1l_2}^{\al_1},$ and the leading term of $\tilde r(s_2)$ is of the form

We finally remark that, by compactness of the interval $I,$ we can cover  the homogeneous domain $D_{l_1}$ by the sub-domains $D_{l_1}^c,c\in \mathcal C_{l_1},$ and a finite number of subdomains $D_{l_1}^{c^j},$ with $c^j\notin \mathcal C_{l_1}.$

Thus we may reduce our studies to considering such sub-domains $D_{l_1}^c,$ for which we shall then, as explained before,  pass to the coordinate $\tilde z$ in place of $z_1.$  The study of the contributions of the corresponding homogeneous domains $\tilde D_{l_1}^c$ will be carried out in Step 2  (and higher) of our algorithm.

§.§ Stopping domains $D^c_{l_1}, c\in  \mathcal C_{l_1},$ for the resolution algorithm at Step 1

We shall stop on the domain $D^c_{l_1}, c=c_{l_1}^{\al_1}\in  \mathcal C_{l_1},$ when the multiplicity
$N\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
of the cluster
$\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
agrees  with the multiplicity of every sub-cluster of it.

Observe that this is equivalent to the condition that the cluster
$\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
contains only one, real root $r$ of multiplicity
$N\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ].
The fact that this root must be real stems from the observation that with any root in
$\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
its complex conjugate must also be in the same cluster, since $\Phi$ is real.

The contribution by such a domain  $D^{c}_{l_1}$ will be covered in a complete way in  Subsection <ref> of the next section.

If $l_1=1,$ we have seen in Remark <ref> (a) that any real root in
$\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
has multiplicity $1,$ so that by (<ref>) we do have
$N\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]=1.
This shows that for $l_1=1$ we stop our resolution algorithm in Step 1.

§  ANALYSIS IN STEP 2 OF THE RESOLUTION ALGORITHM

In this section, we shall proceed in a similar way as in the previous Section <ref>, but our analysis will now be based on the coordinates
$\tilde z$  and the function $\tilde \Phi$ in place of $z_1$ and the function $\Phi.$  Adapting the notation from the previous subsection, recall that

\begin{equation*}%\label{wy}
z_1=\tilde z+ w(s_2),
\end{equation*}

where here $w(s_2)=cs_2^{a_{l_1}}.$  The non-trivial extra term $w(s_2),$ however, will create additional challenges compared to the analysis in the previous section.

Of course, we shall have to apply the same change of coordinates also to our original  phase functions $\breve\phi_1(x_1, s_2)$ and $\Phi_0(x_1,s'):=\breve\phi_1(x_1, s_2)+s_1x_1$ in (<ref>), and therefore put

\begin{eqnarray*}
\tilde\Phi_{0,0}(\tilde z, s_2)&:=& \breve\phi_1(\tilde z+ w(s_2), s_2),\\
\tilde\Phi_{0}(\tilde z, s')&:=& \Phi_{0}(\tilde z+ w(s_2),s_2)=\tilde\Phi_{0,0}(\tilde z, s_2)+s_1\tilde z+s_1w(s_2).
\end{eqnarray*}

Note that then

\begin{equation}\label{tPhi}
\tilde\Phi(\tilde z,s_2):=\pa_{\tilde z}^2\tilde\Phi_{0}(\tilde z, s') =\pa_{\tilde z}^2\tilde\Phi_{0,0}(\tilde z, s_2)
\end{equation}

(compare with (<ref>)).

A crucial tool will again be Lemma <ref>. Its application will here require that $\tilde A_1\ge 1,$ so let us begin by discussing this condition.

§.§ On the condition $\tilde A_1\ge 1$

By Remark  <ref>, we see that  $\tilde A_1\ge 1$ in Case 1, i.e., when $c\notin \mathcal C_{l_1}.$ Assume next that we are in  Case 2 where $c=c_{l_1}^{\al_1}\in \mathcal C_{l_1}.$ Then, if $l_1\ge 2,$ we have that $\tilde A_1=A_1\ge 1.$

But, in Remark <ref> we have seen that for $l_1=1$ we can stop our resolution algorithm already at Step 1, so that henceforth we may and shall assume that $l_1\ge 2,$   hence $\tilde A_1=A_1\ge 1.$

For the sake of completeness, let us finally observe that when $l_1=1,$ then by   Remark <ref> (a)   we must be in Subcase (i), and by (<ref>) we have

\Phi_{\ka^{1}}(z_1,s_2)= C_1z_1^{m-2} \big(z_1^{n-m}+ C_2 s_2 \big)
=C_1z_1^{m-2}\prod_\al \Big(z_1-c^\al_{l_1} s_2^{a_{1}}\Big),

with  nontrivial real constants  $C_1,C_2.$ This implies that

\tilde\Phi_{\ka^{1} }(\tilde z,s_2)=C_1\tilde z\, (\tilde z+c_{l_1}^{\al_1}s_2^{a_1})^{m-2} \prod_{\al\ne \al_1} \Big(\tilde z -(c^\al_{l_1}-c_{l_1}^{\al_1})\ s_2^{a_{1}}\Big),

where the product over the $\al$s such that $\al\ne \al_1$ will consist of $n-m-1\ge 1$ different factors, since all non-trivial roots  of
$\Phi_{\ka^{1}}$  have multiplicity 1.
By (<ref>) this implies that
$\tilde A_1=(n-3)a_1=(n-3)/(n-m),$  and $\tilde B_1=1,$ so that $\tilde \ga_1=[(1,(n-3)/(n-m)),(n-2,0)].$

Thus, $\tilde A_1<1$ if and only if  $m=2 .$

(a) Notice that  by Remark <ref>  and our discussion in Subsection <ref> we see that   we shall have dealt in a complete way with the contribution by the homogeneous domain $D_1$ and therefore will be allowed to assume that $l_1\ge 2$ in Step 3 and higher.

(b) We shall see later (see Section <ref>)  that our resolution algorithm will stop whenever we have arrived at a narrow subdomain  containing only one  sub-cluster of roots consisting of just  one  root   (possibly with multiplicity) whose leading jet $w(s_2)$ is real,  i.e., when no further “fine splitting” of roots can occur. In such a  case
we shall be able to argue in a very similar way as we are doing here.

Recall also that since the edges $\tilde\ga_1$ and $\ga_1$ are lying on the same line $L_1,$ we have $\tilde a_1=a_1,$ and

\begin{equation}\label{keyestl=1}
\frac{\tilde A_1-1}{\tilde a_1}+\tilde B_1+2=m.
\end{equation}

For later use, note that this identity remains valid even  when $\tilde A_1<1.$ And,  if $m\ge 3,$  so that  $\tilde A_1\ge 1,$ then  by Lemma <ref> we also have

\begin{equation}\label{keyestlg1}
\frac{\tilde A_l-1}{\tilde a_l}+\tilde B_l+2\le m,\quad  l=1,\dots, \tilde L.
\end{equation}

Clearly, this holds true also for $l_1\ge 2.$

§.§ Domain decomposition and dyadic localization in $\tilde z$ and $s_2$

After having fixed $l_1\in\{1,\dots,L\},$  according to our discussion in the previous section we can cover the domain $\tilde D_{l_1}^c$ by $\tilde \ka^l$- homogeneous  domains  $\tilde D_l$ of the form

\tilde D_l:=\{(\tilde z,s_2):   2^{-M} s_2^{\tilde a_l}\le |\tilde z|\le 2^{M} s_2^{\tilde a_l}\}, \quad l\le \tilde L,

and  transition domains of the form

\begin{eqnarray*}
%E_{0}&:=&\{(z_1,s_2):   2^M s_2^{a_{1}}<|z_1|\}, \\
\tilde E_l&:=&\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l+1}}<|\tilde z|<2^{-M} s_2^{\tilde a_l}\}, \quad l\le \tilde L-1,\\
\tilde E_{\tilde L}&:=&\{(\tilde z,s_2):  |\tilde z|<2^{-M} s_2^{\tilde a_{\tilde L}}\} .
\end{eqnarray*}

More precisely, in Case 1 where $c\notin \mathcal C_{l_1}, $ according to Remark <ref> we just need the transition domain $\tilde E_{\tilde L},$ where $\tilde L=l_1.$

In Case 2, where $c\in \mathcal C_{l_1},$ we need to distinguish two subcases: if we are in  Subcase (i), then
according to  Remark <ref>   (a) we shall need the transition domains $\tilde E_l$ with $l\ge l_1\ge 1$ and the homogeneous domains  $\tilde D_l$ with $l\ge l_1+1\ge 2.$ Moreover, according to Remark <ref> (a), we shall treat  the case $l_1=1$
separately (in Subsection <ref>), so that we shall here even assume that $l_1\ge 2.$

In Subcase (ii) of Case 2, according to Remark <ref> we may assume that $l_1\ge 2$ and shall then need the transition domains $\tilde E_l$ with $l\ge l_1-1\ge 1$ and the homogeneous domains  $\tilde D_l$ with $l\ge l_1\ge 2.$

Moreover, our discussion in Subsection <ref> shows that under these assumptions we always have that  $\tilde A_1\ge 1.$

Recall also that in all cases that had appeared we had seen that the edges $\tilde \ga_1$ and $\ga_1$ were supported on the same line $\tilde L_1=L_1$ of “slope” $\tilde a_1=a_1,$ so that  the identity (<ref>) and  the inequalities (<ref>) will hold true.

Having fixed $l_1$ and $l$  according to these assumptions, we shall put

a:= a_{l_1}, \quad \tilde a:= \tilde a_l.

Note that if we are in Case 1, or Subcase (i) of Case 2, then $\tilde a\ge \tilde a_{l_1}=a,$ but in Subcase (ii) it may also happen that $\tilde a<a$ (for instance if $l=l_1-1,$ where $\tilde a=a_{l_1-1}<a_{l_1}=a$).

Note also that, in analogy to  (<ref>), by Lemma <ref> we also have

\begin{equation}\label{tkeyestlg1}
\frac{\tilde A_l-1}{\tilde a_l}+\tilde B_l+2\le m,\quad  l=1,\dots, \tilde L,
\end{equation}

since $\tilde A_1\ge 1.$

As in Section <ref>, we shall  again  localize to  these domains $\tilde D_l$ and $\tilde E_l$ in a smooth way by means of the corresponding  functions $\tilde\rho_l$ respectively $\tilde\tau_l$  given by

\tilde\rho_l(\tilde z,s_2):=\chi_0\Big(\frac{ \tilde z}{2^M s_2^{\tilde a_l}}\Big) -\chi_0\Big(\frac{ \tilde z}{2^{-M} s_2^{\tilde a_l}}\Big),

\tilde\tau_l(\tilde z,s_2):=\chi_0\Big(\frac{ \tilde z}{2^{-M} s_2^{\tilde a_l}}\Big) \, -\chi_0\Big(\frac{ \tilde z}{2^{M} s_2^{\tilde a_{l+1}}}\Big).

The corresponding functions such as $\mu^\la, J(\la,s)$ and $F(\la,s_2,s_3,y_1),$  will be designated  by means of an extra upper index $\tilde \rho_l$ respectively $\tilde \tau_l,$ such as
$\mu^{\tilde\rho_l,\la},J^{\tilde \rho}(\la,s),$ or $F^{\tilde \tau_l}(\la,s_2,s_3,y_1).$

Similarly as before, let us here  localize to

|\tilde z|\sim 2^{-j}, \ s_2\sim 2^{-k},

where we may assume that $j,k$ are sufficiently large integers $\gg 1.$ To this end, we define scaled coordinates $z, \si_2$ by writing
$\tilde z=2^{-j} z, \ s_2=2^{-k} \si_2,$ so that

|z|\sim 1, \ \sigma_2\sim 1.

Accordingly, let us here put

\begin{eqnarray*}
\tilde J_{j,k}(\la, s)&:=&\chi_1(2^k s_2)\int e^{-i\la s_3\tilde\Phi_0(\tilde z, s')} \chi_1(2^j\tilde z)\,\eta d\tilde z\\
&=&\chi_1(\si_2)\, 2^{-j} \int  e^{-i\la s_3\tilde\Phi_0^s(z, s_1,\si_2)}  \chi_1(z)\, \eta dz,
\end{eqnarray*}

where we have put

\tilde\Phi_0^s(z, s_1,\si_2):=\tilde\Phi_0(2^{-j}z, s_1,2^{-k}\si_2),

\begin{eqnarray*}
\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) &:=& 2^{-j}\iint e^{-i\la s_3(\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1)} \chi_0(s_1)\chi_1(z)  \,\eta dz ds_1.
\end{eqnarray*}

The corresponding contribution to $\mu^\la$ is here given by

\begin{eqnarray}\nonumber
\mu_{j,k}^\la(y+\Ga)&:=&\la^{\frac52}2^{-k}\times \\
&&\iint  \tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) e^{-i\la s_3(s_2^2B(s_2)-2^{-k}\si_2 y_2-y_3)} \chi_1(\si_2) d\si_2 \chi_1(s_3) ds_3. \label{tmujk}
\end{eqnarray}

Of course, we are following here the same conventions as in Subsection <ref>. Applying the same reasoning as in that subsection, in analogy to (<ref>), (<ref>) and (<ref>) we see here that
for the contribution by the transition domain $\tilde E_l,$ i.e., for $ J^{\tilde \tau_l},$ we may assume that $J^{\tilde\tau_l}_{j,k}\equiv 0,$ unless

\begin{equation}\label{tEjk1}
j\ge k\tilde a_l+M/2, \qquad \text{if} \quad l=1,\dots \tilde L.
\end{equation}

Similarly,  for the contribution by the homogeneous domain $\tilde D_l, 2\le l\le\tilde  L,$ i.e., for $ J^{\tilde \rho_l},$ we  may assume that $J^{\tilde \rho_l}_{j,k}\equiv 0,$ unless

\begin{equation}\label{tDjk1}
|j-k\tilde a_l|\le M/2.
\end{equation}

For the range of $j$s and $k$s described by (<ref>), respectively (<ref>), we are then also allowed to assume for simplicity that

\begin{eqnarray*}
J^{\tilde \tau_l}_{j,k}(\la, s)&=&J^{\tilde \rho_l}_{j,k}(\la, s)=\tilde J_{j,k}(\la, s).
\end{eqnarray*}

The corresponding maximal operators  will be designated  by means of an extra lower index $j,k,$ such as
$\M_{j,k}^\la,$ or $\M^{\rho_l}_{j,k}.$

§.§ Resolution of singularity on the transition domain $\tilde E_l$

Fix $\tilde E_l$ with $l\ge l_1,$ respectively $l\ge l_1-1\ge 1$  if we are in the Subcase (ii) of Case 2.
Arguing as in Subsection <ref>, we may write

\begin{equation}\label{tElreso}
\tilde\Phi(\tilde z,s_2)=\tilde V(\tilde z,s_2)\, s_2^{\tilde A_l}\tilde z^{\tilde B_l},\qquad (\tilde z,s_2)\in \tilde E_l,
\end{equation}

where $\tilde V$ is analytic in $\tilde z$ and fractional analytic in $s_2$ away from the coordinate axes and $|\tilde V|\sim 1.$
Moreover, by means of Lemma <ref> we see again that there is a function $\tilde U(\tilde z,s_2)$  having similar properties as $\tilde V(\tilde z,s_2)$ such that

\begin{equation*}%\label{tprimitiv2}
\pa_{\tilde z}^2\big(\tilde U(\tilde z,s_2)s_2^{\tilde A_l} \tilde z^{\tilde B_l+2}\big)=\tilde\Phi(\tilde z,s_2)=\pa_{\tilde z}^2\tilde\Phi_{0,0}(\tilde z,s_2).
\end{equation*}

In particular, $|\tilde U|\sim 1.$
Again the function $\tilde U(\tilde z,s_2)s_2^{\tilde A_l} \tilde z^{\tilde B_l+2}$ is analytic in $\tilde z$ and vanishes of second order at $\tilde z=0.$ Consequently, we see that (compare also (<ref>))

\begin{equation*}%\label{tbphionEl}
\tilde\Phi_{0,0}(\tilde z,s_2)=\tilde U(\tilde z,s_2)s_2^{\tilde A_l} \tilde z^{\tilde B_l+2}+\tilde z \tilde g(s_2)+\tilde h(s_2),
\end{equation*}

with fractional analytic functions $\tilde g(s_2),\tilde h(s_2).$ Since $\tilde z=0$ iff $z_1=w(s_2)$  (recall that
$\tilde z=z_1-w(s_2)$), these  are explicitly given by

\begin{eqnarray}\label{tgh}
\begin{split}
\tilde h(s_2)&= \breve\phi_1(w(s_2), s_2), \\
\tilde g(s_2)&=\pa_{z_1}\breve\phi_1(w(s_2), s_2).
\end{split}
\end{eqnarray}

\begin{equation*}%\label{tPhi0onEl}
\tilde\Phi_{0}(\tilde z, s')=\tilde U(\tilde z,s_2)s_2^{\tilde A_l} \tilde z^{\tilde B_l+2}+ s_1(\tilde z+w(s_2))+\tilde z \tilde g(s_2)+\tilde h(s_2).
\end{equation*}

Let us finally again apply our changes of coordinates $\tilde z=2^{-j} z, \ s_2=2^{-k} \si_2.$  Then, if we put $A:=\tilde A_l, B:= \tilde B_l$ and

assuming that (<ref>) holds true, we obtain

\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1=d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2}+ s_1(2^{-j} z+w(s_2)-y_1)+2^{-j} z \tilde g(s_2)
+\tilde h(s_2),

where $U$ has similar properties as in (<ref>).

Note that we have picked up in particular the new term $s_1w(s_2),$ where

|w(s_2)|\sim s_2^{a}\sim 2^{-ka}

(recall that  $a=a_{l_1}$). The same will in fact apply in all subsequent steps of our resolution algorithm, where in Step $p$  we will have to choose for $w(s_2)$ the leading jet

c_{l_1\cdots l_p}^{\al_1\cdots \al_p}s_2^{a_{l_1\cdots
l_p}^{\al_1\cdots \al_{p-1}}}.

Let us then write

w(s_2)=2^{-ka} \si_2^ a\tilde w(\si_2),

so that $|\tilde w|\sim 1,$ whereas all derivatives of $\tilde w$ can be assumed to be very small. We shall use the following  abbreviations:

\begin{eqnarray*}
Y_1:&=& 2^{ka} y_1,\\
\gamma(\si_2,Y_1)&:=&\si_2^ a\tilde w(\si_2)-Y_1,\\
\end{eqnarray*}

These allow us  to re-write

\begin{eqnarray}\nonumber
&&\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1\\
&=&d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2}+ 2^{-j} s_1\big(K\ga(\si_2,Y_1)+z\big)+2^{-j} z \tilde g(s_2)+\tilde h(s_2),\label{tPhi0s}
\end{eqnarray}

where $z\sim 1\sim \si_2.$ We can then re-write

\begin{eqnarray}\nonumber
&&\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) = 2^{-j}\iint e^{-i\la s_3(\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1)} \chi_0(s_1)\chi_1(z)  \,\eta dz ds_1\\ \label{tFjk}
&&=2^{-j} e^{-i\la s_3\tilde h(s_2)} \,\int \vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\\ \nonumber
&&\hskip5cm \times e^{-i\la s_3\big(d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2} +2^{-j}z \tilde g(2^{-k}\si_2)\big)}\chi_1(z)\eta dz,
\end{eqnarray}

where $\vp:=\widehat {\chi_0}\in \S(\RR)$ is rapidly decaying.

§.§ $L^2$ - estimation of $\M_{j,k}^\la$ for the contribution by $\tilde E_l$

In the same way as we had  argued in Subsection <ref>, we obtain the same kind of $L^2$-estimate:

\begin{equation}\label{tL2Eljk}
\|\M^\la_{j,k}\|_{2\to 2}\lesssim 2^{-j}(1+\la d_{j,k})^{-\frac 12}.
\end{equation}

§.§ $L^{1+\ve}$ - estimates of $\M_{j,k}^\la$ for the contribution by $\tilde E_l$ when $\la\ge 2^j$

§.§.§ Reduction to $K|\gamma(\si_2,Y_1)|\lesssim 1$

a) Let us first  consider the $\si_2$ -region where

\begin{equation}\label{Kgag1}
K|\gamma(\si_2,Y_1)|\gg 1.
\end{equation}

Then $K\ga(\si_2,Y_1)+z\sim K\ga(\si_2,Y_1).$

Case 1: $\la d_{j,k}\ge 1$. Note that this implies that

\la\ge 2^{k A+j(B+2)}\ge 2^{2j},\quad \text{hence} \quad 2^j\le \la^{\frac 12}.

Given any $0<\de\ll 1,$ we may then assume that $K|\gamma(\si_2,Y_1)|\ge \la^{\de-1} 2^j.$
For, if

1\ll K|\gamma(\si_2,Y_1)|< \la^{\de-1} 2^j,

then we would have $1\ll \la^{\de-1/2},$ contradicting our assumption that $\la\gg1,$ if we choose $\de<1/2.$

But, if $K|\gamma(\si_2,Y_1)|\ge \max\{1, \la^{\de-1} 2^j\},$ then
|\la 2^{-j} [K\ga(\si_2,Y_1)+z]|\ge \la^{\de},
so that by (<ref>) and (<ref>) we see that the contribution to $\mu_{j,k}$ by this $\si_2$-region is a small error term of order $O(\la^{-N})$  for every $N\in \NN.$

Case 2: $\la d_{j,k}< 1$.  Here, we may ignore the term   $d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2}$ of the phase in (<ref>), since we can include the factor $e^{-i\la s_3(d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2})} $ in the amplitude. Let us thus assume that

\begin{eqnarray*}
\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)
&=&2^{-j} e^{-i\la s_3\tilde h(s_2)} \,\int \vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\\
&&\hskip 3cm \times e^{-i\la s_3 \big(2^{-j}z \tilde g^s(\si_2)+\tilde h^s(\si_2)\big)}\chi_1(z)\eta dz,
\end{eqnarray*}

where we have put

\tilde g^s(\si_2):=\tilde g(2^{-k}\si_2),\quad  \tilde h^s(\si_2):=\tilde h(2^{-k}\si_2).

This case will require a more careful analysis. Observe first that by  (<ref>) and (<ref>), we have

\begin{eqnarray*}
\tilde h(s_2)&=& w(s_2)^n\beta(w(s_2))+s_2w(s_2)^m\omega(w(s_2))+s_2^2w(s_2) q(w(s_2), s_2),\\
\tilde g(s_2)&=& w(s_2)^{n-1}\tilde\beta(w(s_2))+s_2w(s_2)^{m-1}\tilde\omega(w(s_2))+s_2^2\tilde q(s_2)
=:\tilde g_0(s_2)+s_2^2\tilde q(s_2).
\end{eqnarray*}

Note that the term $s_2^2w(s_2) q(w(s_2), s_2)$ can be absorbed  by the term $s_2^2B(s_2)$ in (<ref>), by slightly modifying the definition of $B(s_2), $ still keeping the condition $|B(s_2)|\sim 1.$  Let us therefore henceforth simply assume that

\tilde h(s_2)= w(s_2)^n\beta(w(s_2))+s_2w(s_2)^m\omega(w(s_2)).

For every $\al\in\NN,$ the following estimate holds true:

|\pa_{\si_2}^\al \tilde h^s(\si_2)|\lesssim 2^{-k(1+ma)},
\qquad |\pa_{\si_2}^\al \tilde g_0^s(\si_2)|\lesssim 2^{-k(1+(m-1)a)},
\quad \text {if} \quad \si_2\sim 1.

Note that

na-(1+ma)=(n-m)a-1\ge (n-m)a_1-1=0,

so that $na\ge 1+ma$ and also $(n-1)a\ge 1+(m-1)a.$
Since $|w(s_2)|\sim s_2^a\sim 2^{-ka},$ the claimed estimates now follow easily.

Subcase 2.a: $\la 2^{-ka}\lesssim 1.$ Here,  by Lemma <ref>, $\la \tilde h^s(\si_2)=O(2^{-k})$ and
$\la \tilde g_0^s(\si_2)=O(2^{-k}),$ since $m\ge 2.$ We may thus ignore the terms $\tilde h^s(\si_2)$ and  $2^{-j}z \tilde g_0(\si_2)$ in the amplitude of $\tilde F_{j,k}.$

Moreover, since  we shall here not use any oscillation from the $z$-integration, we may also  assume that the term $2^{-j} zs_2^2 \tilde q(s_2)$ stemming from our decomposition of $\tilde g(s_2)$ is absorbed by the term $s_2^2B(s_2)$ in (<ref>), so that

\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) = 2^{-j}\iint e^{-i\la s_32^{-j} s_1[K\ga(\si_2,Y_1)+z]} \chi_0(s_1)\chi_1(z)  \,\eta dz ds_1.

Note  also that

\la2^{-j}[K\ga(\si_2,Y_1)+z]=\la2^{-ka}\si_2^a \tilde w(\si_2) -\la 2^{-ka}Y_1 +\la 2^{-j}z.

The first  term $\la 2^{-ka}\si_2^a \tilde w(\si_2)$ can later on be included into the amplitude in  (<ref>), since $\la 2^{-ka}\lesssim 1,$ so that we can ignore this term.

Assume first that $|Y_1|\gg 1.$ Then
$1\ll |K\ga(\si_2,Y_1)|\sim K|Y_1|,$ hence

\begin{equation}\label{Kgasize}
\la 2^{-j} |K\ga(\si_2,Y_1)+z|\sim \la 2^{-ka} |Y_1|.
\end{equation}

We can then proceed in a similar way as we did in  the first part of  Subsection <ref>.

If  $|Y_1|\gg1 $ and $|Y_1|\ge \la^{\de-1} 2^{ka},$  integrations by parts in $s_1$ show that
$F_{j,k}=O(\la^{-N})$ for every $N\in\NN,$ and thus also $\mu_{j,k}^\la(y+\Ga)=O(\la^{-N})$ for every $N\in\NN,$ which shows that this leads just to a small error term.

So, assume next that

1\ll |Y_1|\lesssim \la^{\de-1} 2^{ka}.

We then decompose this region  into the regions

\Delta_{\iota_1}:= \{y:|Y_1|\sim 2^{\iota_1}\},

where clearly we may assume that $1\ll\iota_1\lesssim k.$ We further decompose $\Delta_{\iota_1}$ into the subregions

\begin{eqnarray*}
\Delta_{\iota_1,\iota_2}&:=& \{y:|Y_1|\sim 2^{\iota_1}, |Y_2|\sim 2^{\iota_2}\}, \quad 1\ll \iota_2\le k,\\
\Delta_{\iota_1,0}&:=& \{y:|Y_1|\sim 2^{\iota_1}, |Y_2|\lesssim 1\}.
\end{eqnarray*}

|\Delta_{\iota_1,\iota_2}|\lesssim 2^{\iota_1+\iota_2-ka-k}.

For $y\in \Delta_{\iota_1},$ then we may integrate by parts in $s_1.$
Moreover, if $\iota_2\gg1,$  and if $\la 2^{-2k+\iota_2}\ge 1,$ then  our preceding discussion and (<ref>) show that we can also integrate by parts in $\si_2,$ and altogether this leads to the following estimate for $\mu_{k,j,\iota}^\la:$

|\mu_{j,k,\iota}^\la(y+\Ga)|\lesssim \la^{\frac 52} 2^{-j-k} (\la 2^{-ka} 2^{\iota_1})^{-1}(\la 2^{-2k} 2^{\iota_2})^{-1}=\la^{\frac 12} 2^{-j+k+ka-\iota_1-\iota_2}.

Thus, Lemma <ref> shows that, for every $\ve >0,$

\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12}2^{-j}.

And, if $\iota_2=0,$ we can again apply van der Corput's estimate of order 2  in $\si_2,$ which leads to

|\mu_{j,k,\iota}^\la(y+\Ga)|\lesssim \la^{\frac 52} 2^{-j-k} (\la 2^{-ka} 2^{\iota_1})^{-1}(\la 2^{-2k})^{-\frac 12}
=\la 2^{-j+ka-\iota_1},

\|\M_{j,k,(\iota_1,0)}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}.

These estimate have to be summed over the  $\iota_2$s, which will produce an extra factor of order $O(k).$ However, later on we have to sum over the $j$s and $k$s such that $k\lesssim j/\tilde a,$ so effectively by summing we shall only pick up a factor of order $O(2^{\ve j}),$ where we may choose $\ve>0$ as small as we need;  this will cause no harm.

Assume next that $|Y_1|\lesssim 1.$ Here we decompose into  the regions

\begin{eqnarray*}
\Delta_{\iota_2}&:=& \{y:|Y_1|\lesssim 1, |Y_2|\sim 2^{\iota_2}\}, \quad 1\ll \iota_2\le k,\\
\Delta_{0}&:=& \{y:|Y_1|, |Y_2|\lesssim 1\}.
\end{eqnarray*}

|\Delta_{\iota_2}|\lesssim 2^{\iota_2-ka-k}.

We can now only gain from integration by parts (respectively an application of van der Corput's estimate) in $\si_2,$ and thus obtain

|\mu_{j,k,\iota_2}^\la(y+\Ga)|\lesssim \la^{\frac 52} 2^{-j-k} (\la 2^{-2k} 2^{\iota_2})^{-1}=\la^{\frac 32} 2^{-j+k-\iota_2}\lesssim
\la^{\frac 12} 2^{-j+k+ka-\iota_2},

where we have used our assumption $\la 2^{-ka}\lesssim 1.$

|\mu_{j,k,0}^\la(y+\Ga)|\lesssim \la^{\frac 52} 2^{-j-k} (\la 2^{-2k})^{-\frac 12}
=\la^2 2^{-j}\lesssim \la 2^{-j+ka}.

This shows that we arrive at the same estimates as  before, if we choose $\iota_1=0$ therein.

Subcase 2.b: $\la 2^{-ka}\gg1.$ Assume again first that $|Y_1|\gg 1,$ so that (<ref>) holds true.

If  $|Y_1|\ge\la^\de$ for some $\de $ such that $0<\de\ll 1,$ then $\la 2^{-j} |K\ga(\si_2,Y_1)+z|\gtrsim \la^{\de},$ hence

\begin{equation}\label{rdecay}
\mu_{j,k}^\la(y+\Gamma)=O(\la^{-N})\qquad\text{for every}\quad N\in \NN,
\end{equation}

since $\vp$ is rapidly decaying, so this case contributes again only a small error term.

So, assume next w.l.o.g.  that $1\ll |Y_1|\le \la^\de.$ Then

\begin{equation}\label{y1size}
|y_1|\lesssim \la^\de2^{-ka},
\end{equation}

and, by (<ref>),

\begin{equation}\label{phisize}
|\vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)|\lesssim (\la2^{-ka} |Y_1|)^{-N}\le  (\la2^{-ka})^{-N}
\end{equation}

for every $N\in\NN.$

Again, if $\la 2^{-ka}\ge \la^\de,$ this implies (<ref>), so let us assume that $\la 2^{-ka}\le \la^\de.$
Then, by Lemma <ref>, since $m\ge 2,$

|\la \tilde h^s(\si_2)|\lesssim \la 2^{-k(1+ma)}=\la^{1-m} (\la 2^{-ka})^m 2^{-k}\ll \la^{1-m} \la^{\de m}\le\la^{\de m-1}\ll1,

if we choose $\de>0$ sufficiently small. Similarly,

|\la 2^{-j} z\tilde g_0^s(\si_2)|\lesssim 2^{-j}\la 2^{-k(1+(m-1)a)}\le 2^{-j}(\la 2^{-ka})2^{-k}.

Let us next write $\tilde a=da,$ where $d>0.$  Since we are on $\tilde E_l,$ by (<ref>) we then have that  $j\ge k\tilde a=kda.$  Thus, for every $\ve >0,$

\begin{eqnarray*}
|\la 2^{-j} z\tilde g_0^s(\si_2)|&\lesssim&  2^{-kda}(\la 2^{-ka})2^{-k}= \la^{-\ve} (\la 2^{-ka})^{1+\ve} 2^{-k(1+da-a\ve)}\\
&\le& \la^{-\ve +\de(1+\ve)} 2^{-k(1+da-a\ve)}.
\end{eqnarray*}

Choosing first $\ve>0$ sufficiently small, and then $\de>0$ sufficiently small, we see that also $|\la 2^{-j} z\tilde g_0^s(\si_2)|\ll 1.$

We may thus again ignore the terms $\tilde h^s(\si_2)$ and  $2^{-j}z \tilde g_0(\si_2)$ in the amplitude of $\tilde F_{j,k},$ so that we may assume that

\begin{eqnarray}\nonumber
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-j-k}\iint \vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)  \,e^{i\la s_3 y_3} (1-\chi_0(K\ga(\si_2,Y_1)))\\
&&\hskip 0.2cm \times  e^{-i\la s_32^{-2k}\Big(\si_2^2\big(B(2^{-k}\si_2)+2^{-j}z \tilde q^s(\si_2)\big)-\si_2 Y_2\Big)}\chi_1(z)  \chi_1(\si_2) \chi_1(s_3) \eta dz d\si_2  ds_3.\label{tmujk2}
\end{eqnarray}

Here, we assume that $\chi_0\equiv 1$ on a sufficiently large neighborhood of the origin, and we have put
Y_2:= 2^k y_2.

Reduction to $|Y_2|\lesssim 1.$ Note that here

\begin{eqnarray*}
\Big|\pa_{\si_2} \big(\vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\big)\Big|&\lesssim& \la 2^{-ka}\Big| \vp'\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\Big|\\
&\lesssim& \Big| \tilde \vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\Big|,
\end{eqnarray*}

where $\tilde \vp\in \S(\RR)$, and where we have used (<ref>) and $1\le |Y_1|$  for the last inequality. Note that also the $\si_2$-derivative of  $\chi_0(K\ga(\si_2,Y_1))$ can be controlled in a similar way, since  $ K=2^j2^{-ka}\le \la2^{-ka}.$

Thus, the $\si_2$-derivative  of the amplitude in (<ref>) behaves in the same way as the factor itself.
In a similar way as in Subsection <ref>, let us consider now the subregions

\begin{eqnarray*}
\Delta_{\iota}&:=& \{y: 1\ll |Y_1|\le \la^\de, |Y_2|\sim 2^{\iota}\}, \quad 1\ll \iota_2\le k.
\end{eqnarray*}

Since for $y\in \Delta_{\iota}$

\Big|\pa_{\si_2}\Big(\si_2^2\big(B(2^{-k}\si_2)+2^{-j}z \tilde q^s(\si_2)\big)-\si_2 Y_2\Big)\Big|\sim 2^\iota,

an integration by parts in $\si_2,$  using also (<ref>) with $N=1,$  shows that

|\mu_{j,k}^\la(y+\Ga)|=\la^{\frac52}2^{-j-k} (\la2^{-ka})^{-1}\, (\la2^{-2k}2^\iota)^{-1}=\la^{\frac 12}2^{-j}2^{k(a+1)} 2^{-\iota}.

And, by (<ref>), we find that
|\Delta_\iota|\lesssim \la^\de2^{-ka} 2^{-k} 2^{\iota},
so that we conclude that

\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-j},

provided we choose $\de>0$ sufficiently small.
Here, $\M_{j,k,\iota}^\la$ denotes the contribution  by $\Delta_\iota$  to $\M_{j,k}^\la.$ Thus, we obtain the following analogue of

\sum\limits_{1\ll \iota\le k}\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-j}.

We have thus reduced ourselves to the region

\Delta_{0}:= \{y: 1\ll |Y_1|\le \la^\de, |Y_2|\lesssim 1\}.

Here,  we apply van der Corput's estimate of order $2$ (Lemma <ref>) in $\si_2$ and obtain

|\mu_{j,k}^\la(y+\Ga)|=\la^{\frac52}2^{-j-k} (\la2^{-ka})^{-1}\, (\la2^{-2k})^{-\frac 12}=\la2^{-j}2^{ka},

whereas $|\Delta_0|\lesssim \la^\de2^{-ka} 2^{-k},$ so that we shall here obtain  the estimate

\|\M_{j,k,0}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}

as in (<ref>) of Section <ref>, and altogether we arrive also here at  the estimate (<ref>) for the contribution by the region where $|Y_1|\gg 1.$

We have thus reduced ourselves to estimating the contribution by the region where  $|Y_1|\lesssim1.$
Recall also that we are still assuming (<ref>) and $\la2^{-j}\ge 1.$

We next observe that we can further reduce to the case where $1\ll |K\ga(\si_2,Y_1)|\lesssim 2^j\la^{\de-1},$ again with $0<\de\ll1,$ for if $|K\ga(\si_2,Y_1)|\gg 2^j\la^{\de-1},$ we see from (<ref>) that the contribution to $\mu_{j,k}$ by this $\si_2$-region is a small error term of order $O(\la^{-N})$  for every $N\in \NN.$

So, let us assume that

\si_2\in \Sigma:=\{\si_2\sim 1: |\ga(\si_2,Y_1)|\lesssim \la^{\de-1}2^{ka}\}.

Then $|\Sigma|\lesssim \la^{\de-1}2^{ka},$ and thus

\iint \limits_\Sigma\big|\vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\big|\, \chi_1(\si_2) |\eta| dz d\si_2
\lesssim |\Sigma|(\la2^{-j})^{-1}=\la^{\de-2}2^{j+ka},

hence, by (<ref>),

|\mu_{j,k}^\la(y+\Ga)|=\la^{\frac52}2^{-j-k} \la^{\de-2}2^{j+ka}=\la^{\frac12+\de}2^{k(a-1)}.

Moreover, since here $y\in \Delta:=\{y: 1\ll |Y_1|\lesssim 1\},$ where $|\Delta|\lesssim 2^{-ka},$ by choosing $\de>0$ sufficiently small  we  may conclude that

\begin{equation*}%\label{L1Y1<1}
\|\M_{j,k,|Y_1|\lesssim 1}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-k}.
\end{equation*}

Note that this type of estimate had not yet appeared in Section <ref>.

b) We have thus reduced ourselves to estimating the contribution by the $\si_2$ -region where

\begin{equation}\label{Kgal1}
K|\gamma(\si_2,Y_1)|\lesssim 1.
\end{equation}

A look at the phase $\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1$ in (<ref>) then shows that we can now again apply Lemma <ref> in the variables $(s_1,z)$ in place of $(\eta,x)$ to the oscillatory integral $\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)$ in (<ref>), with $N:=\la 2^{-j}\ge 1.$
Moreover, observe that $s_1$ and $z$   result  from our original coordinates $s_1$ and $z_1=x_1$ by means of  a smooth change of coordinates.  Passing back to our original coordinate, according to Remarks <ref>  we therefore must evaluate the original phase $\breve\phi_1(x_1, s_2)+s_1x_1-s_1y_1$ in (<ref>) at $x_1=y_1,$ and thus get

\begin{equation*}%\label{tapplemma5.2}
\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)   = (\la2^{-j})^{-1} 2^{-j}e^{-i\la s_3(\breve\phi_1(y_1, s_2)}    \eta(y_1,2^{-k}\si_2).
\end{equation*}

This implies that

\begin{eqnarray*}
\mu_{j,k,K|\ga|\lesssim1}^\la(y+\Ga)&=&\la^{\frac32}2^{-k}
\int  e^{-i\la s_3\big(\breve\phi_1(y_1, 2^{-k}\si_2)+2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3\big)} \\
&&\hskip2cm \times \chi_1(\si_2) \chi_1(s_3) \, \chi_0(K\ga(\si_2,Y_1))\, \eta d\si_2ds_3
\end{eqnarray*}

for the contribution by the region where $K|\ga(\si_2,Y_1)|\lesssim 1.$
By (<ref>), we have

\breve\phi_1(y_1, 2^{-k}\si_2):=y_1^n\beta(y_1)+2^{-k}\si_2y_1^m\omega(y_1)+2^{-2k}\si_2^2 y_1 q(y_1, 2^{-k}\si_2).

The  last term  can be absorbed by the term $2^{-2k}\si_2^2B(2^{-k}\si_2),$ by slightly modifying $B(s_2)$ to $B(s_2,y_1),$ which allows us  to assume  for simplicity that $q\equiv 0.$  Then we may re-write

\begin{eqnarray*}
\mu_{j,k,K|\ga|\lesssim1}^\la(y+\Ga)&=&\la^{\frac32}2^{-k}
\int  e^{-is_3\la \big(2^{-k}\Phi_1(\si_2, y)+y_1^n\beta(y_1)-y_3\big)} \\
&&\hskip2cm \times \chi_1(\si_2) \chi_1(s_3) \, \chi_0(K\ga(\si_2,Y_1))\, \eta d\si_2ds_3,
\end{eqnarray*}

where the phase $\Phi_1(\si_2, y)$ is given by

\Phi_1(\si_2, y):=-\si_2[y_2-y_1^m\omega(y_1)]+2^{-k}\si_2^2B(2^{-k}\si_2,y_1).

We shall now distinguish two subcases:

The subcase where $K\lesssim 1.$ We claim that then our assumption (<ref>) implies that  $|y_1|\lesssim 2^{-j}.$

To this end, note first that the condition $K\lesssim 1$ is equivalent to $2^{-ka}\lesssim 2^{-j}.$ Thus, if $|Y_1|\lesssim 1,$ then $|y_1|\lesssim 2^{-ka}\lesssim 2^{-j}.$ And, if $|Y_1|\gg 1,$  then
$1\gtrsim K|\ga(\si_2,Y_1))|\sim K|Y_1|=2^j|y_1|,$ so that again  $|y_1|\lesssim 2^{-j}.$

Observe next that here all $\si_2$-derivatives of $\chi_0(K\ga(\si_2,Y_1))$ are uniformly bounded. This shows that we can now just follow the arguments from  the last part of Subsection <ref> (after identity (<ref>)), by either integrating by parts in $\si_2,$ or applying van der Corput's estimates, in order to arrive again at  estimates of the form (<ref>) for the corresponding contributions to $\M_{j,k}^\la$ (notice that in Subsection <ref> the quantity $Y_1$ had been  defined in a different way, so that there
$|Y_1|\lesssim 1$ had been equivalent to $|y_1|\lesssim 2^{-j}$).

The subcase where $K\gg 1.$ Here we see that  necessarily $|Y_1|\lesssim 1.$  We shall again try to follow our reasoning from  Subsection <ref>, but shall see that the case $\iota_2=0$ will still require a more refined reasoning.

Let us again decompose the region where  $|Y_1|\lesssim 1$ into the subregions

\begin{eqnarray*}
\Delta_{\iota_2}&:=& \{y:|Y_1|\lesssim 1,2^k |y_2 -y_1^m\omega(y_1)|\sim 2^{\iota_2}\}, 1\ll \iota_2\le k,\\
\Delta_{0}&:=& \{y:|Y_1|\lesssim 1, 2^k|y_2 -y_1^m\omega(y_1)|\lesssim 1\}.
\end{eqnarray*}

|\Delta_{\iota_2}|\lesssim 2^{-ka} 2^{\iota_2-k}.

Assume first that $\iota_2\gg 1.$ Observe first that

\int \Big|\pa_{\si_2}\big( \chi_0(K\ga(\si_2,Y_1)) \chi_1(\si_2)  \eta \big)\Big|\,d\si_2
\lesssim K\int \big| \chi'_0(K\ga(\si_2,Y_1))\big| \chi_1(\si_2)\,d\si_2 \lesssim 1,

since here $\si_2\sim 1,$ so that we may use $\tau_2:= \si_2^a\tilde w(\si_2)$ as a new variable of integration in place of $\si_2.$
Thus, we may  again integrate by parts in $\si_2$ and obtain that

|\mu_{j,k,K|\ga|\lesssim1,\iota_2}^\la(y+\Ga)|\lesssim \la^{\frac32}2^{-k}(\la 2^{-2k+\iota_2})^{-1}=\la^{\frac 12}2^{k-\iota_2},

\begin{equation}\label{L1K>1<iota}
\|\M_{j,k,K|\ga|\lesssim1,\iota_2}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-ka}.
\end{equation}

Note that this type of estimate had not yet appeared in Section <ref>.

Assume finally  that $\iota_2=0.$  Then a direct application of  van der Corput's estimate of order 2 in $\si_2$ would lead in a similar way to the estimate

\|\M_{j,k,K|\ga|\lesssim1,0}^\la\|_{1+\ve\to 1+\ve}\lesssim \la 2^{-k} 2^{-ka},

which turns out to be insufficient. Indeed, we shall need to gain a factor $K^{-1}=2^{ka} 2^{-j}$ over  this estimate, which will require a more refined argument.

Note first that
$|\pa_{\si_2}\ga(\si_2,Y_1)|\sim 1.$  This allows us to  write, for $\si_2\sim 1,$

\ga(\si_2,Y_1)	= (\sigma_2-\sigma_2^c) W(\sigma_2,Y_1),

where $\si_2^c=\si_2(Y_1)$ and $ W$ are analytic functions with $|\si_2^c|\sim 1\sim |W|.$
It is then natural to change coordinates from $\si_2$ to

\tau_2:=K(\si_2-\si_2^c),\quad \text{i.e.,}\quad \si_2=\si_2^c+K^{-1}\tau_2.

Then, for $y\in \Delta_0,$ we may assume that, with respect to our new variables of integration,

\begin{eqnarray}\nonumber
\mu_{j,k,K|\ga|\lesssim1,0}^\la(y+\Ga)&=&\la^{\frac32}2^{-k}K^{-1}
\int  e^{-is_3\la \big(2^{-k}\Phi_2(\tau_2, y)+y_1^n\beta(y_1)-y_3\big)} \\
&&\hskip2cm \times \chi_1(\si_2^c+K^{-1}\tau_2) \chi_1(s_3) \, \chi_0(\tau_2)\, \eta d\tau_2ds_3,\label{tmutau2}
\end{eqnarray}

\begin{eqnarray*}
\Phi_2(\tau_2,y):=\Phi_1(\si_2^c+K^{-1}\tau_2,y)=-\big(\si_2[y_2-y_1^m\omega(y_1)]+2^{-k}\si_2^2B(2^{-k}\si_2,y_1)\big)
\Big|_{\si_2:=\si_2^c+K^{-1}\tau_2}.
\end{eqnarray*}

Since $y\in \Delta_0,$ we may further write

y_2-y_1^m\omega(y_1)=:2^{-k}Y_2, \quad \text{where}\quad |Y_2|\lesssim 1,

so that

\Phi_2(\tau_2,y)=\Phi_1(\si_2^c+K^{-1}\tau_2,y)=2^{-k}\big(-\si_2Y_2+\si_2^2B(2^{-k}\si_2,y_1)\big)
\Big|_{\si_2:=\si_2^c(Y_1)+K^{-1}\tau_2}.

By performing a  Taylor expansion of the phase  function $\Phi_1(\si_2^c+K^{-1}\tau_2,y)$ with respect to $K^{-1}\tau_2,$ we see that we may write

\la2^{-k}\Phi_2(\tau_2,y)
= \la2^{-2k}\Big(H_0(Y)+ K^{-1} \tau_2\,H_1(Y)+
K^{-2} \tau_2^2\, H_2(Y,K^{-1}\tau_2)\Big),

where the $H_j$ are analytic functions of their very variables, and where

\begin{equation}\label{onH}
|H_2|\sim 1, \qquad H_1(Y)=-(Y_2-A(y_1))=-2^k\big(y_2-y_1^m\om(y_1)-2^{-k}A(y_1)\big).
\end{equation}

$A(y_1)=2\si_2^c(Y_1)B(0,y_1)+D(y_1),$ with an error  term $D(y_1)$ of order $O(2^{-k}).$ Note also that

\big|\pa_{\tau_2}^2\big[\tau_2^2H_2(Y,K^{-1}\tau_2)\big]\big|\sim 1, \qquad \big|\pa_{\tau_2}\big[\tau_2^2H_2(Y,K^{-1}\tau_2)\big]\big|\lesssim 1,

since $K\gg 1.$

We are thus led to decomposing the $y$-region $\Delta_0$  into the following subsets:

\begin{eqnarray*}
\Delta_{0,\nu}&:=&\{y\in \Delta_0: |H_1(Y)|\sim  2^\nu K^{-1}\}, \qquad 1\ll \nu \lesssim  \log K\le \log\la;\\
\Delta_{0,0}&:=&\{y\in \Delta_0: |H_1(Y)|\lesssim K^{-1}\}.
\end{eqnarray*}

As usually, the corresponding measures will be denoted by $\mu_{j,k,\nu}^\la,$ and so forth.
Since $2^{ka}|y_1|=|Y_1|\lesssim 1$ on $\Delta_0,$ in combination with (<ref>) we see that

\begin{equation}\label{tsizeDeltanu}
|\Delta_{0,\nu}|\lesssim 2^{-ka} 2^{-k} 2^\nu K^{-1}.
\end{equation}

Assume first that $\nu\gg 1.$ Then an integration by parts with respect  to  $\tau_2$ in (<ref>) leads to

\la^{\frac32}2^{-k}K^{-1}\big[\la 2^{-2k} \, 2^\nu K^{-2}\big]^{-1}=\la^{\frac 12} 2^{k-\nu} K,

hence, in combination with (<ref>),

\|\M_{j,k,\nu}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-ka},

i.e., the same type of estimate as in (<ref>).

Assume next  that $\nu=0.$ Here, we can again apply van der Corput's estimate of order 2 in $\tau_2$ and obtain that

\la^{\frac32}2^{-k}K^{-1}\big[\la 2^{-2k} \,K^{-2}\big]^{-\frac 12}=\la,

\|\M_{j,k,0}^\la\|_{1+\ve\to 1+\ve}\lesssim \la 2^{-ka} 2^{-k} K^{-1}=\la^{1+\ve} 2^{-j-k},

which is  again of the form (<ref>).

Combining all the estimates from this subsection, we see that in analogy with the estimate (<ref>) which we had established for    Step 1, for  Step 2 we obtain that for every $\ve>0$ we have the following estimate:

\begin{equation}\label{tMjk}
\|\M_{j,k}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}+  \la^{\frac 12+\ve}2^{-j} +\la^{\frac 12+\ve}2^{-k}+\la^{\frac 12+\ve}2^{-ka}.
\end{equation}

§.§ $L^p$ - estimates of $\M_{j,k}^\la$  for the contribution by $E_l$ when $\la\ge 2^{j}$

Interpolation between the $L^2$ - estimate (<ref>) and the $L^{1+\ve}$ - estimate (<ref>) leads to

\begin{eqnarray}\nonumber
\|\M_{j,k}^\la\|_{p\to p}&\lesssim&
(\la^{1+\ve}2^{-k})^{\frac 2p -1}(1+\la d_{j,k})^{-\frac 12 (2-\frac 2p)} 2^{-j}
+ (\la^{\frac 12+\ve})^{\frac 2p -1}(1+\la d_{j,k})^{-\frac 12 (2-\frac 2p)} 2^{-j}\\ \nonumber
&+&(\la^{\frac 12+\ve}2^{-k})^{\frac 2p -1}\big(2^{-j}(1+\la d_{j,k})^{-\frac 12}\big)^{2-\frac 2p}\\ \nonumber
&+& (\la^{\frac 12+\ve}2^{-ka})^{\frac 2p -1}\big(2^{-j}(1+\la d_{j,k})^{-\frac 12}\big)^{2-\frac 2p}\\ \nonumber
\la^{\ve'} (\la d_{j,k})^{\frac 2p -1} (1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 2p-1}\, 2^{-k(\frac 2p -1)}\,2^{-j} \\ \nonumber
&&+\, \la^{\ve'}  (\la d_{j,k})^{\frac 1p -\frac 12}(1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 1p-\frac 12}  2^{-j}\\
&&+ \la^{\ve'}
(\la d_{j,k})^{\frac 1p -\frac 12}(1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 1p-\frac 12}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}\nonumber\\
&&+ \la^{\ve'}
(\la d_{j,k})^{\frac 1p -\frac 12}(1+\la d_{j,k})^{\frac 1p-1} \, (d_{j,k}^{-1})^{\frac 1p-\frac 12}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}, \nonumber
\end{eqnarray}

if $1<p< 2.$
Arguing as in Subsection <ref>, we  see that we can sum over all dyadic $\la$s and obtain that, for every $\ve'>0,$

\begin{eqnarray}\nonumber
\sum\limits_{\la\gg 1}\|\M_{j,k}^\la\|_{p\to p}&\lesssim& (d_{j,k}^{-1}2^{-k})^{\frac 2p-1+\ve'}\,2^{-j}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\\  \label{tsuminla1}
&+&(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}\\
&+&(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}. \nonumber
\end{eqnarray}

Recall next that $A=\tilde A_l, B=\tilde B_l,$ so that by (<ref>)

\frac{A-1}{\tilde a}+ B+2\le m, \quad  \text{and} \quad d_{j,k}^{-1}=2^{kA+j(B+2)}.

Moreover, since we are on $\tilde E_l,$ by (<ref>) we have $j\ge k\tilde a.$
This shows that we can argue in exactly the same way as in Subsection <ref>  in order to show that we can sum the first two terms on the right-hand side of (<ref>) over all $k$s and $j$s such that $j\ge k\tilde a.$

This leaves us with showing that

\begin{equation}\label{tsuminjk1}
\sum\limits_{j,k\gg1:  j\ge k \tilde a}\Big((d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}\Big)<\infty,
\end{equation}

provided $\ve'>0$ is sufficiently small.

But, if $j\ge k \tilde a,$  since $A\ge 1,$ we see that

\begin{eqnarray*}
(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}
&=& ( 2^{kA+j(B+2)})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}\\
&=& ( 2^{k(A-1)+j(B+2)})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 1p-\frac 12 -\ve')}2^{-j(2-\frac 2p)}\\
&\le& 2^{j\big(\frac{A-1}{\tilde a}+B+2\big)(\frac 1p-\frac 12+\ve')}\, 2^{-k(\frac 1p-\frac 12 -\ve')}2^{-j(2-\frac 2p)}\\
&\le& 2^{jm (\frac 1p-\frac 12+\ve')}\, 2^{-k(\frac 1p-\frac 12 -\ve')}2^{-j(2-\frac 2p)}\\
&=& 2^{j\big[m (\frac 1p-\frac 12+\ve')+\frac 2p -2\big]}\, 2^{-k(\frac 1p-\frac 12 -\ve')}.
\end{eqnarray*}

m (\frac 1p-\frac 12)+\frac 2p -2<m \frac 1{2m} +\frac 2p -2=\frac 2p-\frac 32<0,

since $p>\max\{3/2, h\}.$  Thus we see that if we choose  $\ve'$ sufficiently small, then both the exponent of $2^{j}$ as well as that of $2^k$ in the last estimate are negative, so that we can sum these estimates over all $j,k\ge 1.$

We are thus left with the second sum in (<ref>). But, since $j\ge k\tilde a,$

\begin{eqnarray*}
(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}
&=& ( 2^{kA+j(B+2)})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}\\
&=& ( 2^{k(A-1)+j(B+2)})^{\frac 1p-\frac 12+\ve'}\, 2^{k(\frac 1p-\frac 12 +\ve')-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}\\
&\le &  2^{j\big(\frac {A-1}{\tilde a}+B+2\big)(\frac 1p-\frac 12+\ve')}\, 2^{k(\frac 1p-\frac 12 +\ve'-\frac {2a}p+a)}2^{-j(2-\frac 2p)}\\
&\le &  2^{j\big[m(\frac 1p-\frac 12+\ve')+\frac 2p-2\big]}\, 2^{k(\frac 1p-\frac 12 +\ve'-\frac {2a}p+a)}.
\end{eqnarray*}

Again, we may assume that the exponent of $2^j$ is strictly negative.

Assume first that $\tilde a\ge a.$  Then $j\ge ka,$ hence  after summing first in $j\ge ka,$ we are left with the sum

\begin{eqnarray*}
\sum\limits_{k\ge 1} 2^{ka\big[m(\frac 1p-\frac 12+\ve')+\frac 2p-2\big]}\, 2^{k(\frac 1p-\frac 12 +\ve'-\frac {2a}p+a)}.
\end{eqnarray*}

But, the exponent of $2^k$ can be estimated as follows:

\begin{eqnarray*}
a\big[m(\frac 1p-\frac 12)+\frac 2p-2\big]+\frac 1p-\frac 12 -\frac {2a}p+a&=&(am+1)(\frac 1p-\frac 12)-a\\
<(am+1)\frac 1n-a=\frac{1-(n-m)a}n\le 0,
\end{eqnarray*}

since $p>h$ and $a\ge a_1=1/(n-m).$
By choosing $\ve'$ sufficiently small, we can thus  finally also sum over all $k\ge 1.$

Assume finally  that $\tilde a< a.$  Then $-2a/p+a=-2a(1/p-1/2)<-2\tilde a(1/p-1/2)=-2\tilde a/p+\tilde a,$ so that we can here estimate

\begin{eqnarray*}
(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka(\frac 2p-1)}2^{-j(2-\frac 2p)}
&\le &  2^{j\big[m(\frac 1p-\frac 12+\ve')+\frac 2p-2\big]}\, 2^{k(\frac 1p-\frac 12 +\ve'-\frac {2\tilde a}p+\tilde a)}.
\end{eqnarray*}

Thus, after summing first in $j\ge k\tilde a,$ we are left with the sum

\begin{eqnarray*}
\sum\limits_{k\ge 1} 2^{k\tilde a\big[m(\frac 1p-\frac 12+\ve')+\frac 2p-2\big]}\, 2^{k(\frac 1p-\frac 12 +\ve'-\frac {2\tilde a}p+\tilde a)}.
\end{eqnarray*}

From here on, we can argue as before, only with $a$ replaced by $\tilde a;$  note here that also  $\tilde a\ge a_1.$

This concludes the proof of estimate (<ref>).

§.§ $L^p$ - estimates of $\M_{j,k}^\la$  for the contribution by $E_l$ and $\tilde E_l$  when $\la<2^{j}$

We shall study here both the situations in Step 1 as well as in Step 2.
Let us begin by observing that the condition $\la<2^{j}$ implies that

\begin{equation}\label{la<j}
\la d_{j,k} <2^{-j}\ll1.
\end{equation}

This implies by (<ref>), respectively (<ref>), that here we have the $L^2$-estimate

\begin{equation}\label{L2Eljk2}
\|\M^\la_{j,k}\|_{2\to 2}\lesssim 2^{-j}.
\end{equation}

As for the $L^{1+\ve}$-estimates, let us first look at the Step 1-situation and the contribution by $E_l$:
By (<ref>) and (<ref>), we may here assume that

\Phi_0^s(z,s_1,\si_2)- s_1y_1=-s_1y_1;

the exponentials of all other terms of the phase can be included into the amplitude. Thus,

\begin{eqnarray*}
F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) &=& 2^{-j}\iint e^{-i\la s_3(\Phi_0^s(z,s_1,\si_2)- s_1y_1)} \chi_0(s_1)\chi_1(z)  \,\eta dz ds_1\\
&\sim&2^{-j}\vp(-\la s_3y_1),
\end{eqnarray*}

where $\vp:=\widehat {\chi_0}\in \S(\RR)$ is rapidly decaying.
This shows that, up to a small error term, we can reduce to the region where

|y_1|\le \la^{\de-1},

where again we may later on choose  $\de>0$ as small as we like. But then we obtain

\begin{eqnarray}\nonumber
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-j-k}\\
&&\times \int e^{-i\la s_3(2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3)}\chi_1(\si_2) d\si_2 \chi_1(s_3)\,  \vp(-\la s_3y_1)ds_3.
\label{mujklastep22}
\end{eqnarray}

Let us first assume that $\la\ge 2^k.$
In  a similar way as in Subsection <ref> we  may then  decompose the remaining $y$-region into the subregions

\begin{eqnarray*}
\Delta_{\iota}&:=& \{y: |y_1|\le \la^{\de-1}, |y_2|\sim 2^{\iota-k}\}, \quad 1\ll \iota\le k,\\
\Delta_{0}&:=& \{y:|y_1|\le \la^{\de-1}, |y_2|\lesssim 2^{-k}\}.
\end{eqnarray*}

|\Delta_{\iota}|\lesssim \la^{\de-1} 2^{\iota -k}.

And, as before,  if $\iota\gg1,$  an integration by parts in $\si_2$ shows that

|\mu_{j,k,\iota}^\la(y+\Ga)|\lesssim \la^{\frac52}2^{-j-k} (\la 2^{\iota-2k})^{-1}=\la^{\frac 32} 2^{-j}2^{k-\iota}.

Hence, if we choose $\de$ sufficiently small,

\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 32+\ve/2}2^{-j} 2^{k-\iota}  \la^{\de-1} 2^{\iota -k}\lesssim\la^{\frac 12+\ve}2^{-j},

which leads to the following analogue of (<ref>):

\begin{equation*}\nonumber
\sum\limits_{\iota\le k}\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 12+\ve}2^{-j}.
\end{equation*}

If $\iota=0,$ we can again apply van der Corput's estimate and obtain in a similar way that
\|\M_{j,k,0}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}.
Altogether, we thus obtain the following analogue of (<ref>):

\begin{equation*}%\label{Mjk2}
\|\M_{j,k}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}+ \la^{\frac 12+\ve}2^{-j}.
\end{equation*}

Interpolating the estimates (<ref>) and (<ref>), we find that for $1<p< 2,$

\|\M_{j,k}^\la\|_{p\to p}\lesssim
(\la^{1+\ve}2^{-k})^{\frac 2p -1} 2^{-j}
+ (\la^{\frac 12+\ve})^{\frac 2p -1} 2^{-j}.

As for the first term, summing over all  $k\ge 0$ and all dyadic $\la$s such that $\la<2^j,$ we are left with the sum

\sum\limits_{j\ge 0} 2^{j\big((1+\ve)(\frac 2p-1)-1\big)},

which is easily seen to be convergent if we choose $\ve>0$ sufficiently small.

As for the second term, summing first over all  all $k\le \log\la$  and then all dyadic $\la$s   that $\la<2^j,$ we are left with the sum

\sum\limits_{j\ge  0} 2^{j\big((\frac 12+\ve)(\frac 2p-1)-1\big)},

with an $\ve>0$ which may have to be chosen slightly bigger than before. Again, it is easily seen that the latter series is convergent if we choose $\ve>0$ sufficiently small, since $p>3/2.$

Let us finally assume that $\la< 2^k.$  Then the “oscillatory integral” $\mu_{j,k}^\la(y+\Ga)$ has indeed essentially no oscillation, so that we only get the estimates

|\mu_{j,k}^\la(y+\Ga)|\lesssim \la^{\frac52}2^{-j-k}\quad\text{and}\quad |\Delta|\lesssim\la^{\de-1},

where $\Delta:=\{y:|y_1|\le \la^{\de-1}, |y_2|\ll 1\}.$ Thus,

\begin{equation*}\nonumber
\|\M_{j,k}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 32+\ve}2^{-j-k},
\end{equation*}

\|\M_{j,k}^\la\|_{p\to p}\lesssim (\la^{\frac 32+\ve}2^{-k})^{\frac 2p -1} 2^{-j}.

Summing first over all $j$s such that $2^j>\la$ and all $k$s such that  $2^k>\la,$ we are left with the sum

\sum\limits_{\la\gg 1} \la^{\big( (\frac 32+\ve-1)(\frac 2p-1)-1\big)}.

But, $(3/2-1)(2/p-1)-1=1/p-3/2<-5/6,$ since $p>3/2,$ so that the sum  is again convergent if we choose $\ve>0$ sufficiently small.

Let us next look at the Step 2-situation  and the contribution by $\tilde E_l$:
By (<ref>) and (<ref>), we may here assume that

\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1= s_12^{-ka}\ga(\si_2,Y_1)+\tilde h(s_2),

where $z\sim 1\sim \si_2.$ Putting again $\tilde h^s(\si_2):=\tilde h(2^{-k}\si_2),$ we find that

\begin{equation}\nonumber
\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)
=2^{-j} e^{-i\la s_3\tilde h^s(\si_2)} \,\int \vp\big(s_3\la2^{-ka} \ga(\si_2,Y_1)\big)\chi_1(z)\eta dz,
\end{equation}

where $\vp:=\widehat {\chi_0}\in \S(\RR)$ is rapidly decaying and, by (<ref>),

\begin{eqnarray*}
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-k}\times \\
&&\iint  \tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) e^{-i\la s_3(s_2^2B(s_2)-2^{-k}\si_2 y_2-y_3)} \chi_1(\si_2) d\si_2 \chi_1(s_3) ds_3.
\end{eqnarray*}

Assume first that $\la2^{-ka}\ge 1.$ We then decompose in $y$ into the regions

\begin{eqnarray*}
\Delta_{\iota}&:=& \{y: |Y_1|\sim 2^{\iota}\}, \quad 1\ll \iota\le ka,\\
\Delta_{0}&:=& \{y:|Y_1|\lesssim 1\}, \end{eqnarray*}

and denote the corresponding contributions by $\mu_{j,k,\iota}^\la.$

|\Delta_{\iota}|\lesssim 2^\iota 2^{-ka}.

And, if $\iota\gg1,$ then clearly

|\tilde F_{j,k,\iota}(\la,2^{-k}\si_2,s_3,y_1)|\lesssim 2^{-j}( \la2^{-ka}2^\iota)^{-1},

|\mu_{j,k,\iota}^\la(y+\Ga)|\lesssim \la^{\frac52}2^{-j-k}( \la2^{-ka}2^\iota)^{-1}.

This implies that

\begin{equation}\label{Mjkiota2}
\|\M_{j,k,\iota}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{\frac 32+\ve}2^{-j-k}.
\end{equation}

Finally, if $\iota=0,$ we can  estimate

\begin{eqnarray*}
|\mu_{j,k,0}^\la(y+\Ga)|\lesssim\la^{\frac52}2^{-j-k} \int  \big|\vp\big(s_3\la2^{-ka} \ga(\si_2,Y_1)\big)\big|\chi_1(\si_2) d\si_2 \,\chi_1(s_3)\chi_1(z)dz ds_3.
\end{eqnarray*}

\int \big|\vp\big(s_3\la2^{-ka} (\si_2^ a\tilde w(\si_2)-Y_1)\big)\big|\chi_1(\si_2) d\si_2
\lesssim ( \la 2^{-ka})^{-1},

since  $\si_2\sim 1,$ so that we may use $\tau_2:= \si_2^a\tilde w(\si_2)$ as a new variable of integration in place of $\si_2.$ Thus,

|\mu_{j,k,0}^\la(y+\Ga)|\lesssim \la^{\frac52}2^{-j-k}( \la2^{-ka})^{-1},

so that (<ref>) remains valid also for $\iota=0.$ Interpolation with (<ref>) leads to

\|\M_{j,k,\iota}^\la\|_{p\to p}\lesssim \big(\la^{\frac 32+\ve}2^{-k}\big)^{\frac 2p-1} 2^{-j}.

Summing the estimate in  first in $\iota\le ka,$ then in $k\ge 0$ and next over all  $j$s such that $2^j>\la,$ we are left with the sum

\sum\limits_{\la\gg1} \la^{(\frac 32+\ve)(\frac 2p-1)}\la^{-1}=\la^{\frac 3p-\frac 52+\ve'},

where $\ve'>0$ can be chosen as small as we like. Since $p>3/2,$ we have that  $3/p-5/2<2-5/2<0,$ so that the remaining series in $\la$ is convergent.

Assume next that $\la2^{-ka}<1.$ Here, we must again invoke Lemma <ref>. According to this lemma,
we have that for $\si_2\sim 1,$

|\pa_{\si_2}^\al \tilde h^s(\si_2)|\lesssim 2^{-k(1+ma)}.

This implies that $\la 2^{-k(1+ma)}\ll \la 2^{-ka}<1,$ which shows that we can now include the factor $e^{-i\la s_3\tilde h^s(\si_2)}$ of $\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)$ in the oscillatory integral defining  $\mu_{j,k}^\la(y+\Ga)$ into the amplitude. I.e., we may  simply assume that
$\tilde h^s\equiv 0,$ and thus write

\begin{eqnarray*}
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-j-k}\times \\
&&\hskip-2cm \iint  e^{-i\la s_3(2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3)} \chi_1(\si_2)  \vp\big(s_3\la2^{-ka} \ga(\si_2,Y_1)\big)\eta d\si_2\, \chi_1(s_3) ds_3.
\end{eqnarray*}

But note  that $\la 2^{-ka} \ga(\si_2,Y_1)=\la 2^{-ka}\si_2^ a\tilde w(\si_2)- \la y_1,$ where the first term is harmless, since  $\la 2^{-ka}\le 1.$
A comparison with the identity (<ref>) thus shows that we may now argue exactly as we did for the Step 1- situation.

This concludes our discussion of the case where $\la<2^j.$

§.§ $L^{p}$ - estimates for the contributions by domains  $D_{l_1}^c, c\in \mathcal C_{l_1},$ when the algorithm stops after step 1

Assume now that $c=c^{\al_1}_{l_1}\in \mathcal C_{l_1}$ is such that  the cluster
$\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ]
contains only one, real root $r_0$ of multiplicity
$N_0:=N\left[ \begin{matrix}
\al_1\\
\end{matrix}\right ].
The leading term of $r_0(s_2)$ is then given by $c^{\al_1}_{l_1}s_2^a,$ where again $a:=a_{l_1}.$

We can then modify our change of coordinates by re-defining

\tilde z:=z_1-r_0(s_2),\qquad \text{and} \qquad \tilde\Phi(\tilde z,s_2):=\Phi(\tilde z+r_0(s_2),s_2).

The domain corresponding to $D_{l_1}^c$ is then essentially again of the form

\tilde D_{l_1}^c=\{(\tilde z, s_2):   |\tilde z|<  \ve s_2^{a_{l_1}}\},

with $0<\ve\ll 1.$ We claim that  then also on $\tilde D_{l_1}^c$ we have a good resolution of singularities of $\tilde \Phi,$ namely
\begin{equation}\label{Dlreso}
\tilde\Phi(\tilde z,s_2)=\tilde V(\tilde z,s_2)\, s_2^{\tilde A}\tilde z^{N_0},\qquad (\tilde z,s_2)\in \tilde D_{l_1}^c,
\end{equation}

for some $\tilde A\ge 1,$ where $\tilde V$ has similar properties as in (<ref>).

Indeed, in analogy to (<ref>), we have that

\begin{equation*}%\label{ttildePhifac}
\tilde\Phi(\tilde z,s_2)=\tilde U_0(\tilde z,s_2)(\tilde z+r_0(s_2))^{\nu_1}  \prod_{l=1}^L
\Phi\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ](\tilde z+r_0(s_2),s_2).
\end{equation*}

Since $|\tilde z|\ll |r_0(s_2)|,$ by modifying $\tilde U_0(\tilde z,s_2)$ appropriately, we may replace here
$(\tilde z+r_0(s_2))^{\nu_1}$ by $s_2^{a\nu_1},$ as well as
$\Phi\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ](\tilde z+r_0(s_2),s_2)
by the following:

–  if  $l<l_1,$  then by
s_2^{a_l N\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ]};

–  if  $l=l_1,$  then by
\prod\limits_{\al\ne \al_1} s_2^{a N\left[ \begin{matrix}
\al\\
l_1\end{matrix}\right ]}\, \tilde z^{N_0};

–  if  $l>l_1,$  then by
s_2^{a N\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ]}.

This shows that indeed (<ref>) holds true.  Moreover, a quite similar reasoning shows that the $\ka^{l_1}$-principal part of
$\tilde \Phi$  is given by

\begin{eqnarray*}
\tilde \Phi_{\ka^{l_1}}={\rm constant} \, (\tilde z+cs_2^a)^{\nu_1}\prod\limits_{l<l_1} s_2^{a_l N\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ]} \prod\limits_{\al\ne \al_1} \big(\tilde z-(c_{l_1}^{\al}-c_{l_1}^{\al_1})s_2^a\big)^{N\left[ \begin{matrix}
\al\\
l_1\end{matrix}\right ]}\, \tilde z^{N_0}
\prod\limits_{l>l_1}  (\tilde z+cs_2^a)^{ N\left[ \begin{matrix}
\cdot\\
l\end{matrix}\right ]}.
\end{eqnarray*}

The term with the smallest power of $\tilde z$ in this $\ka^{l_1}$ - homogenous function agrees with  a constant multiple of $s_2^{\tilde A}\tilde z^{N_0},$ which shows that the Newton-Puiseux polyhedron of $\tilde\Phi$ must be of the type sketched in Figure <ref>.

Stopping: comparison between $\N(\Phi)$ and $\N(\tilde\Phi)$

Consequently, we see that $\tilde L=l_1,$ $(\tilde B_{\tilde L},\tilde A_{\tilde L})=(N_0,\tilde A)$  and $\tilde a_{\tilde L}=a_{l_1}=a,$ if $N_0<B_{l_1-1},$
whereas $\tilde L=l_1-1,$ $(\tilde B_{\tilde L},\tilde A_{\tilde L})=(N_0,\tilde A)=(B_{l_1-1}, A_{l_1-1})$  and
$\tilde a_{\tilde L}=a_{l_1-1},$ if $N_0=B_{l_1-1}.$
This shows that the region $ \tilde D_{l_1}^c$ is contained in the transition domain

\tilde E_{\tilde L}:=\{(\tilde z,s_2):  |\tilde z|<2^{-M} s_2^{\tilde a_{\tilde L}}\}

defined as in Subsection <ref>, provided we choose $\ve\ll 2^{-M},$ so that the desired $L^p$-estimates for the contribution by $ \tilde D_{l_1}^c$ follow immediately from our discussions in the previous subsections.

§.§ On the contributions by the homogeneous domains $\tilde D_l$

Recall that we are here in Case 2, i.e.,  $c\in \mathcal C_{l_1},$ and that more precisely  in Subcase (i)  we shall assume that $l\ge l_1+1\ge 2,$ where $l_1\ge 1,$ and in  Subcase (ii)  we assume that $l\ge l_1\ge 2.$

The following notation, which lends for a generalisation to higher steps of the resolution algorithm, will be come useful:
We shall write  $\Phi=:\Phi^{[1]}, D_l=:D_l^{[1]},$ and so forth, and

\tilde \Phi=:\Phi^{[2]},  \quad\tilde z=:z_2=z_1-c^{\al_1}_{l_1}s_2^a,  \quad\tilde D_l=:D_l^{[2]},

and so forth, where as before $a=a_{l_1}.$

Recall next from Remark <ref> (c)  that if we are on $D_l^{[2]}$ and
if $\tilde a_l$ is the leading exponent of a root $\tilde r(s_2)$ of  $\Phi^{[2]}$ lying in  $D_l^{[2]},$ then $\tilde a_l$ is of the form $a_{l_1l_2}^{\al_1},$ and the leading term of $\tilde r(s_2)$ is of the form

Fixing from here on $l_2,$ by replacing $\Phi=:\Phi^{[1]}$ by $ \Phi^{[2]},$ and the set $\mathcal C_{l_1}$ by the set
\mathcal C^{\al_1}_{l_1,l_2}
of all real  coefficients $c_{l_1l_2}^{\al_1\al_2},$
we  can thus from here on proceed  as in Subsection <ref>, changing first
from the variable $z_2$ to

\tilde z_2=z_3:=z_2-cs_2^{a_{l_1l_2}^{\al_1}}=z_1-\big(c^{\al_1}_{l_1}s_2^a+c s_2^{a_{l_1l_2}^{\al_1}}\big),

in order to pass from homogeneous regions

D_l^{[2],c}:=\{(z_2,s_2):   |z_2-cs_2^{a_{l_1l_2}^{\al_1}}|<  \ve s_2^{a_{l_1l_2}^{\al_1}}\},

\tilde D^{[2],c}:=\{(z_3,s_2):   |z_3|<  \ve s_2^{a_{l_1l_2}^{\al_1}}\},

and from $\Phi^{[2]}$ to $\tilde \Phi^{[2]}=: \Phi^{[3]},$
and so forth.
Note also that if $c=c_{l_1l_2}^{\al_1\al_2}\in \mathcal C^{\al_1}_{l_1,l_2},$ then we can write

if we choose for $w(s_2)$ here

w(s_2):=c^{\al_1}_{l_1}s_2^a+c_{l_1l_2}^{\al_1\al_2} s_2^{a_{l_1l_2}^{\al_1}},

i.e., the leading jet of the sub-cluster
$\left[ \begin{matrix} \al_1&\al_2\\
l_1 &l_{2}
\end{matrix}\right ].$
Moreover, we clearly still have that

|w(s_2)|\sim s_2^a

(compare with Remark <ref>).

We are thus led to Step 3 of the resolution of singularities algorithm, in which we can apply the same arguments as in Step 2, now  applied to $\Phi^{[2]}$ in place of $\Phi,$ in order to derive the desired $L^p$-estimates for the associated  maximal operators.

§ HIGHER STEPS OF THE RESOLUTION ALGORITHM AND CONCLUSION OF THE PROOF OF THEOREM <REF>

In the same way as we had passed from Step 1 to Step 2, and  from Step 2 to Step 3, we can now proceed recursively.
Suppose  we have already introduced the functions $\Phi^{[1]}, \dots,\Phi^{[p]}, p\ge 2,$ where

\Phi^{[j]}(z_j,s_2)=\Phi(z_j+w_j(s_2),s_2) \qquad j=2,\dots, p,

and where  $w_j(s_2)$ is the real,  leading jet of a sub-cluster
$\left[ \begin{matrix} \al_1&\cdots &\al_{j-1}\\
l_1&\dots &l_{j-1}
\end{matrix}\right ].
Note that then $z_j=z_1-w_j(s_2).$
The transition domains associated to $\Phi^{[p]}$ can be handled as before in Step 2.
And, by narrowing down in a similar way as in the previous steps to a  homogeneous subdomain $ D^{[p+1],c},$ which corresponds to a narrow subdomain in the original coordinates $z_1$  containing only roots from a sub-cluster
\left[ \begin{matrix} \al_1&\cdots &\al_{p}\\
l_1&\dots &l_{p}
\end{matrix}\right ]
of order $p$ of the previous sub-cluster
\left[ \begin{matrix} \al_1&\cdots &\al_{p-1}\\
l_1&\dots &l_{p-1}
\end{matrix}\right ]
of order $p-1$
whose  associated leading jet

c_{l_1\cdots l_p}^{\al_1\cdots \al_p}s_2^{a_{l_1\cdots
l_{p}}^{\al_1\cdots \al_{p-1}}}

is real, we can  introduce the next function

\Phi^{[p+1]}(z_{p+1},s_2):=\Phi(z_{p+1}+w_{p+1}(s_2),s_2).

Note that then
z_{p+1}=z_p-c_{l_1\cdots l_p}^{\al_1\cdots \al_p}s_2^{a_{l_1\cdots
l_{p}}^{\al_1\cdots \al_{p-1}}},

\Phi^{[p+1]}(z_{p+1},s_2)=\Phi^{[p]}(z_{p+1}+c_{l_1\cdots l_p}^{\al_1\cdots \al_p}s_2^{a_{l_1\cdots
l_{p}}^{\al_1\cdots \al_{p-1}}},s_2).

We have thus arrived at  Step $p+1$ and can bootstrap.

In this way, we shall construct nested sequences of  subclusters
\Big\{\left[ \begin{matrix} \al_1&\cdots &\al_{p}\\
l_1&\dots &l_{p}
\end{matrix}\right ]\Big\}_p.
The corresponding sequences of multiplicities of these sub-clusters are  then decreasing and must therefore eventually become constant, say  after  step $p=P,$ and we are reduced to  a narrow homogeneous domain.  But, arguing as in Subsection <ref>, then the sub-cluster of order $P$ can  contain only one, real root, of some multiplicity $N_0,$ and we can estimate the contribution by  this narrow homogeneous domain in the same way as we did in Subsection <ref>.

This shows that our resolution algorithm will always stop  after a finite number of steps; this completes the proof of Theorem <ref>.

§  FIRST STEPS IN THE PROOF OF THEOREM <REF>

In this section, we assume that $\phi$ is of type $\A_{n-1}^+\setminus\A_{n-1}^e,$  with $n\ge 4.$ Since the coordinates $\x$ in the normal form (<ref>) are in this case already adapted to
$\phi$ (because $n<2m),$  we shall here not really  make use of this normal form, but instead use  Property $(\A_{n-1}^+)$ in these linearly adapted coordinates $\x.$

§.§ Stationary phase in $x_2$

We again  first apply the method of stationary phase to the $x_2$-integration in (<ref>), which  leads to

\begin{eqnarray}\nonumber
\widehat{\mu^\la}(\xi)
&= &e^{-is_3\la} \chi_0(s_3s')\chi_1(s_3) \big[ \la^{-1/2} \int_{\bR} e^{-i\la s_3(s_1x_1+\phi(x_1, x_2^c(x_1,s_2)))}\tilde \eta(x_1,  s_2)\,dx_1\\
&&\hskip 8cm+r(\la,s) \big],\label{9mulahat}
\end{eqnarray}

with a slightly modified cut-off function $\chi_1,$
where $\tilde \eta$ is another smooth bump function supported in a
sufficiently small neighborhood of  the origin,  $r(\la,s)$ is a
remainder term of order

r(\la,s)=O(\la^{-\frac32})\quad \mbox{as} \quad  \la\to+\infty,

and $x_2^c(x_1, s_2)$ denotes  the unique (non-degenerate)
critical point  of the  phase $\phi$ with respect to $x_2.$ Then, similarly  as before,  $\phi(x_1, x_2^c(x_1,s_2))=\breve\phi(x_1,s_2)$ is the Legendre transform of $\phi.$

Following the proof of Theorem <ref>, let us choose line-adapted coordinates  $(y_1,y_2)=(x_1-\alpha (x_2), x_2),$ in which $\phi\x$ is given by $\tilde \phi\y,$ and  decompose

\tilde \phi\y=\tilde\phi(0,y_2)+\tilde\phi_{\rm red}\y.

We then denote by $\tilde \ga_1$ the first non-vertical edge of $\N(\tilde\phi_{\rm red}),$ which has $(n,0)$ as its left endpoint, and choose the weight $\tilde \ka^e =(\tilde \ka^e _1,\tilde\ka^e _2)=(1/n,\tilde\ka^e _2)$ so that $\tilde \ga_1$ is contained in the line $\tilde \ka^e _1t_1+\tilde \ka^e _2t_2=1.$

Assume first  that  $\tilde \ka^e _2>0.$   Then, as we have seen,
we may decompose

\begin{equation*}%\label{9tildedecomp}
\tilde\phi_{\rm red}(y_1,y_2)= p(y_1,y_2) +\tilde\phi_{\rm err}(y_1,y_2),
\end{equation*}

where $ p=p_{\tilde e_1}$ is a $\tilde\ka^e $-homogeneous polynomial of $\tilde\ka^e$-degree $1$ with Taylor support on the first compact edge $\tilde \ga_1$ of $\N(\tilde\phi_{\rm red}),$ consisting of at least two monomial terms, one of them being of the form $cy_1^{n},$   and where $\tilde\phi_{\rm err}$ consists of terms of higher $\tilde\ka^e $-degree.

If we put $z_1:=x_1-\tilde\al(s_2),$  with $\tilde\al(s_2):= \al(s_2w_0(s_2)),$  then we had seen in  the proofs of Lemma <ref> and Theorem <ref> that  the new coordinates  $(z_1,s_2)$  are line-adapted to $\breve\phi(x_1,s_2).$ Moreover,  if we assume  without loss of generality (for simplicity) that $w_0(0)=1,$ then if   $\breve\phi(x_1,s_2)$ is represented  by the function $\breve\phi^{\rm la}(z_1,s_2)$ in these line-adapted coordinates, then by  (<ref>) we have the following analogue of (<ref>):

\begin{equation*}%\label{9legendrephila}
\breve\phi^{\rm la}(z_1,s_2)=s_2^2B(s_2)+ \breve\phi_1(z_1, s_2)
\end{equation*}

\begin{equation}\label{9newphi}
\breve\phi_1(z_1, s_2)=p(z_1,s_2) +\breve\phi_{\rm err}(z_1,s_2),
\end{equation}

with the same polynomial $p$ as in (<ref>) and  an error term $\breve\phi_{\rm err}$ consisting  of terms of  $\tilde\ka^e $-degree $>1,$ and where $p(0,s_2)\equiv 0\equiv \breve\phi_{\rm err}(0,s_2).$
Note that this defines here the function $\breve\phi_1 $ for the present setting.

On the other hand, if $\tilde \ka^e _2=0,$  then we have seen in the proof of Theorem <ref> that $\N(\breve\phi^{\rm la}_{\rm red})$ is contained in the region where $t_1\ge n.$ Since $\phi$ is analytic, this implies that

\begin{equation}\label{9legendrephila3}
\breve\phi^{\rm la}(z_1,s_2)=s_2^2B(s_2)+ \breve\phi_1(z_1, s_2)=s_2^2B(s_2)+z_1^n \big(G(s_2) +\breve\phi_{\rm err}(z_1,s_2)\big),
\end{equation}

where   $G$ and $ \breve\phi_{\rm err}$ are analytic  and  $G(0)\ne 0,$ whereas $\breve\phi_{\rm err}(0,s_2)\equiv 0.$

The identity (<ref>) shows that  the function $\breve\phi^{\rm la}$  has already a good resolution of singularity when   $\tilde \ka^e _2=0,$ and that here $n_e=n.$ We shall therefore assume that  $\tilde \ka^e _2>0$ in the subsequent sections,  and  only briefly comment on the case $\tilde \ka^e _2=0$ in Subsection <ref>.

The contribution of the error term $r(\la,s)$ to $\widehat{\mu^\la}$ in (<ref>)
and   the corresponding maximal operator are again easily estimated, and we shall henceforth ignore this error term.

After changing from the coordinate $x_1$ to the coordinate $z_1$ in the integral in (<ref>), we then we may finally assume that

\begin{eqnarray*}\nonumber
\widehat{\mu^\la}(\xi)
&=& \la^{-\frac 12} \chi_0(s_3s')\chi_1(s_3) e^{-i\la s_3\big(s_2^2B(s_2)+s_1\tilde\al(s_2)+1\big)}\\
&&\hskip3cm  \times \int_{\bR} e^{-i\la s_3(s_1z_1+ \breve\phi_1(z_1, s_2))} \eta\,dz_1 \\ %\label{9mulahat2} \\
&=& \la^{-\frac 12} \chi_0(s_3s')\chi_1(s_3) e^{-i\la s_3\big(s_2^2B(s_2)+s_1\tilde\al(s_2)+1\big)}J(\la, s),\nonumber
\end{eqnarray*}

where again

\begin{equation*}%\label{Jla}
J(\la, s):=\int e^{-i\la s_3(\breve\phi_1(z_1, s_2)+s_1z_1)}  \eta\,dz_1.
\end{equation*}

In order to defray the notation, we have simply dropped the tilde from $\eta.$
Introducing also

\begin{eqnarray}\nonumber
F(\la,s_2,s_3,y_1) &:=&\int  J(\la, s) e^{i\la s_3s_1(y_1-\tilde\al(s_2))} \chi_0(s_3s')\chi_0(s_1)\,ds_1\\  \label{9defiF}
&=&\iint e^{-i\la s_3\big(\breve\phi_1(z_1, s_2)+s_1(z_1+\tilde\al(s_2)-y_1)\big)}  \eta(z_1,  s_2)  \chi_0(s_3s')\chi_0(s_1)\,  \eta dz_1 ds_1,
\end{eqnarray}

by  Fourier inversion and a change from the coordinates $\xi$ to $s,$ we then  have that (with slightly modified functions $\chi_0, \chi_1$)

\begin{eqnarray*}\nonumber
\mu^\la(y+\Ga)
\iint F(\la,s_2,s_3,y_1) e^{-i\la s_3(s_2^2B(s_2)-s_2 y_2-y_3)}\\ % \label{9mula}  \\
&&\hskip2cm\times  \chi_0(s_2) \, \eta ds_2 \chi_1(s_3) ds_3,
\end{eqnarray*}

where we have  again put $\Gamma:=(0,0,1).$

Observe that (<ref>) again allows to write $\breve\phi_1(z_1, s_2)$ in a similar way as we did this in (<ref>):

\begin{equation}\nonumber
\breve\phi_1(z_1, s_2)=z_1^n\beta(z_1)+s_2z_1^{m'}\omega(z_1)+s_2^2z_1 q(z_1, s_2),
\end{equation}

where $x_1^{m'}\omega(z_1)$ stands for a finite type function, if $m'$ is finite, and a flat function, if $m'=\infty.$ Note also that here $\tilde \ka^e _2+m'/n\ge 1.$

A comparison between the identities (<ref>) and (<ref>) for $\breve\phi_1$ suggests that at this stage we could as well have
introduced the notion of effective multiplicity $n_e$ for functions $\phi$ of type $\A^-_{n-1},$  namely by means of  the identity (<ref>) for $\breve\phi_1.$  Indeed, for such $\phi,$ the edge $\breve\ga_1$ of the Newton polyhedron $\N(\breve\phi_1)$ with right vertex $(n,0)$ lies on the line $\ka^e_1t_1+\ka^e_2t_2=1,$ with the weight

\ka^e:=\big(\frac 1n, \frac {n-m} n\big),

and if we define the number $n_e$ in same way  by means of the edge $\ga_1$ as we did this for functions of type
$\A^+_{n-1}$ in Subsection <ref>, we find that here $n_e=m,$ since the point $(m,1)$ lies on the edge $e_1.$ The according exponent $p_e$ would then be given by $p_e=2m/(m+1).$ But, note that here $p_e\le h=2n/(n+2),$ since $2m\le n,$ so $p_e$ is not really relevant for $\A^-_{n-1}$-type singularities.  This was the reason why we had introduced the notion of effective multiplicity only for $\A^+_{n-1}$-singularities, for which  in addition a direct definition of $n_e$ in terms of Newton polyhedra associated to $\phi$ is available.

Recall, however, that  in contrast to Section <ref>, we now have

\begin{equation}\label{kae2<1}
\tilde\ka^e_2<1/2.
\end{equation}

Recall also  that since $\phi$ is of type $\A_{n-1}^+\setminus\A_{n-1}^e,$  $p$ does satisfy the following two assumptions:

(A1) $\pa_{z_1}^2 p(z_1,s_2)$ consists of  at least two distinct monomial terms, one of them of course being $ n(n-1)z_1^{n-2}.$

(A2) If $\pa_{z_1}^2 p(z_1,s_2)$  does not  vanish of maximal possible order $n-2$ along a real, non-trivial root of $\pa_{z_1}^2 p(z_1,s_2);$
more precisely,  $\pa_{z_1}^2 p(z_1,s_2)$ is not  of the form

\begin{equation}\label{maxvan2}
\pa_{z_1}^2 p(z_1,s_2)=n(n-1)(z_1-cs_2^d)^{n-2},
\end{equation}

with $c\in \RR\setminus \{0\}$ and an integer exponent $d\in \NN_{\ge 1}.$

Let us also recall from Remark <ref> that

\begin{equation}\label{nne}
n<2n_e, \quad\text{in particular}\quad n_e>2.
\end{equation}

With our new function $\breve\phi_1$ as defined in (<ref>), we  can now  essentially follow the proof of Theorem <ref>, however, with  some important  modifications and  changes, and we shall therefore mostly concentrate on those.

To this end, we shall adopt  the same notation, putting for instance again (compare (<ref>),

\Phi_0(z_1,s'):=\breve\phi_1(z_1, s_2)+s_1z_1, \qquad \Phi(z_1,s_2):=\pa_{z_1}^2\Phi_0(z_1,s') =\pa_{z_1}^2\breve\phi_1(z_1, s_2),

and shall use the same notation for the associated  oscillatory integrals $J$ and $F,$ Newton polyhedra, and so forth.

It is important to observe that condition (A1) ensures that the Newton polyhedra of $\breve\phi_1$ and of $\Phi$ are closely related:

$\N(\Phi)$ is obtained by translating $\N(\breve\phi_1)$  by $-2$ in horizontal direction and then considering only the part with $t_1\ge 0$ (compare Figure <ref>).

Comparison between $\N(\breve\phi_1)$ and $\N(\Phi)$

In particular, since $n_e>2,$ the first edge $\ga_1$ of $\N(\Phi)$, whose endpoints are  $(B_0,A_0)=(n-2,0)$ and $(B_1,A_1),$ where $A_1\ge 1,$ and the first edge $\ga^e_1$ of $\N(\breve\phi_1),$ whose right endpoint is $(n,0),$ are having the same  modulus of slope

\begin{equation}\label{9a1}
a_1=\frac {\ka_1^1}{\ka^1_2}=\frac 1{n-n_e}.
\end{equation}

Actually, a similar result holds true also for the other edges  of $\N(\phi).$ To this end, let us denote by $\breve \ka^l$ the weight associated to the $l$th edge $\ga^e_l$ of $\N(\breve \phi_1).$

a_l=\frac {\ka_1^l}{\ka^l_2}=\frac {\breve\ka_1^l}{\breve\ka^l_2}\ge a_1, \qquad l=1,\dots, L.

In particular, in a similar way as  in Subsection <ref>, by  Lemma <ref> this will now imply (for Step 1) that

\begin{equation}\label{9leqm}
\frac{A_l-1}{a_l}+B_l+2\le n_e
\end{equation}

for $l=1,\dots, L.$
This is the analogue of the estimate (<ref>), which had been crucial for the proof of Theorem <ref>.

One should therefore basically think of simply replacing $m$ by $n_e$ in our  proof of Theorem <ref> in order to get a proof  of Theorem <ref>. However, this is, of course, an oversimplification, and  there are also some important difference to be noticed, on which we shall  concentrate subsequently.

§ ANALYSIS IN  STEP 1 OF THE RESOLUTION ALGORITHM

§.§ $L^2$ - estimation of $\M_{j,k}^\la$ for the contribution by $E_l$

Since our arguments in Subsection <ref> did not make use of the special form of $\breve\phi_1,$  the $L^2$-estimates
(<ref>) remain valid also here.

§.§ $L^{1+\ve}$ - estimates of $\M_{j,k}^\la$ for the contribution by $E_l, l=1,\dots,L,$ when $\la\ge 2^j$

§.§.§ The case where $\tilde\al(s_2)\not\equiv 0$
This case behaves more like a simplified version of the Step 2 - situation. We shall therefore defer its discussion to the end of Section <ref>.

§.§.§ The case where $\tilde\al(s_2)\equiv 0$

Here, we can basically follow our discussion of the Step 1 - situation in Section <ref>.
Observe first that all estimates up to (<ref>) will again remain valid. However, after having applied Lemma <ref>, the identity (<ref>) for $F_{j,k,|Y_1|\lesssim1}(\la,s_2,s_3,y_1)$ makes  explicit use of  the form of the phase $\breve\phi_1,$ as well as the oscillatory integral giving $\mu_{j,k,|Y_1|\lesssim1}^\la(y+\Ga).$ We shall see that this part will indeed require a different discussion here. The complete phase is now given by

\begin{eqnarray*}
&& 2^{-2k}\big[\si_2^2 B(2^{-k}\si_2) -2^{k}\si_2 y_2 \big]  +\breve\phi_1(y_1,2^{-k}\si_2) -y_3\\
&=&2^{-2k}\big[\si_2^2 B(2^{-k}\si_2) -2^{k}\si_2 y_2 \big]  +p(y_1,2^{-k}\si_2) +\breve\phi_{\rm err}(y_1,2^{-k}\si_2)-y_3.
\end{eqnarray*}

Recall also that here $Y_1:=2^jy_1,$ so that the condition $|Y_1|\lesssim 1$ is equivalent to $|y_1|\lesssim 2^{-j}.$

But, since $p$ is $\tilde\ka^e$-homogeneous of degree $1,$ we have

\begin{equation}\label{pdil}
p(y_1,2^{-k}\si_2)=2^{-k/\tilde \ka^e_2}p(2^{ka_1}y_1,\si_2).
\end{equation}

And, by (<ref>), we may write $2^{-k/\tilde \ka^e_2}=2^{-\ve_0 k}2^{-2k},$ for some $\ve_0>0.$ Moreover,

|2^{ka_1}y_1|=|2^{-j+ka_1}Y_1|\lesssim 2^{-j+ka_1}\le 2^{-(j-ka_l)},

and since we are on $E_l,$  by (<ref>) we have that $j-ka_l\gtrsim 1.$
Thus, $|2^{ka_1}y_1|\lesssim 1.$
These estimates show that the term $p(y_1,2^{-k}\si_2) +\breve\phi_{\rm err}(y_1,2^{-k}\si_2)$ can be absorbed into  the term
$2^{-2k}\si_2^2 B(2^{-k}\si_2),$ i.e., we may assume that the complete phase is of the form

\begin{equation*}%\label{cphase}
2^{-2k}\big[\si_2^2 B(2^{-k}\si_2,Y_1) -2^{k}\si_2 y_2 \big]  -y_3,
\end{equation*}

where $B(2^{-k}\si_2,Y_1)$ has similar properties as  $B(2^{-k}\si_2),$ in particular $|B(2^{-k}\si_2,Y_1)|\sim 1.$

From here on, we can thus again follow the arguments from Subsection <ref>, if we assume there that $\om(y_1)\equiv 0,$ and arrive at  the same $L^{1+\ve}$-estimates as in that subsection.

§.§ $L^p$ - estimation  of $\M^{\tau_l}$  for the contribution by $E_l,l=1,\dots,L,$ when $\la\ge 2^j$

The previous discussions show that  also the $L^p$-estimates (<ref>)  remain valid. Since, for the summation in $\la$ of these estimates, only the condition $p>3/2$ was needed, and since the subsequent summation over all $k\le j/a,$  with $a:=a_l\ge a_1$  works as well,
we can thus eventually reduce to proving  that the estimate

\begin{equation}\label{9crucsum}
\sum\limits_{j\gg1}\Big( 2^{j\big[(\frac{A-1}a+B+2)(\frac 2p-1+\ve')-1\big]}
+  2^{j\big[\frac Aa+B+2)(\frac 1p-\frac 12+\ve')-1\big]}\Big)<\infty
\end{equation}

still  holds true.

But, by (<ref>),

\frac{A-1}a+B+2\le n_e \quad \text{and} \quad \frac 2p-1< \frac 1{n_e},

since $1/p<1/p_e=1/2+1/(2n_e),$ so that we may assume that the exponent of the first summand in (<ref>) is strictly negative, and thus we can sum in $j.$

Similarly, by (<ref>) and (<ref>),

\frac{A}a+B+2\le n_e +\frac 1a\le n_e+\frac 1{a_1}=n_e+(n-n_e)=n\quad \text{and} \quad \frac 1p-\frac 12<\frac 1{2n_e}<\frac 1n,

so that we may assume that the exponent of the second summand in (<ref>) is strictly negative  too and we can again  sum in $j.$

§.§ $L^p$-estimation of $\M^{\tau_0}$ when $\la\ge 2^j$

Here we need to argue in a different way in order to see that the $L^{1+\ve}$- estimate (<ref>) still remains true.

Since $p$ is a $\tilde\ka^e$-homogeneous polynomial function, we can write  it in the form

p(y_1,s_2)=y_1^n\beta(y_1)+s_2\tilde \rho(y_1) +s_2^2\tilde q(y_1,s_2),

where $\tilde \rho$ and $\tilde q$ are $\tilde\ka^e$-homogeneous polynomial functions. Suppose we had $\tilde q(0,0)\ne 0,$ so that $\tilde q$ would be a constant function. Then $s_2^2\tilde q(y_1,s_2)$ would be $\tilde\ka^e$-homogeneous of degree $2\tilde\ka^e_2=1,$ which would contradict (<ref>).
Hence we must have $\tilde q(0,0)=0.$

Essentially in the same way we can show that

\breve \phi_1(y_1,s_2)=y_1^n\beta(y_1)+s_2 \rho(y_1) +s_2^2q(y_1,s_2),

with analytic functions $\rho$ and $q$ such that $q(0,0)=0.$ Thus, the complete phase can be written as

2^{-2k}\big[\si_2^2 B(2^{-k}\si_2) +\si_2^2q(y_1,2^{-k}\si_2)-2^{k}\si_2 (y_2-\rho(y_1) \big]  +y_1^n\beta(y_1)  -y_3,

and since we may assume that $|q|\ll 1,$ the term $\si_2^2q(y_1,2^{-k}\si_2)$ can again be absorbed into the first term. Thus, we can indeed now
argue as in Subsection <ref>, simply replacing the term $y_1^m\om(y_1)$ in this argument by $\rho(y_1).$

As in  Subsection <ref>, what then remains to show is that

\begin{equation*}%\label{9suminjk2}
\sum\limits_{j,k\gg1:  k\ge j/a_1}\Big(( 2^{jn-k})^{\frac 2p-1+\ve'}\, 2^{-j}
+  ( 2^{jn})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\Big)<\infty.
\end{equation*}

But, we  can now essentially follow the arguments from  Subsection <ref>.
For instance, in the first summand, we can first sum over all $k\ge j/a_1$ and arrive at the sum

\sum\limits_{j\gg 1} 2^{j\big[(n-\frac 1{a_1})(\frac 2p-1+\ve')-1\big]}.

n-\frac 1{a_1}=n_e, \quad \text{and} \quad \frac 2p-1< \frac 1{n_e},

so that by choosing $\ve'$ sufficiently small we can indeed  sum in $j.$ The remaining arguments from Subsection <ref> can easily be adapted in a similar way, since for $p>p_e$ we still have

n (\frac 1p-\frac 12)<n\frac 1{2n_e}<n\frac 1n=1.

Before we come to discuss the contributions by the homogeneous domains $D_l,$ we need to  prove a crucial lemma on multiplicities of roots.

§.§ On the condition $\tilde A_1\ge 1:$ a multiplicity lemma

Let $P(y_1,y_2)$ be a polynomial function which is $\ka^1$-homogeneous  of degree 1 and satisfies $P(y_1,0)=y_1^{n-2},$  where $0<\ka^1_2\le1.$ Define
$n_e$ as in Figure <ref>, i.e.,

Note that then $a:=\ka^1_1/\ka^1_2$ is the  modulus of the slope of the line $L_1.$
Assume that $P$ has a non-trivial real root of multiplicity strictly greater then $n_e-2.$

Then, either $P$ is of the form

\begin{equation}\label{Pfac}
P(y_1,y_2)=(y_1-\la y_2^p)^{n-2}, \quad\text{where}\quad p\in\NN_{>0}, \la\in \RR\setminus\{0\},
\end{equation}

or all non-trivial real roots of $P$ have multiplicity 1, and, more precisely, $P(y_1,y_2)$ is of the form

P(y_1,y_2)=y_1^{\nu_1} (y_1^q-\la y_2^p), \quad \text{where}\quad p,q \in\NN_{>0}, q\ge 2, \frac pq=a, \la\in \RR\setminus\{0\}.

According to <cit.>,
$P$ can be written in the form

\begin{equation}\label{Pfac2}
P(y_1,y_2)= y_1^{\nu_1}\prod_{k} (y_1^q-\la_k y_2^p)^{n_k},
\end{equation}

with distinct $\la_k\in \CC\setminus\{0\}$ and multiplicities $n_k\in \bN\setminus\{0\},$ with  $\nu_1\in\NN.$

Let us put $N:=\sum\limits_k n_k,$ so that $n-2=\nu_1+Nq,$  and note that $a=p/q.$ Note also that the  monomial of lowest degree in $y_1$ in $P$ is then of the form $cy_1^{\nu_1} y_2^{N p},$ which implies that $(B_1,A_1)=(\nu_1,Np).$ But then

\ka^1_1\nu_1+\ka^1_2 Np=1, \quad \ka^1_1=a\ka^1_2,

\ka^1_2= 1/(a\nu_1+Np).
This implies that

\begin{equation}\label{ne-2}
n_e-2=\frac{1-\ka^1_2}{\ka^1_1}=\frac{1/\ka^1_2-1}a=\frac {a\nu_1+Np-1}a.
\end{equation}

Now, by our assumption, there is some $k'$ such that $\la_{k'}\in \RR\setminus\{0\}$ and $n_{k'}>n_e-2.$ In particular, (<ref>) implies that

\begin{equation}\label{aN}
N\ge n_{k'} >\nu_1+\frac {Np-1}a,
\end{equation}

\begin{equation}\label{paN}
\end{equation}

(i) If $q=1,$ then $a=p,$ so that by (<ref>) we have  $p\nu_1<1,$ hence $\nu_1=0.$ Thus, by (<ref>)

P(y_1,y_2)= \prod_{k} (y_1-\la_k y_2^p)^{n_k}.

But, by (<ref>), $n_{k'}>N-1/p,$ hence $n_{k'}=N=n-2.$ This implies

P(y_1,y_2)=(y_1-\la_{k'} y_2^p)^{n-2}.

(i) If $q\ge 2,$ then by (<ref>),

N<\frac {1-a\nu_1}{p-a}\le \frac 1{p-a}=\frac 1{p(1-1/q)}\le \frac 1{p/2}=\frac 2p\le 2.

Thus, $N=1,$ and without loss of generality we may assume that $N=n_1=1.$ This implies that

P(y_1,y_2)= y_1^{\nu_1} (y_1^q-\la_1 y_2^p),

hence all non-trivial real  roots of $P$ are simple.

§.§ On the contributions by the homogeneous domains $D_l, l=1,\dots,L$

As in Subsection <ref>, the case $l=1$ requires special considerations. To this end,
recall that the $\ka^e$-principal part of $\breve\phi_1(z_1,s_2)$ is the polynomial $p(z_1,s_2),$  and accordingly

is the $\ka^1$-principal part of $\Phi(z_1,s_2)=\pa_{z_1}^2\breve\phi_1(z_1,s_2),$  which is associated to the first edge $\ga_1$ of $\N(\Phi).$
Moreover, if $a=a_1,$ then every non-trivial  real root of $P$ will be of the form $cs_2^a,$  with $c\ne 0.$

Now, suppose again that we change in a narrow homogeneous neighborhood $D_1^c$ of this root to the coordinates $\tilde z:=z_1-cs_2^a,$ so that we pass from $\Phi$ to $\tilde \Phi(\tilde z,s_2):=\Phi(\tilde z+cs_2^a,s_2),$  and from $P$ to $\tilde P(\tilde z,s_2):=P(\tilde z+cs_2^a,s_2).$

Observe first that Subcase (ii) of Case 2 from Subsection <ref> will appear  here if and only if $\tilde P$ will be of the form $c_0z_1^{n-2},$ i.e., if the edge $\ga_1$ will “shrink” to its right vertex $(\tilde B_0,\tilde A_0)=(B_0, A_0)=(n-2,0)$  under the change of coordinates. But, this would mean that

However, this is excluded by our assumption (A2) (cf. (<ref>)).

Thus, again, only Subcase (i)  of Case 2 can appear (compare Remark <ref> (a)), so that the first compact edge  $\tilde\ga_1$ of  $\N(\tilde\Phi)$
lies on the same line $L_1$ as $\ga_1.$ Denote again by $(\tilde B_0,\tilde A_0)=(B_0, A_0)=(n-2,0)$ its right endpoint, and by  $(\tilde B_1,\tilde A_1)$ its left endpoint. Then $\tilde \ga_1$ is associated to $\tilde P.$

Assume now that we had $\tilde \nu_1=\tilde B_1>n^e-2,$ where $\tilde \nu_1$ denotes the power of $y_1$ of the first factor in (<ref>), if we look at
$\tilde P$ in place of $P.$ But that would mean that $P(z_1,s_2)$ would have a  non-trivial real root of order  strictly greater than $n^e-2,$  so that we could   apply Lemma <ref> to $P(z_1,s_2)$  and conclude that  either

P(z_1,s_2)=(z_1-c s_2^p)^{n-2},

(compare (<ref>)), or  all non-trivial real roots of $P$ have multiplicity 1. The first case is, however, again excluded by assumption  (A1).

In conclusion, we see that either all of the non-trivial real roots of $P$ have multiplicity  1 (as it had been in Section <ref>), or we must have
$\tilde B_1\le n^e-2.$ But, this is equivalent to having $\tilde A_1\ge 1$ (compare Figure <ref>).

From here on, all of the remaining arguments from Section <ref> carry over to our present situation.

§  ANALYSIS IN STEP 2 OF THE RESOLUTION ALGORITHM

Fixing again $l_1\in\{1,\dots,L\}$ as in Section <ref>, putting  accordingly

a:=a_{l_1}\ge a_1,
and proceeding in the same way as before,  we are again led to considering  changes of coordinates of the form

\tilde z:=z_1-w(s_2),

where $w(s_2)=cs_2^a$ (or a full root $r_0$ with this leading term). The function $\Phi(z_1)$ will then lead to the function $\tilde \Phi(\tilde z).$ Note, however, one difference in the effect on the function $F:$

Due to the presence of the extra term $\tilde\al(s_2)$  in the phase for  $F(\la,s_2,s_3,y_1)$ in (<ref>), we should  here indeed better  define
$\Phi_0(x_1,s'):=\breve\phi_1(x_1, s_2)+s_1(x_1+\tilde \al(s_2))$ , and therefore

\begin{eqnarray*}
\tilde\Phi_{0,0}(\tilde z, s_2)&:=& \breve\phi_1(\tilde z+ w(s_2), s_2),\\
\tilde\Phi_{0}(\tilde z, s')&:=& \Phi_{0}(\tilde z+ w(s_2),s_2)=\tilde\Phi_{0,0}(\tilde z, s_2)+s_1\tilde z+s_1\big(w(s_2)+\tilde \al(s_2)\big).
\end{eqnarray*}

We shall therefore also introduce the function

W(s_2):=w(s_2)+\tilde \al(s_2),

besides the function $w(s_2),$ in Step 2, and similarly  also in every higher order step.

Now, either $W(s_2)\equiv 0,$ but then we can proceed in Step 2 (or higher) essentially  in the same way as in Step 1, so that we shall ignore this case in the sequel.

Or, $W(s_2)\not\equiv 0,$ and then

|W(s_2)|\sim s_2^{a'}

for some rational $a'>0,$ since $W(s_2)$ is fractionally analytic.

We remark that the value of $a'$ can depend on the  step of the resolution algorithm, since in higher steps of our  algorithm we shall have to replace the function $w(s_2)$ by the leading jet of some  higher-order sub-cluster of roots.  Observe also: since $\tilde \al(s_2)$ is analytic, we have that
$|\tilde \al(s_2)|\sim s_2^{a_0}$ for some integer $a_0\ge 1.$  The following lemma is then immediate:

Either $a'\ge 1,$ or $a'<1,$ and then $a<1$ and hence $1>a'=a\ge a_1.$

Next,  as in Subsection <ref>, we can reduce to estimating the contributions by the $\tilde \ka^l$- homogeneous  domains
$\tilde D_l$ of the form

\tilde D_l:=\{(\tilde z,s_2):   2^{-M} s_2^{\tilde a_l}\le |\tilde z|\le 2^{M} s_2^{\tilde a_l}\}, \quad l\le \tilde L,

and  the transition domains of the form

\begin{eqnarray*}
%E_{0}&:=&\{(z_1,s_2):   2^M s_2^{a_{1}}<|z_1|\}, \\
\tilde E_l&:=&\{(\tilde z,s_2):   2^M s_2^{\tilde a_{l+1}}<|\tilde z|<2^{-M} s_2^{\tilde a_l}\}, \quad l\le \tilde L-1,\\
\tilde E_{\tilde L}&:=&\{(\tilde z,s_2):  |\tilde z|<2^{-M} s_2^{\tilde a_{\tilde L}}\} .
\end{eqnarray*}

More precisely, in Case 1 where $c\notin \mathcal C_{l_1}, $ according to Remark <ref> we just need the transition domain $\tilde E_{\tilde L},$ where $\tilde L=l_1.$

In Case 2, where $c\in \mathcal C_{l_1},$ we need to distinguish two subcases: if we are in  Subcase (i), then
according to  Remark <ref>   (a) we shall need the transition domains $\tilde E_l$ with $l\ge l_1\ge 1$ and the homogeneous domains  $\tilde D_l$ with $l\ge l_1+1\ge 2.$ Moreover,  in analogy with  Remark <ref> (a), we may and shall   treat  the case where $l_1=1$ and where  all non-trivial real roots of $P$ have multiplicity 1 separartely,
so that we shall here even assume that $l_1\ge 2.$

In Subcase (ii) of Case 2, according to Remark <ref> we may assume that $l_1\ge 2$ and shall then need the transition domains $\tilde E_l$ with $l\ge l_1-1\ge 1$ and the homogeneous domains  $\tilde D_l$ with $l\ge l_1\ge 2.$

Moreover, our discussion in Subsection <ref> shows that under these assumptions we may assume that  $\tilde A_1\ge 1,$
and that the edges $\tilde \ga_1$ and $\ga_1$ are supported on the same line $\tilde L_1=L_1$ of “slope” $\tilde a_1=a_1,$ so that, by Lemma <ref>, we now have  (in analogy with  the identities (<ref>) and  (<ref>)) that

\begin{equation}\label{9keyestl=1}
\frac{\tilde A_1-1}{\tilde a_1}+\tilde B_1+2=n_e,
\end{equation}

(even when $\tilde A_1<1$),  and
\begin{equation}\label{9keyestlg1}
\frac{\tilde A_l-1}{\tilde a_l}+\tilde B_l+2\le n_e,\quad  l=1,\dots, \tilde L
\end{equation}

(compare with (<ref>)).

Having fixed $l_1$ and $l$  according to these assumptions, we shall put

\tilde a:= \tilde a_l.

Note that if we are in Case 1, or Subcase (i) of Case 2, then $\tilde a\ge \tilde a_{l_1}=a,$ but in Subcase (ii) it may also happen that $\tilde a<a$ (for instance if $l=l_1-1,$ where $\tilde a=a_{l_1-1}<a_{l_1}=a$).

§.§ Resolution of singularity on the transition domain $\tilde E_l$

Fix $\tilde E_l$ with $l\ge l_1,$ respectively $l\ge l_1-1\ge 1$  if we are in the Subcase (ii) of Case 2. Arguing as in Subsection <ref>, we here find that

\begin{equation}\label{9tPhi0onEl}
\tilde\Phi_{0}(\tilde z, s')=\tilde U(\tilde z,s_2)s_2^{\tilde A_l} \tilde z^{\tilde B_l+2}+ s_1(\tilde z+W(s_2))+\tilde z \tilde g(s_2)+\tilde h(s_2).
\end{equation}

And, after applying the change of  coordinates $\tilde z=2^{-j} z, \ s_2=2^{-k} \si_2$  putting  $A:=\tilde A_l, B:= \tilde B_l$ and

assuming that (<ref>) holds true, we obtain

\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1=d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2}+ s_1(2^{-j} z+W(s_2)-y_1)+2^{-j} z \tilde g(s_2)
+\tilde h(s_2),

where $U$ has similar properties as in (<ref>). Let us then write

W(s_2)=2^{-ka'} \si_2^{a'}\tilde W(\si_2),

so that $|\tilde W|\sim 1,$ whereas all derivatives of $\tilde W$ can be assumed to be very small. We shall use the following  abbreviations:

\begin{eqnarray}\nonumber
Y_1:&=& 2^{ka'} y_1,\\
\gamma(\si_2,Y_1)&:=&\si_2^{a'}\tilde W(\si_2)-Y_1,  \label{9Y1K}\\
\end{eqnarray}

These allow us  to re-write again as in (<ref>) and (<ref>)

\begin{eqnarray*}\nonumber
&&\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1\\
&=&d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2}+ 2^{-j} s_1\big(K\ga(\si_2,Y_1)+z\big)+2^{-j} z \tilde g(s_2)+\tilde h(s_2)
\end{eqnarray*}

\begin{eqnarray*}\nonumber
&&\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) = 2^{-j}\iint e^{-i\la s_3(\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1)} \chi_0(s_1)\chi_1(z)  \,\eta dz ds_1\\
&&=2^{-j} e^{-i\la s_3\tilde h(s_2)} \,\int \vp\big(s_3\la2^{-j} [K\ga(\si_2,Y_1)+z]\big)\\ \nonumber
&&\hskip5cm \times e^{-i\la s_3\big(d_{j,k} U^s(z,\si_2) \si_2^{A} z^{B+2} +2^{-j}z \tilde g(2^{-k}\si_2)\big)}\chi_1(z)\eta dz,
\end{eqnarray*}

where $z\sim 1\sim \si_2,$ and where $\vp:=\widehat {\chi_0}\in \S(\RR)$ is rapidly decaying.

§.§ $L^2$ - estimation of $\M_{j,k}^\la$ for the contribution by $\tilde E_l$

The $L^2$-estimates (<ref>) again remain valid here.

§.§ $L^{1+\ve}$ - estimates of $\M_{j,k}^\la$ for the contribution by $\tilde E_l$ when $\la\ge 2^j$

Again, we can essentially follow the discussions from Subsection <ref>, and shall only highlight places where the arguments require  modifications. Recall from (<ref>) that

\tilde h(s_2)= \breve\phi_1(w(s_2), s_2), \quad  \tilde g(s_2)=\pa_{z_1}\breve\phi_1(w(s_2), s_2).

Thus,  if we  put again

\tilde g^s(\si_2):=\tilde g(2^{-k}\si_2),\quad  \tilde h^s(\si_2):=\tilde h(2^{-k}\si_2),

we see that

\tilde h^s(\si_2)= \breve\phi_1(2^{-ka}\si_2^a\tilde w(\si_2), 2^{-k}\si_2), \quad  \tilde g(\si_2)=\pa_{z_1}\breve\phi_1(2^{-ka}\si_2^a\tilde w(\si_2), 2^{-k}\si_2).

But,  (<ref>) implies that here

p(2^{-ka}\si_2^a\tilde w(\si_2), 2^{-k}\si_2)=2^{-k/\tilde \ka^e_2}p(2^{-k(a-a_1)}\si_2^a\tilde w(\si_2),\si_2),

where $a-a_1\ge 0,$ and, by (<ref>), $2^{-k/\tilde \ka^e_2}=2^{-\ve_0 k}2^{-2k},$ for some $\ve_0>0.$

Similarly, since $\pa_{y_1}p$ is $\tilde\ka^e$-homogeneous of degree $1-\tilde \ka^e_1,$ we see that

\pa_{y_1}p(2^{-ka}\si_2^a\tilde w(\si_2), 2^{-k}\si_2)=2^{-k\frac {1-\tilde \ka^e_1}{\tilde \ka^e_2}}\pa_{y_1}p(2^{-k(a-a_1)}\si_2^a\tilde w(\si_2),\si_2).

And, if we again assume that $j\gtrsim k\tilde a,$ recalling that $\tilde a\ge a_1,$ we see that also

2^{-j}2^{-k\frac {1-\tilde \ka^e_1}{\tilde \ka^e_2}} \lesssim  2^{-k/\tilde \ka^e_2}.

These estimates imply that

\tilde h^s(\si_2)=O(2^{-\ve_0 k}2^{-2k}), \quad 2^{-j} \tilde g^s(\si_2)=O(2^{-\ve_0 k}2^{-2k}) \text{   for  } j\gtrsim k\tilde a,

which means that we can absorb these terms (as functions of $\si_2$) into  the term $s_2^2B(s_2)=2^{-2k}\si_2^2B(2^{-k}\si_2)$ of the amplitude of the oscillatory integral (<ref>).

From here on, we can simply  follow the arguments for the  reduction to the case $K|\ga(\si_2,Y_1)|\lesssim 1$ in Subsection
<ref>, distinguishing now between the Subcase 2.a: $\la 2^{-ka'}\lesssim 1$, and
Subcase 2.b: $\la 2^{-ka'}\gg  1,$ and replacing accordingly $a$ by $a'$ in these steps of proof.

Note that the main difference compared to the previous discussion lies in how we get rid of the terms $\tilde h^s(\si_2)$ and
$2^{-j} \tilde g^s(\si_2),$ which turns out to be even a bit easier here than in Subsection <ref>.

Next, we can also easily adapt the arguments which led to (<ref>) and the identity

\begin{eqnarray*}
\mu_{j,k,K|\ga|\lesssim1}^\la(y+\Ga)&=&\la^{\frac32}2^{-k}
\int  e^{-i\la s_3\big(\breve\phi_1(y_1, 2^{-k}\si_2)+2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3\big)} \\
&&\hskip2cm \times \chi_1(\si_2) \chi_1(s_3) \, \chi_0(K\ga(\si_2,Y_1))\, \eta d\si_2ds_3.
\end{eqnarray*}

But, due to (<ref>), we may absorb the term $\breve\phi_1(y_1, 2^{-k}\si_2)$ into $2^{-2k}\si_2^2B(2^{-k}\si_2)$ within the amplitude, so that without loss of generality we may assume that

\begin{eqnarray*}
\mu_{j,k,K|\ga|\lesssim1}^\la(y+\Ga)&=&\la^{\frac32}2^{-k}
\int  e^{-i\la s_3\big(2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3\big)} \\
&&\hskip2cm \times \chi_1(\si_2) \chi_1(s_3) \, \chi_0(K\ga(\si_2,Y_1))\, \eta d\si_2ds_3.
\end{eqnarray*}

This is essentially as in Subsection <ref>, if we assume there that $\om(y_1)\equiv 0,$ and replace $a$ by $a'$ in these arguments.

In this way,  combining all these estimates, we see that we here obtain the following analogue to estimate (<ref>):

\begin{equation*}%\label{9tMjk}
\|\M_{j,k}^\la\|_{1+\ve\to 1+\ve}\lesssim \la^{1+\ve}2^{-j-k}+  \la^{\frac 12+\ve}2^{-j} +\la^{\frac 12+\ve}2^{-k}+\la^{\frac 12+\ve}2^{-ka'}.
\end{equation*}

By means of interpolation with our $L^2$ - estimate  (<ref>) and summation in $\la$ we arrive then at the following analogue to
Arguing as in Subsection <ref>, we  see that we can sum over all dyadic $\la$s and obtain that, for every $\ve'>0,$

\begin{eqnarray}\nonumber
\sum\limits_{\la\gg 1}\|\M_{j,k}^\la\|_{p\to p}&\lesssim& (d_{j,k}^{-1}2^{-k})^{\frac 2p-1+\ve'}\,2^{-j}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-j}\\  \label{9tsuminla1}
&+&(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}\\
&+&(d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka'(\frac 2p-1)}2^{-j(2-\frac 2p)}. \nonumber
\end{eqnarray}

Putting again  $A:=\tilde A_l, B:=\tilde B_l,$ we recall from (<ref>) and (<ref>) that

\frac{A-1}{\tilde a}+ B+2\le n_e, \quad  \text{and} \quad d_{j,k}^{-1}=2^{kA+j(B+2)}.

Moreover, since we are on $\tilde E_l,$ by (<ref>) we have $j\ge k\tilde a.$

This shows that we can argue in exactly the same way as in Subsection <ref>  in order to show that we can sum the first two terms on the right-hand side of (<ref>) over all $k$s and $j$s such that $j\ge k\tilde a.$

We are thus left with showing that

\begin{equation}\label{9tsuminjk1}
\sum\limits_{j,k\gg1:  j\ge k \tilde a}\Big((d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-k(\frac 2p-1)}2^{-j(2-\frac 2p)}
+  (d_{j,k}^{-1})^{\frac 1p-\frac 12+\ve'}\, 2^{-ka'(\frac 2p-1)}2^{-j(2-\frac 2p)}\Big)<\infty,
\end{equation}

provided $\ve'>0$ is sufficiently small.

The first sum in (<ref>) can be estimated in the same way as in  Subsection <ref>  - we just have to replace $m$ be $n_e$ in the argument.

As for the second sum, note that the terms in the second sum can be estimated by the terms of the first sum, provided $a'\ge 1.$

This leaves as with the case where $a'<1.$ Then, by Lemma <ref>, $a'=a\ge a_1,$  and as before we can distinguish between the cases $\tilde a\ge a,$ and $\tilde a<a.$

In the first case, we can follow the estimates  from Subsection <ref>, with $m$ replaced by $n_e,$ and only need to verify that $(n-n_e)a\ge 1.$ But, $ a\ge a_1=1/(n-n_e)$ by (<ref>), so that the desired estimate holds true.

In the second case where  $\tilde a<a,$  we can again argue as in Subsection <ref>, since we also have that $\tilde a\ge a_1.$

This completes the proof of (<ref>).

§.§ $L^p$ - estimates of $\M_{j,k}^\la$  for the contribution by $E_l$ and $\tilde E_l$  when $\la<2^{j}$

Arguing in a similar way as in Subsection <ref>, we again  have the  $L^2$-estimate (<ref>).

As for the $L^{1+\ve}$-estimates, let us first look at the Step 1-situation and the contribution by $E_l$.  Again, we shall first assume that $\tilde\al(s_2)\equiv 0.$

Arguing in a similar way as in Subsection <ref>, we may here again assume that

\Phi_0^s(z,s_1,\si_2)- s_1y_1=-s_1y_1.

We can therefore adopt our discussion from Subsection <ref> verbatim,  in which we only made use of the assumption $p>3/2.$

Let us next look at the Step 2-situation  and the contribution by $\tilde E_l$:
By (<ref>), we can here assume that

\tilde\Phi_0^s(z,s_1,\si_2)- s_1y_1= s_12^{-ka'}\ga(\si_2,Y_1)+\tilde h(s_2),

where $z\sim 1\sim \si_2,$  with $Y_1,K$ and $\ga(\si_2,Y_1)$ now defined by (<ref>). But, the term  $\tilde h(s_2)$ can again be absorbed into  the term $s_2^2B(s_2)$ and shall henceforth be dropped.

We then  find that

\begin{equation}\nonumber
\tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1)
=2^{-j} \,\int \vp\big(s_3\la2^{-ka'} \ga(\si_2,Y_1)\big)\chi_1(z)\eta dz,
\end{equation}

where $\vp:=\widehat {\chi_0}\in \S(\RR)$ is rapidly decaying, and

\begin{eqnarray*}
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-k}\times \\
&&\iint  \tilde F_{j,k}(\la,2^{-k}\si_2,s_3,y_1) e^{-i\la s_3(s_2^2B(s_2)-2^{-k}\si_2 y_2-y_3)} \chi_1(\si_2) d\si_2 \chi_1(s_3) ds_3.
\end{eqnarray*}

Assume first that $\la2^{-ka'}\ge 1.$  We can here argue in the same way as in Subsection  <ref>, only with $a$ replaced by $a',$ since we only made use of the assumption $p>3/2.$

Assume next that $\la2^{-ka'}<1.$  Since we  had already seen that we may  assume that $\tilde h^s\equiv 0,$ we can again follow the reasoning from Subsection  <ref>, and  write

\begin{eqnarray*}
\mu_{j,k}^\la(y+\Ga)&=&\la^{\frac52}2^{-j-k}\times \\
&&\hskip-2cm \iint  e^{-i\la s_3(2^{-2k}\si_2^2B(2^{-k}\si_2)-2^{-k}\si_2 y_2-y_3)} \chi_1(\si_2)  \vp\big(s_3\la2^{-ka'} \ga(\si_2,Y_1)\big)\eta d\si_2\, \chi_1(s_3) ds_3.
\end{eqnarray*}

But note  that $\la 2^{-ka'} \ga(\si_2,Y_1)=\la 2^{-ka'}\si_2^ {a'}\tilde W(\si_2)- \la y_1,$ where the first term is harmless, since  $\la 2^{-ka'}\le 1.$
Thus we may again argue as in the Step 1 situation discussed before.

This concludes our discussion of the case where $\la<2^j.$

§.§.§ Completion of Step 1: the case where $\tilde\al(s_2)\not\equiv 0$
This case can be handled like a Step 2 case, where
formally $a=+\infty$ and $W(s_2):=\tilde\al(s_2),$ so that $a'=a_0\ge 1,$  since $a_0\in \NN.$ We leave the details to the interested reader.

§.§ On the contributions by the homogeneous domains $\tilde D_l$

These can easily  be handled in analogy to our discussions in Subsection <ref>, with the obvious modifications of the functions $W(s_2)$ as indicated at the beginning of Section <ref>.

§.§  The case where $\tilde\ka^e_2=0$
In view of  Remark <ref>, this case behaves very much like the cases  explained before  where we would have stopped our resolution algorithm, after applying a change of coordinates by subtracting a complete real roots. It can therefore be handled in a similar way as we did in such situations. We leave the details to the interested reader.

§ HIGHER STEPS OF THE RESOLUTION ALGORITHM AND CONCLUSION OF THE PROOF OF THEOREM <REF>

Here, we can again just proceed as we did in the corresponding Section <ref> for the proof of Theorem <ref>; this completes the proof of Theorem <ref>.

§ PROOF OF THE IOSEVICH-SAWYER-SEEGER CONJECTURE FOR SURFACES OF CLASS $\A^E_{N-1}$

In this final section, we shall briefly sketch how the proof of Theorem <ref> can be modified in order to prove the following

Assume that $S$ is the graph of $1+\phi,$ and accordingly $x^0=(0,0,1),$  where $\phi$ is  analytic and of type
$\A_{n-1}^e$. Then, if  the density $\rho$ is supported in a sufficiently small neighborhood  of $x^0,$  the  condition
$p>\max\{ 3/2, h, {2n}/(n+1)\}=max\{ 3/2,  {2n}/(n+1)\}$  is  sufficient for $\M$ to be $L^p$-bounded.

Details are again left to the interested reader.  Recall  that this result will complete the proof of the Iosevich-Sawyer-Seeger conjecture for arbitrary analytic hypersurfaces in $\RR^3.$

We again assume that $(y_1,y_2)$ are line-adapted coordinates in which (<ref>) and (<ref>) hold true, so that

p\y=y_1^n +c_1 y_1 y_2^{\beta+2}, \qquad \text{with} \ \beta\in\NN.

Since $\pa_{y_1}^2 p\y=n(n-1) y_1^{n-2}, $ we now find that the first edge $\ga_1$ of $\N(\Phi)$ has a modulus of slope $a_1'$ such that $+\infty\ge a_1'>a_1=1/(n-n_e).$ This first edge will intersect the line $s_2=1$ at a point $(n'_e-2,1),$
so that $a'_1=1/(n-n'_e),$ and

\begin{equation}\label{ne'}
\frac n2<n_e<n'_e\le n.
\end{equation}

In Step 1 of the resolution algorithm, we can then proceed as in the proof of Theorem <ref>, only with $a_1$ replaced by $a'_1,$ and $n_e$ replaced by $n'_e.$ Note that now $a\ge a'_1,$ and $1/p-1/2<1/(2n)\le 1/(2n'_e),$  so that the discussion in Subsection <ref> carries over directly. Similarly, since now

n-\frac 1{a'_1}=n-(n-n'_e)=n'_e, \quad \text{and} \quad \frac 2p-1< \frac 1{n'_e},

also  the discussion in Subsection <ref> carries over, since now we can therein assume that $k\ge j/a'_1.$

In Step 2 (and higher later on), recall  that we had put $z_1:=y_1.$ We first note that  here we apply changes of coordinates of the form

\tilde z:= z_1-w(s_2),

where now $|w(s_2)|\sim s_2^a,\quad\text{with} \quad a\ge a'_1.$ It is important that the control of the functions $\tilde g$ and $\tilde h$ that we had discussed in Subsection <ref> still works here: recall that

p(2^{-ka}\si_2^a\tilde w(\si_2), 2^{-k}\si_2)=2^{-k/\tilde \ka^e_2}p(2^{-k(a-a_1)}\si_2^a\tilde w(\si_2),\si_2),

and since $a-a_1\ge a'_1-a_1>0,$ we again can conclude that $2^{-k(a-a_1)}\lesssim 1.$ Similarly, since now $j\ge ka\ge ka'_1\ge ka_1,$ we still have that

2^{-j}2^{-k\frac {1-\tilde \ka^e_1}{\tilde \ka^e_2}} \lesssim  2^{-k/\tilde \ka^e_2}.

Thus, we can basically proceed as before, but there is one major difference:

In our application of the Multiplicity Lemma <ref> for the study of the contribution by the domain $D_1,$ we can no longer exclude the first possibility (<ref>), i.e., it may happen that

P(z_1,s_2)=(z_1-c_1 s_2^p)^{n-2},\qquad c_1\in\RR\setminus\{0\}, p\in\NN_{\ge 1},

vanishes of maximal order $n-2$ along a nontrivial real root. Note that here we would perform a change of coordinates
$\tilde z:= z_1-w(s_2),$ with $w(s_2):=c_1s_2^{a'_1},$ where $a'_1=p$ is a natural number, in order to pass from $\Phi$ to
$\tilde \Phi.$ This is now an analytic change of coordinates, so that $\N(\tilde\Phi)$ is still a Newton polyhedron (and not just Newton-Puiseux), which implies that automatically we will have $\tilde A_1\ge 1.$

However, since in the new coordinates, $\tilde P(\tilde z,s_2)=\tilde z^{n-2},$ the first edge $\tilde \ga_1$ of $\N(\tilde\Phi)$ will have a (modulus of) slope $a''_1$ such that $\infty \ge a''_1>a'_1>a_1,$ and we shall be reduced to a region of the form

\tilde D_1^{c_1}:=\{|\tilde z|<\ve s_2^{a'_1}\},

in which we shall again have to devise transition domains $\tilde E_l$ and homogeneous domains $\tilde D_l,$  including the case $l=1.$

For  all other cases in our application of Lemma <ref>, the discussion at the end of Subsection <ref> carries over.

Now, if we proceed to higher steps of our resolution algorithm, we shall either no longer encounter the first possibility (<ref>)  from a certain step on, so that we can directly adapt our discussion from the proof of Theorem <ref>,
or  the first possibility (<ref>)  will arise at every Step j, within the corresponding domain $D_1^{(j)}.$
However, when $j$ is sufficiently large, say when $j\ge j_0,$ then $D_1^{(j)}$ will contain exactly one real root, of maximal multiplicity $n-2,$ (compare the discussion in Section <ref>), if we consider this domain in our original coordinates $(z_1,s_2),$ and we see that this root $r(s_2)$ will have a Taylor series expansion of the form

r(s_2)=c_1s_2^{a'_1}+c_2s_2^{a'_2}+\cdots +c_js_2^{a'_j}+\cdots,

with natural numbers $a'_1<a'_2<\cdots <a'_j<\cdots,$ and real coefficients $c_j.$  Moreover, we may then write
$\Phi(z_1,s_2)=V(z_1,s_2)(z_1-r(s_2))^{n-2},$ with $|V|\sim 1.$ As what we always did   in such a situation, we can now  perform the analytic change of coordinates

\tilde z:= z_1-r(s_2),

and  find that in these new coordinates $(\tilde z, s_2),$ we have

\tilde\Phi(\tilde z,s_2)=\tilde V(\tilde z,s_2)\tilde z^{n-2}.

As in earlier situations where we had stopped the resolution of singularities algorithm, we now see that we have arrived at a good resolution of singularities and can apply our usual arguments to prove $L^p$ - boundedness of $\M$ for
$p>\max\{ 3/2,2n/(n+1)\}$ (note that here $n'_e=n$ if we define $n'_e$ with respect to these new coordinates).

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Arnol'd, V.I.,  Gusein-Zade, S.M. and  Varchenko, A.N.,
Singularities of differentiable maps.  Vol. II,  Monodromy and asymptotics of integrals,
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Birkhäuser, Boston Inc., Boston, MA, 1988.

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Estimations de certaines fonctions maximales.
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