#  A unified approach to degenerate problems in the half-space

G. Metafune L. Negro C. Spina Dipartimento di Matematica e Fisica “Ennio De
Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy. -mail:
<EMAIL_ADDRESS>di Matematica e Fisica “Ennio De
Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy. email:
<EMAIL_ADDRESS>di Matematica e Fisica“Ennio De Giorgi”,
Università del Salento, C.P.193, 73100, Lecce, Italy. e-mail:
<EMAIL_ADDRESS>
###### Abstract

We study elliptic and parabolic problems governed by the singular elliptic
operators

$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right),\qquad\alpha_{1},\alpha_{2}\in\mathbb{R}$

in the half-space $\mathbb{R}^{N+1}_{+}=\\{(x,y):x\in\mathbb{R}^{N},y>0\\}$.

Mathematics subject classification (2020): 35K67, 35B45, 47D07, 35J70, 35J75.

Keywords: degenerate elliptic operators, boundary degeneracy, vector-valued
harmonic analysis, maximal regularity.

##  1 Introduction

In this paper we study solvability and regularity of elliptic and parabolic
problems associated to the degenerate operators

$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right)\quad{\rm
and}\quad D_{t}-\mathcal{L}$

in the half-space $\mathbb{R}^{N+1}_{+}=\\{(x,y):x\in\mathbb{R}^{N},y>0\\}$ or
in $(0,\infty)\times\mathbb{R}^{N+1}_{+}$.

Here $b,\ c$ are constant real coefficients and we use
$L_{y}=D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}.$ Note that singularities in
the lower order terms appear when either $b$ or $c$ is different from 0. When
$b=0$, then $L_{y}$ is a Bessel operator and we shall denote it by $B_{y}$.

The real numbers $\alpha_{1},\alpha_{2}$ satisfy $\alpha_{2}<2$ and
$\alpha_{2}-\alpha_{1}<2$ but are not assumed to be nonnegative. The reasons
for these restrictions will be explained later in this introduction.

$\mathcal{L}$ is the sum of a degenerate diffusion $y^{\alpha_{1}}\Delta_{x}$,
tangential to $\partial\mathbb{R}^{N+1}_{+}$, and of a 1d degenerate normal
diffusion $y^{\alpha_{2}}L_{y}$ which commute only when $\alpha_{1}=0$. It
satisfies the scaling property

$I_{s}^{-1}\mathcal{L}I_{s}=s^{2-{\alpha_{2}}}\mathcal{L},\quad
I_{s}u(x,y)=u(s^{1-\frac{(\alpha_{2}-\alpha_{1})}{2}}x,sy).$

When $\alpha_{1}=\alpha_{2}=0$,
$\mathcal{L}=\Delta_{x}+\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right)$
reduces to the so-called Caffarelli-Silvestre extension operators, studied in
detail in [21] under Dirichlet and Neumann boundary conditions. We refer the
reader also to [4], [5] for the case $b=0$ and with variable coefficients.

The case $\alpha_{1}=0$ and $\alpha_{2}$ arbitrary (even without the
restriction $\alpha_{2}<2$) can be easily reduced to that above, using the
methods of this paper and, in particular, the transformation of Section 3.

The case $\alpha_{1}=\alpha_{2}=1$ and $b=0$, namely
$\mathcal{L}=y\left(\Delta_{x}+D_{yy}\right)+cD_{y}$, is also widely treated
in the literature on degenerate problems. A comparison of our results with
those already known is done in section 7.4

When $\alpha_{2}=0$ our operators generalize the class of Baouendi-Grushin
operators $\mathcal{L}=y^{\alpha}\Delta_{x}+D_{yy}$, to which they reduce when
$c=b=0$. A comparison with known results is done in Section 7.3 and here we
only point out that we allow also negative $\alpha$.

Finally, we mention that kernel estimates for operators in divergence form and
with normal and tangential degeneracy on the hyperplane $\\{y=0\\}$ have been
obtained in [30, 31].

The aim of this paper is to provide a unified approach which allows to prove
elliptic and parabolic $L^{p}$ estimates and solvability of the associated
problems. In the language of semigroup theory, we prove that $\mathcal{L}$
generates an analytic semigroup, characterize its domain as a weighted Sobolev
space and show that it has maximal regularity, which means that both $D_{t}v$
and $\mathcal{L}v$ have the same regularity as $(D_{t}-\mathcal{L})v$.

Surprisingly enough, the case $\alpha_{1}=\alpha_{2}$ implies all other cases
by a change of variables, as described in Section 3. However this modifies the
underlying measure and the procedure works if one is able to deal with the
simpler case $\mathcal{L}=y^{\alpha}(\Delta_{x}+L_{y})$ in all the scale of
$L^{p}_{m}$ spaces, where $L^{p}_{m}=L^{p}(\mathbb{R}^{N+1}_{+};y^{m}dxdy)$. A
similar simplification holds also for the 1d operator $L_{y}$: it is
sufficient to deal in full generality with the case where $b=0$, that is when
$L_{y}$ is a Bessel operator. Finally, it is sufficient to deal only with
Neumann boundary conditions, since the case of Dirichlet boundary conditions
is again deduced by a change of variables.

The operators $\mathcal{L}$, $D_{t}-\mathcal{L}$ are studied through estimates
like

$\|y^{\alpha}\Delta_{x}u\|_{p,m}+\|y^{\alpha}B_{y}u\|_{p,m}\leq
C\|\mathcal{L}u\|_{p,m},\quad$ (1)

and

$\|D_{t}u\|_{p,m}+\|\mathcal{L}u\|_{p,m}\leq
C\|(D_{t}-\mathcal{L})u\|_{p,m},\quad$ (2)

where the $L^{p}$ norms are taken over $\mathbb{R}_{+}^{N+1}$ and on
$(0,\infty)\times\mathbb{R}_{+}^{N+1}$ respectively. This kind of estimates
are quite natural in this context but not easy to prove. Of course they imply
$\|y^{\alpha}D_{x_{i}x_{j}}u\|_{p,m}\leq C\|\mathcal{L}u\|_{p,m}$, by the
Calderón-Zygmund inequalities in the $x$-variables, and can be restated by
saying that $\mathcal{L}$ is closed on $D(y^{\alpha}\Delta_{x})\cap
D(y^{\alpha}B_{y})$ or that $y^{\alpha}\Delta_{x}\mathcal{L}^{-1}$ is bounded.
Note that the weaker inequality (1) with $\|\mathcal{L}u\|_{p,m}+\|u\|_{p,m}$
on the right hand side implies the homogeneous one, as stated, by scaling.

Let us explain how to obtain (1). Assuming that
$y^{\alpha}(\Delta_{x}u+B_{y}u)=f$ and taking the Fourier transform with
respect to $x$ (with covariable $\xi$) we obtain
$-|\xi|^{2}\hat{u}(\xi,y)+B_{y}\hat{u}(\xi,y)=y^{-\alpha}\hat{f}(\xi,y)$ and
then
$y^{\alpha}|\xi|^{2}\hat{u}(\xi,y)=-y^{\alpha}|\xi|^{2}(|\xi|^{2}-B_{y})^{-1}y^{-\alpha}\hat{f}(\xi,y)$.
Denoting by ${\cal F}$ the Fourier transform with respect to $x$ we get

$y^{\alpha}\Delta_{x}\mathcal{L}^{-1}=-{\cal
F}^{-1}\left(y^{\alpha}|\xi|^{2}(|\xi|^{2}-B_{y})^{-1}y^{-\alpha}\right){\cal
F}$

and the boundedness of $y^{\alpha}\Delta_{x}\mathcal{L}^{-1}$ is equivalent to
that of the multiplier $\xi\in\mathbb{R}^{N}\to
M(\xi)=y^{\alpha}|\xi|^{2}(|\xi|^{2}-B_{y})^{-1}y^{-\alpha}$ in
$L^{p}(\mathbb{R}^{N};L^{p}_{m}(0,\infty))=L^{p}_{m}(\mathbb{R}_{+}^{N+1})$.

We prove this by a vector valued Mikhlin multiplier theorem which rests on
square function estimates for the family $M(\xi)$ and its derivatives. The
strategy for proving (2) is similar after taking the Fourier transform with
respect to $t$.

Both the elliptic and parabolic estimates above share the name “maximal
regularity” even though this term is often restricted to the parabolic case.
We refer to [15] and the new books [12], [13] for the functional analytic
approach to maximal regularity we use. The whole theory relies on a deep
interplay between harmonic analysis and structure theory of Banach spaces but
largely simplifies when the underlying Banach spaces are $L^{p}$ spaces, by
the use of classical square function estimates. This last approach has been
employed extensively in [3], showing that uniformly parabolic operators have
maximal regularity, under very general boundary conditions.

However the a-priori estimates (1) and (2) are not sufficient for the
solvability of the equation $\lambda u-\mathcal{L}u=f$. In fact, $\mathcal{L}$
is not dissipative unless additional restrictions on the parameters and on the
underlying measure are assumed, see Section 9.1, and approximation methods
with uniformly parabolic operators do not need to converge.

In order to prove existence results, or generation results in the language of
semigroups, we use that the operator valued map $\xi\in\mathbb{R}^{N}\to
N(\xi)=(\lambda+y^{\alpha}|\xi|^{2}-y^{\alpha}B_{y})^{-1}$,
$\lambda\in\mathbb{C}_{+}$, is a Fourier multiplier in
$L^{p}(\mathbb{R}^{N};L^{p}_{m}(0,\infty))=L^{p}_{m}(\mathbb{R}_{+}^{N+1})$,
see [23, Section 8] where the relevant one dimensional degenerate operators
are studied in detail.

Before describing the content of the sections, let us explain the meaning of
the restrictions $\alpha_{2}<2$, $\alpha_{2}-\alpha_{1}<2$.

Let us first consider the case where $\alpha_{1}=\alpha_{2}=\alpha$, so that
the unique requirement is $\alpha<2$. It turns out that when $\alpha\geq 2$
the problem is easily treated in the strip $\mathbb{R}^{N}\times[0,1]$ in the
case of the Lebesgue measure, see [7], and all problems are due to the strong
diffusion at infinity. The case $\alpha\geq 2$ in the strip
$\mathbb{R}^{N}\times[1,\infty[$ requires therefore new investigation even
though the operator $y^{\alpha}L_{y}$ alone can be treated for any
$\alpha\in\mathbb{R}$, by the similarity transformation of Section 3.

When $\alpha_{1}\neq\alpha_{2}$, the change of variables of Section 3, namely
$T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}$, transforms
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}$ into
$y^{\alpha}(\Delta_{x}+\tilde{B}_{y})$,
$\alpha=\frac{2\alpha_{1}}{\alpha_{1}-\alpha_{2}+2}$. However, the strip
$\mathbb{R}^{N}\times[0,1]$ is mapped into itself only when
$\alpha_{2}-\alpha_{1}<2$. Under this condition it is possible, though not
treated in this paper, that the restriction $\alpha_{2}<2$ can be removed, at
least when the operator is studied in $\mathbb{R}^{N}\times[0,1]$ rather than
in $\mathbb{R}^{N+1}_{+}$. But dealing with the case
$\alpha_{2}-\alpha_{1}\geq 2$ requires further investigation, as explained
above.

Assuming, in addition, that $\alpha_{1}\geq 0$, the range of parameters for
which we prove solvability is optimal, since it coincides with that of
$L_{y}$. However, when $\alpha_{1}<0$ it can happen that $\mathcal{L}$
generates in a range of parameters for which the domain is less regular. We
discuss these phenomena in Section 9.2, see in particular Example 9.10,
without demanding for completeness.

Most of the result of this paper can be extended to operators with variable
coefficients

$\mathcal{L}=y^{\alpha_{1}}\sum_{i,j=1}^{N}a_{ij}(t,x,y)D_{x_{i}x_{j}}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right)$

assuming uniform ellipticity and appropriate continuity of the matrix
$(a_{ij})$. In fact, the case of constant coefficients $(a_{ij})$ follows by a
linear change of the $x$-variables and allows to use perturbation methods. The
situation is easier in a finite strip $\mathbb{R}^{N}\times[0,1]$ and for
positive $\alpha_{1},\alpha_{2}$ since the powers
$y^{\alpha_{1}},y^{\alpha_{2}}$ are bounded. First order terms like
$y^{\frac{\alpha_{1}}{2}}b(t,x,y)\cdot\nabla_{x}$ with $b$ bounded can be also
added by perturbation. We shall deal with these consequences in a subsequent
paper in order to treat operators which degenerate near the boundary of a
domain with (possibly) different rates along normal and tangential directions.

The paper is organized as follows. In Section 2 we briefly recall the harmonic
analysis background needed in the paper, as square function estimates,
$\mathcal{R}$-boundedness and a vector valued multiplier theorem.

In Section 3, we exploit an elementary change of variables, in a functional
analytic setting, to reduce our operators to the simpler case where
$\alpha_{1}=\alpha_{2}$.

In Sections 4 and 5 we recall some preliminary results concerning anisotropic
weighted Sobolev spaces and one-dimensional Bessel operators.

In Section 6, which is the core of the paper, we prove generation results,
maximal regularity and domain characterization for the operator
$y^{\alpha}\Delta_{x}+y^{\alpha}B_{y}^{n}$ where $B_{y}^{n}$ is the Bessel
operator with Neumann boundary conditions. The general case both with Neumann
and Dirichlet boundary conditions will be deduced in Sections 7 and 8 using
the isometry of Section 3.

In Section 9, we complement our results by characterizing the contractivity
range and investigating uniqueness. We show that many examples of degenerate
operators, like the Baouendi-Grushin operators, are special cases of ours and
that our results improve those already existing in the literature.

Notation. For $N\geq 0$,
$\mathbb{R}^{N+1}_{+}=\\{(x,y):x\in\mathbb{R}^{N},y>0\\}$. For
$m\in\mathbb{R}$ we consider the measure $y^{m}dxdy$ in $\mathbb{R}^{N+1}_{+}$
and we write $L^{p}_{m}(\mathbb{R}_{+}^{N+1})$ for
$L^{p}(\mathbb{R}_{+}^{N+1};y^{m}dxdy)$ and often only $L^{p}_{m}$ when
$\mathbb{R}^{N+1}_{+}$ is understood. Similarly
$W^{k,p}_{m}(\mathbb{R}^{N+1}_{+})=\\{u\in
L^{p}_{m}(\mathbb{R}^{N+1}_{+}):\partial^{\alpha}u\in
L^{p}_{m}(\mathbb{R}^{N+1}_{+})\quad|\alpha|\leq k\\}$.

$\mathbb{C}^{+}=\\{\lambda\in\mathbb{C}:\textrm{\emph{Re}\,}\lambda>0\\}$ and,
for $|\theta|\leq\pi$, we denote by $\Sigma_{\theta}$ the open sector
$\\{\lambda\in\mathbb{C}:\lambda\neq 0,\ |Arg(\lambda)|<\theta\\}$.

##  2 Vector-valued harmonic analysis

Regularity properties for
$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}$ follow once we
prove the estimate

$\|y^{\alpha_{1}}\Delta_{x}u\|_{p}+\|y^{\alpha_{2}}L_{y}u\|_{p}\leq
C\|\mathcal{L}u\|_{p}$ (3)

where the $L^{p}$ norms are taken over $\mathbb{R}^{N+1}_{+}$ on a
sufficiently large set of functions $u$. This is equivalent to saying that the
domain of $\mathcal{L}$ is the intersection of the domain of
$y^{\alpha_{1}}\Delta_{x}$ and $y^{\alpha_{2}}L_{y}$ (after appropriate
tensorization) or that the operator
$y^{\alpha_{1}}\Delta_{x}{\mathcal{L}}^{-1}$ is bounded. This strategy arose
first in the study of maximal regularity of parabolic problems, that is for
the equation $u_{t}=Au+f,u(0)=0$ where $A$ is the generator of an analytic
semigroup on a Banach space $X$. Estimates like

$\|u_{t}\|_{p}+\|Au\|_{p}\leq\|f\|_{p}$

where now the $L^{p}$ norm is that of $L^{p}([0,T[;X)$ can be interpreted as
closedness of $D_{t}-A$ on the intersection of the respective domains or,
equivalently, boundedness of the operator $A(D_{t}-A)^{-1}$ in
$L^{p}([0,T[;X)$.

Nowadays this strategy is well established and relies on Mikhlin vector-valued
multiplier theorems. Let us state the relevant definitions and main results we
need, referring the reader to [3], [29] or [15].

Let ${\cal S}$ be a subset of $B(X)$, the space of all bounded linear
operators on a Banach space $X$. ${\cal S}$ is $\mathcal{R}$-bounded if there
is a constant $C$ such that

$\|\sum_{i}\varepsilon_{i}S_{i}x_{i}\|_{L^{p}(\Omega;X)}\leq
C\|\sum_{i}\varepsilon_{i}x_{i}\|_{L^{p}(\Omega;X)}$

for every finite sum as above, where $(x_{i})\subset X,(S_{i})\subset{\cal S}$
and $\varepsilon_{i}:\Omega\to\\{-1,1\\}$ are independent and symmetric random
variables on a probability space $\Omega$. The smallest constant $C$ for which
the above definition holds is the $\mathcal{R}$-bound of $\mathcal{S}$,
denoted by $\mathcal{R}(\mathcal{S})$. It is well-known that this definition
does not depend on $1\leq p<\infty$ (however, the constant
$\mathcal{R}(\mathcal{S})$ does) and that $\mathcal{R}$-boundedness is
equivalent to boundedness when $X$ is an Hilbert space. When $X$ is an $L^{p}$
space (with respect to any $\sigma$-finite measure), testing
$\mathcal{R}$-boundedness is equivalent to proving square functions estimates,
see [15, Remark 2.9 ].

###### Proposition 2.1

Let ${\cal S}\subset B(L^{p}(\Sigma))$, $1<p<\infty$. Then ${\cal S}$ is
$\mathcal{R}$-bounded if and only if there is a constant $C>0$ such that for
every finite family $(f_{i})\in L^{p}(\Sigma),(S_{i})\in{\cal S}$

$\left\|\left(\sum_{i}|S_{i}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}.$

The best constant $C$ for which the above square functions estimates hold
satisfies $\kappa^{-1}C\leq\mathcal{R}(\mathcal{S})\leq\kappa C$ for a
suitable $\kappa>0$ (depending only on $p$). The proposition above
$\mathcal{R}$-boundedness follows from domination.

###### Corollary 2.2

Let ${\cal S},{\cal T}\subset B(L^{p}(\Sigma))$, $1<p<\infty$ and assume that
$\cal T$ is $\mathcal{R}$ bounded and that for every $S\in\cal S$ there exists
$T\in\cal T$ such that $|Sf|\leq|Tf|$ pointwise, for every $f\in
L^{p}(\Sigma)$. Then ${\cal S}$ is $\mathcal{R}$-bounded.

Let $(A,D(A))$ be a sectorial operator in a Banach space $X$; this means that
$\rho(-A)\supset\Sigma_{\pi-\phi}$ for some $\phi<\pi$ and that
$\lambda(\lambda+A)^{-1}$ is bounded in $\Sigma_{\pi-\phi}$. The infimum of
all such $\phi$ is called the spectral angle of $A$ and denoted by $\phi_{A}$.
Note that $-A$ generates an analytic semigroup if and only if
$\phi_{A}<\pi/2$. The definition of $\mathcal{R}$-sectorial operator is
similar, substituting boundedness of $\lambda(\lambda+A)^{-1}$ with
$\mathcal{R}$-boundedness in $\Sigma_{\pi-\phi}$. As above one denotes by
$\phi^{R}_{A}$ the infimum of all $\phi$ for which this happens; since
$\mathcal{R}$-boundedness implies boundedness, we have
$\phi_{A}\leq\phi^{R}_{A}$.

The $\mathcal{R}$-boundedness of the resolvent characterizes the regularity of
the associated inhomogeneous parabolic problem, as we explain now.

An analytic semigroup $(e^{-tA})_{t\geq 0}$ on a Banach space $X$ with
generator $-A$ has maximal regularity of type $L^{q}$ ($1<q<\infty$) if for
each $f\in L^{q}([0,T];X)$ the function $t\mapsto
u(t)=\int_{0}^{t}e^{-(t-s)A})f(s)\,ds$ belongs to $W^{1,q}([0,T];X)\cap
L^{q}([0,T];D(B))$. This means that the mild solution of the evolution
equation

$u^{\prime}(t)+Au(t)=f(t),\quad t>0,\qquad u(0)=0,$

is in fact a strong solution and has the best regularity one can expect. It is
known that this property does not depend on $1<q<\infty$ and $T>0$. A
characterization of maximal regularity is available in UMD Banach spaces,
through the $\mathcal{R}$-boundedness of the resolvent in a suitable sector
$\omega+\Sigma_{\phi}$, with $\omega\in\mathbb{R}$ and $\phi>\pi/2$ or,
equivalently, of the scaled semigroup $e^{-(A+\omega^{\prime})t}$ in a sector
around the positive axis. In the case of $L^{p}$ spaces it can be restated in
the following form, see [15, Theorem 1.11]

###### Theorem 2.3

Let $(e^{-tA})_{t\geq 0}$ be a bounded analytic semigroup in $L^{p}(\Sigma)$,
$1<p<\infty$, with generator $-A$. Then $T(\cdot)$ has maximal regularity of
type $L^{q}$ if and only if the set
$\\{\lambda(\lambda+A)^{-1},\lambda\in\Sigma_{\pi/2+\phi}\\}$ is
$\mathcal{R}$\- bounded for some $\phi>0$. In an equivalent way, if and only
if there are constants $0<\phi<\pi/2$, $C>0$ such that for every finite
sequence $(\lambda_{i})\subset\Sigma_{\pi/2+\phi}$, $(f_{i})\subset L^{p}$

$\left\|\left(\sum_{i}|\lambda_{i}(\lambda_{i}+A)^{-1}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}$

or, equivalently, there are constants $0<\phi^{\prime}<\pi/2$, $C^{\prime}>0$
such that for every finite sequence $(z_{i})\subset\Sigma_{\phi^{\prime}}$,
$(f_{i})\subset L^{p}$

$\left\|\left(\sum_{i}|e^{-z_{i}A}f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}\leq
C^{\prime}\left\|\left(\sum_{i}|f_{i}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{p}(\Sigma)}.$

Finally we state a version of the operator-valued Mikhlin multiplier theorem
in the N-dimensional case, see [3, Theorem 3.25] or [15, Theorem 4.6].

###### Theorem 2.4

Let $1<p<\infty$, $M\in C^{N}(\mathbb{R}^{N}\setminus\\{0\\};B(L^{p}(\Sigma))$
be such that the set

$\left\\{|\xi|^{|\alpha|}D^{\alpha}_{\xi}M(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\alpha|\leq N\right\\}$

is $\mathcal{R}$-bounded. Then the operator $T_{M}={\cal F}^{-1}M{\cal F}$ is
bounded in $L^{p}(\mathbb{R}^{N},L^{p}(\Sigma))$, where $\cal{F}$ denotes the
Fourier transform.

##  3 Degenerate operators and similarity transformations

We consider first the 1d operators

$L=D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}},\qquad B=D_{yy}+\frac{c}{y}D_{y}$

on the half line $\mathbb{R}_{+}=]0,\infty[$. Note that $B$ (which stands for
Bessel) is nothing but $L$ when $b=0$. Often we write $L_{y},B_{y}$ to
indicate that they act with respect to the $y$ variable.

The equation $Lu=0$ has solutions $y^{-s_{1}}$, $y^{-s_{2}}$ where
$s_{1},s_{2}$ are the roots of the indicial equation $f(s)=-s^{2}+(c-1)s+b=0$

$s_{1}:=\frac{c-1}{2}-\sqrt{D},\quad s_{2}:=\frac{c-1}{2}+\sqrt{D}$ (4)

where

$D:=b+\left(\frac{c-1}{2}\right)^{2}.$ (5)

The above numbers are real if and only if $D\geq 0$. When $D<0$ the equation
$u-Lu=f$ cannot have positive distributional solutions for certain positive
$f$, see [26]. When $b=0$, then $\sqrt{D}=|c-1|/2$ and $s_{1}=0,s_{2}=c-1$ for
$c\geq 1$ and $s_{1}=c-1,s_{2}=0$ for $c<1$.

Next we consider, for $\alpha_{1},\alpha_{2}\in\mathbb{R}$, the operators

$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}$

(keeping the assumption $D\geq 0$ on $L_{y}$) in the space
$L^{p}_{m}=L^{p}_{m}(\mathbb{R}^{N+1}_{+})$.

We investigate when these operators can be transformed one into the other by
means of change of variables and multiplications.

For $k,\beta\in\mathbb{R}$, $\beta\neq-1$ let

$\displaystyle T_{k,\beta\,}u(x,y)$
$\displaystyle:=|\beta+1|^{\frac{1}{p}}y^{k}u(x,y^{\beta+1}),\quad(x,y)\in\mathbb{R}^{N+1}_{+}.$
(6)

Observe that

$T_{k,\beta\,}^{-1}=T_{-\frac{k}{\beta+1},-\frac{\beta}{\beta+1}\,}.$

###### Proposition 3.1

Let $1\leq p\leq\infty$, $k,\beta\in\mathbb{R}$, $\beta\neq-1$. The following
properties hold.

  * (i)

For every $m\in\mathbb{R}$, $T_{k,\beta\,}$ maps isometrically
$L^{p}_{\tilde{m}}$ onto $L^{p}_{m}$ where

$\tilde{m}=\frac{m+kp-\beta}{\beta+1}.$

  * (ii)

For every $u\in W^{2,1}_{loc}\left(\mathbb{R}^{N+1}_{+}\right)$ one has

    * 1.

$y^{\alpha}T_{k,\beta\,}u=T_{k,\beta\,}(y^{\frac{\alpha}{\beta+1}}u)$, for any
$\alpha\in\mathbb{R}$;

    * 2.

$D_{x_{i}x_{j}}(T_{k,\beta\,}u)=T_{k,\beta}\left(D_{x_{i}x_{j}}u\right)$,
$D_{x_{i}}(T_{k,\beta\,}u)=T_{k,\beta}\left(D_{x_{i}}u\right)$;

    * 3.

$D_{y}T_{k,\beta\,}u=T_{k,\beta\,}\left(ky^{-\frac{1}{\beta+1}}u+(\beta+1)y^{\frac{\beta}{\beta+1}}D_{y}u\right)$,
$D_{yy}(T_{k,\beta\,}u)=T_{k,\beta\,}\Big{(}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)(2k+\beta)y^{\frac{\beta-1}{\beta+1}}D_{y}u+k(k-1)y^{-\frac{2}{\beta+1}}u\Big{)}$.

    * 4.

$D_{xy}T_{k,\beta\,}u=T_{k,\beta\,}\left(ky^{-\frac{1}{\beta+1}}D_{x}u+(\beta+1)y^{\frac{\beta}{\beta+1}}D_{xy}u\right)$

Proof. The proof of (i) follows after observing the Jacobian of
$(x,y)\mapsto(x,y^{\beta+1})$ is $|1+\beta|y^{\beta}$. To prove (ii) one can
easily observe that any $x$-derivatives commutes with $T_{k,\beta}$. Then we
compute

$\displaystyle D_{y}T_{k,\beta\,}u(x,y)=$
$\displaystyle|\beta+1|^{\frac{1}{p}}y^{k}\left(k\frac{u(x,y^{\beta+1})}{y}+(\beta+1)y^{\beta}D_{y}u(x,y^{\beta+1})\right)$
$\displaystyle=$ $\displaystyle
T_{k,\beta\,}\left(ky^{-\frac{1}{\beta+1}}u+(\beta+1)y^{\frac{\beta}{\beta+1}}D_{y}u\right)$

and similarly

$\displaystyle D_{yy}T_{k,\beta\,}u(x,y)=$ $\displaystyle
T_{k,\beta\,}\Big{(}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)(2k+\beta)y^{\frac{\beta-1}{\beta+1}}D_{y}u+k(k-1)y^{-\frac{2}{\beta+1}}u\Big{)}.$

###### Proposition 3.2

Let $T_{k,\beta\,}$ be the isometry above defined. The following properties
hold.

  * (i)

For every $u\in W^{2,1}_{loc}\left(\mathbb{R}^{N+1}_{+}\right)$ one has

$T_{k,\beta\,}^{-1}\Big{(}y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}\Big{)}T_{k,\beta\,}u=\Big{(}y^{\frac{\alpha_{1}}{\beta+1}}\Delta_{x}+(\beta+1)^{2}y^{\frac{\alpha_{2}+2\beta}{\beta+1}}\tilde{L}_{y}\Big{)}u$

where $\tilde{L}$ is the operator defined as in (1) with parameters $b,c$
replaced, respectively, by

$\displaystyle\tilde{b}$
$\displaystyle=\frac{b-k\left(c-1+k\right)}{(\beta+1)^{2}},$
$\displaystyle\tilde{c}$
$\displaystyle=\frac{c+2k+\beta\left(c+1+2k+\beta\right)}{(\beta+1)^{2}}.$ (7)

  * (ii)

The discriminant $\tilde{D}$ and the parameters $\tilde{s}_{1,2}$ of
$\tilde{L}$ defined as in (11), (4) are given by

$\displaystyle\tilde{D}$ $\displaystyle=\frac{D}{(\beta+1)^{2}},$ (8)

and

$\displaystyle\tilde{s}_{1,2}=\frac{s_{1,2}+k}{\beta+1}\quad(\beta+1>0),\qquad\tilde{s}_{1,2}$
$\displaystyle=\frac{s_{2,1}+k}{\beta+1}\quad(\beta+1<0).$ (9)

Proof. Using Proposition 3.1 we can compute

$\displaystyle{L}_{y}T_{k,\beta\,}u(x,y)$
$\displaystyle=T_{k,\beta\,}\Big{[}(\beta+1)^{2}y^{\frac{2\beta}{\beta+1}}D_{yy}u+(\beta+1)(2k+\beta)y^{\frac{\beta-1}{\beta+1}}D_{y}u+k(k-1)y^{-\frac{2}{\beta+1}}u$
$\displaystyle\hskip
38.74988pt+cky^{-\frac{2}{\beta+1}}u+c(\beta+1)y^{\frac{\beta-1}{\beta+1}}D_{y}u-by^{-\frac{2}{\beta+1}}u\Big{]}$
$\displaystyle=T_{k,\beta\,}\Bigg{[}y^{\frac{2\beta}{\beta+1}}\Bigg{(}(\beta+1)^{2}D_{yy}u+\frac{(\beta+1)\left(2k+\beta+c\right)}{y}D_{y}u\Bigg{.}$
$\displaystyle\Bigg{.}\hskip
64.58313pt-\Big{(}b-k(c+k-1)\Big{)}\frac{u}{y^{2}}\Bigg{)}\Bigg{]}=T_{k,\beta\,}\left(y^{\frac{2\beta}{\beta+1}}\tilde{L_{y}}u\right)$

which implies

$\displaystyle
T_{k,\beta\,}^{-1}\left(y^{\alpha_{2}}{L}_{y}\right)T_{k,\beta\,}u$
$\displaystyle=y^{\frac{\alpha_{2}+2\beta}{\beta+1}}\tilde{L_{y}}u.$

Similarly one has
$y^{\alpha_{1}}\Delta_{x}T_{k,\beta\,}u=T_{k,\beta\,}\left(y^{\frac{\alpha_{1}}{\beta+1}}\Delta_{x}u\right)$.
Adding the last equalities yields (i). The remaining properties follow
directly from the definitions (4), (11).

##  4 Weighted Sobolev spaces

Let $p>1$, $m,\alpha_{1}\in\mathbb{R}$, $\alpha_{2}<2$. In order to describe
the domain of the operator $y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}$, we
collect in this section the main results concerning anisotropic weighted
Sobolev spaces, referring to [22] for further details and all the relative
proofs. We define the Sobolev space

$\displaystyle W^{2,p}(\alpha_{1},\alpha_{2},m)$ $\displaystyle=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):\ u,\ y^{\alpha_{1}}D_{x_{i}x_{j}}u,\
y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u,y^{\alpha_{2}}D_{yy}u,\
y^{\frac{\alpha_{2}}{2}}D_{y}u\in L^{p}_{m}\right\\}$

which is a Banach space equipped with the norm

$\displaystyle\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}=$
$\displaystyle\|u\|_{L^{p}_{m}}+\sum_{i,j=1}^{n}\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\sum_{i=1}^{n}\|y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u\|_{L^{p}_{m}}+\|y^{\alpha_{2}}D_{yy}u\|_{L^{p}_{m}}+\|y^{\frac{\alpha_{2}}{2}}D_{y}u\|_{L^{p}_{m}}.$

Next we add a Neumann boundary condition for $y=0$ in the form
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$ and set

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):\ y^{\alpha_{2}-1}D_{y}u\ \in L^{p}_{m}\\}$

with the norm

$\|u\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-1}D_{y}u\|_{L^{p}_{m}}.$

We consider also an integral version of the Dirichlet boundary condition,
namely a weighted summability requirement for $y^{-2}u$ and introduce

$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):y^{\alpha_{2}-2}u\in L^{p}_{m}\\}$

with the norm

$\|u\|_{W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)}=\|u\|_{W^{2,p}(\alpha_{1},\alpha_{2},m)}+\|y^{\alpha_{2}-2}u\|_{L^{p}_{m}}.$

The symbol $\mathcal{R}$ stands for ”Rellich”, since Rellich inequalities
concern with the summability of $y^{-2}u$.

We consider only the case $\alpha_{2}<2$. Analogous results can be recovered
for $\alpha_{2}>2$ via the similarity transformation of Lemma 4.3.

We have made the choice not to include the mixed derivatives in the definition
of $W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$ to simplify
some arguments. However the following result follows from Theorem 7.1.

###### Proposition 4.1

If $\alpha_{2}-\alpha_{1}<2$ and $\alpha_{1}^{-}<\frac{m+1}{p}$ then for every
$u\in W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$

$\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|u\|_{W^{2,p}_{\cal N}(\alpha_{1},\alpha_{2},m)}.$

###### Remark 4.2

With obvious changes we consider also the analogous Sobolev spaces
$W^{2,p}(\alpha_{2},m)$ and $W^{2,p}_{\cal N}(\alpha_{2},m)$ on
$\mathbb{R}_{+}$. For example we have

$W^{2,p}_{\mathcal{N}}(\alpha,m)=\left\\{u\in W^{2,p}_{loc}(\mathbb{R}_{+}):\
u,\ y^{\alpha}D_{yy}u,\ y^{\frac{\alpha}{2}}D_{y}u,\ y^{\alpha-1}D_{y}u\in
L^{p}_{m}\right\\}.$

All the results of this section will be valid also in $\mathbb{R}_{+}$
changing (when it appears) the condition $\alpha_{1}^{-}<\frac{m+1}{p}$ to
$0<\frac{m+1}{p}$.

The next proposition shows how these spaces transform under the map of Section
3.

###### Proposition 4.3

Let $p>1$, $m,\alpha_{1},\alpha_{2}\in\mathbb{R}$ with $\alpha_{2}<2$. Then
one has

$\displaystyle
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=T_{0,\beta}\left(W^{2,p}_{\mathcal{N}}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})\right),\qquad\tilde{\alpha}_{1}=\frac{\alpha_{1}}{\beta+1},\quad\tilde{\alpha}_{2}=\frac{\alpha_{2}+2\beta}{\beta+1}.$

In particular, by choosing $\beta=-\frac{\alpha_{2}}{2}$ one has

$\displaystyle
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=T_{0,-\frac{\alpha_{2}}{2}}\left(W^{2,p}_{\mathcal{N}}(\tilde{\alpha},0,\tilde{m})\right),\qquad\tilde{\alpha}=\frac{2\alpha_{1}}{2-\alpha_{2}},\quad\tilde{m}=\frac{m+\frac{\alpha_{2}}{2}}{1-\frac{\alpha_{2}}{2}}.$

###### Remark 4.4

It is essential to deal with $W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$:
in general the map $T_{0,\beta}$ does not transform
$W^{2,p}(\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{m})$ into
$W^{2,p}(\alpha_{1},\alpha_{2},m)$.

The next result clarifies in which sense the condition
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}$ is a Neumann boundary condition.

###### Proposition 4.5

The following assertions hold.

  * (i)

If $\frac{m+1}{p}>1-\alpha_{2}$, then
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}(\alpha_{1},\alpha_{2},m)$.

  * (ii)

If $\frac{m+1}{p}<1-\alpha_{2}$, then

$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=\\{u\in
W^{2,p}(\alpha_{1},\alpha_{2},m):\lim_{y\to 0}D_{y}u(x,y)=0\ {\rm for\ a.e.\
x\in\mathbb{R}^{N}}\\}.$

In both cases (i) and (ii), the norm of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ is equivalent to that of
$W^{2,p}(\alpha_{1},\alpha_{2},m)$.

We provide an equivalent description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$, adapted to the operator
$D_{yy}+cy^{-1}D_{y}$.

###### Proposition 4.6

Let $c\in\mathbb{R}$ and $\frac{m+1}{p}<c+1-\alpha_{2}$. Then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m}\right.$ $\displaystyle\left.\hskip
43.05542pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;\;and\;\;}\lim_{y\to 0}y^{c}D_{y}u=0\right\\}$

and the norms $\|u\|_{W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)}$ and

$\|u\|_{L^{p}_{m}}+\|y^{\alpha_{1}}\Delta_{x}u\|_{L^{p}_{m}}+\|y^{\alpha_{2}}(D_{yy}u+cy^{-1}D_{y}u)\|_{L^{p}_{m}}$

are equivalent on $W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$. Finally,
when $0<\frac{m+1}{p}\leq c-1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\right\\}.$

The following equivalent description of
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$ involves a Dirichlet, rather
than Neumann, boundary condition, in a certain range of parameters.

###### Proposition 4.7

Let $c\geq 1$ and $\frac{m+1}{p}<c+1-\alpha_{2}$. The following properties
hold.

  * (i)

If $c>1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m},\right.$ $\displaystyle\left.\hskip
47.36096pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}y^{c-1}u=0\right\\}.$

  * (ii)

If $c=1$ then

$\displaystyle W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=$
$\displaystyle\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u\in L^{p}_{m},\right.$ $\displaystyle\left.\hskip
47.36096pty^{\alpha_{2}}\left(D_{yy}u+c\frac{D_{y}u}{y}\right)\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}u(x,y)\in\mathbb{C}\right\\}.$

The next results show the density of smooth functions in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$. Let

$\mathcal{C}:=\left\\{u\in
C_{c}^{\infty}\left(\mathbb{R}^{N}\times[0,\infty)\right),\ D_{y}u(x,y)=0\
{\rm for}\ y\leq\delta\ {\rm and\ some\ }\delta>0\right\\},$ (10)

its one dimensional version

$\mathcal{D}=\left\\{u\in C_{c}^{\infty}([0,\infty)),\ D_{y}u(y)=0\ {\rm for}\
y\leq\delta\ {\rm and\ some\ }\delta>0\right\\}$ (11)

and finally (finite sums below)

$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}=\left\\{u(x,y)=\sum_{i}u_{i}(x)v_{i}(y),\
u_{i}\in C_{c}^{\infty}(\mathbb{R}^{N}),\ v_{i}\in\cal
D\right\\}\subset\mathcal{C}.$

###### Theorem 4.8

If $\frac{m+1}{p}>\alpha_{1}^{-}$ then
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is dense in
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

Note that the condition $(m+1)/p>\alpha_{1}^{-}$, or $m+1>0$ and
$(m+1)/p+\alpha_{1}>0$, is necessary for the inclusion
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subset
W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

###### Proposition 4.9

Let $\frac{m+1}{p}>\alpha_{1}^{-}$. The following properties hold for any
$u\in W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

  * (i)

If $\frac{m+1}{p}>1-\frac{\alpha_{2}}{2}$ then

$\displaystyle\|y^{\frac{\alpha_{2}}{2}-1}u\|_{L^{p}_{m}}\leq
C\|y^{\frac{\alpha_{2}}{2}}D_{y}u\|_{L^{p}_{m}}.$

  * (ii)

If $\alpha_{2}-\alpha_{1}<2$ and
$\frac{m+1}{p}>1-\frac{\alpha_{1}+\alpha_{2}}{2}$,
$\frac{m+1}{p}>\alpha_{1}^{-}$ then

$\displaystyle\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}-1}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{y}\nabla_{x}u\|_{L^{p}_{m}}.$

Finally, we investigate some relationships between
$W^{2,p}(\alpha_{1},\alpha_{2},m)$,
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ and
$W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)$.

###### Proposition 4.10

The following properties hold.

  * (i)

if $u\in W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$ then
$y^{\alpha_{2}-1}D_{y}u\in L^{p}_{m}.$

  * (ii)

If $\alpha_{2}-\alpha_{1}<2$ and $\frac{m+1}{p}>2-\alpha_{2}$, then

$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)=W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}(\alpha_{1},\alpha_{2},m),$

with equivalence of the corresponding norms. In particular,
$C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ is dense in
$W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$.

We clarifies the action of the multiplication operator $T_{k,0}:u\mapsto
y^{k}u$. The following lemma is the companion of Lemma 4.3 which deals with
the transformation $T_{0,\beta}$.

###### Lemma 4.11

Let $\alpha_{2}-\alpha_{1}<2$ and $\frac{m+1}{p}>2-\alpha_{2}$. For every
$k\in\mathbb{R}$

$\displaystyle T_{k,0}:W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)\to
W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$

is an isomorphism (we shall write
$y^{k}W^{2,p}_{\mathcal{N}}(\alpha_{1},\alpha_{2},m)=W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m-kp)$).

##  5 One dimensional degenerate operators

In this section we summarize the main results proved in [23] for the one
dimensional operator $y^{\alpha}B_{y}-\mu
y^{\alpha}=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)-\mu y^{\alpha}$,
$\mu\geq 0$, in $L^{p}_{m}$. To characterize the domain for $\mu>0$, we denote
by

$D(y^{\alpha})=\left\\{u\in L^{p}_{m}:\ y^{\alpha}u\in L^{p}_{m}\right\\}$

the domain of the potential $V(y)=y^{\alpha}$ in $L^{p}_{m}$.

###### Theorem 5.1

Let $\alpha<2$, $c\in\mathbb{R}$ and $1<p<\infty$.

  * (i)

If $0<\frac{m+1}{p}<c+1-\alpha$, then the operator $y^{\alpha}B$ endowed with
domain $W^{2,p}_{\mathcal{N}}(\alpha,m)$ generates a bounded positive analytic
semigroup of angle $\pi/2$ on $L^{p}\left(\mathbb{R}_{+},y^{m}dy\right)$.

  * (ii)

If $\mu>0$ and $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ then the operator
$y^{\alpha}B-\mu y^{\alpha}$ endowed with domain
$W^{2,p}_{\mathcal{N}}(\alpha,m)\cap D(y^{\alpha})$ generates a bounded
analytic semigroup in $L^{p}_{m}$ which has maximal regularity.

In both cases the set $\mathcal{D}$ defined in (11) is a core.

We shall use $y^{\alpha}B^{n}$, $n$ stands for Neumann, for $y^{\alpha}B$ with
domain $W^{2,p}_{\mathcal{N}}(\alpha,m)$ and similarly for
$y^{\alpha}B^{n}-\mu y^{\alpha}$. Note that the condition
$\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ is equivalent to
$0<\frac{m+1}{p}<c+1-\alpha$ and $-\alpha<\frac{m+1}{p}<c+1-\alpha$. The first
guarantees that $y^{\alpha}B^{n}$ is a generator in $L^{p}_{m}$ and the second
that $B^{n}$ is a generator in $L^{p}_{m+\alpha p}$.

In the next proposition we show that the multipliers

$\displaystyle\xi\in\mathbb{R}^{N}\to N_{\lambda}(\xi)$
$\displaystyle=\lambda(\lambda-y^{\alpha}B_{y}+y^{\alpha}|\xi|^{2})^{-1},$
$\displaystyle\xi\in\mathbb{R}^{N}\to M_{\lambda}(\xi)$
$\displaystyle=|\xi|^{2}y^{\alpha}(\lambda-y^{\alpha}B_{y}+y^{\alpha}|\xi|^{2})^{-1}$

satisfy the hypothesis of Theorem 2.4. $M_{\lambda}$ is used in Section 6 to
characterize the domain of $\mathcal{L}=y^{\alpha}(\Delta_{x}+B_{y})$ whereas
$N_{\lambda}$ to prove that $\mathcal{L}=y^{\alpha}(\Delta_{x}+B_{y})$
generates an analytic semigroup.

###### Proposition 5.2

Assume that $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. Then the families

$\displaystyle\left\\{|\xi|^{|\beta|}D^{\beta}_{\xi}(M_{\lambda})(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\
|\beta|\leq N,\lambda\in\mathbb{C}^{+}\right\\},$
$\displaystyle\left\\{|\xi|^{|\beta|}D^{\beta}_{\xi}(\lambda
N_{\lambda})(\xi):\xi\in\mathbb{R}^{N}\setminus\\{0\\},\ |\beta|\leq
N,\lambda\in\mathbb{C}^{+}\right\\}$

are $\mathcal{R}$-bounded in $L^{p}_{m}$.

##  6 Domain and maximal regularity for
$y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y}$

Let $c,m\in\mathbb{R}$ and $p>1$. In this section we prove generation results,
maximal regularity and domain characterization for the degenerate operators

$\mathcal{L}:=y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y},\quad\alpha<2$

in $L^{p}_{m}\left(\mathbb{R}^{N+1}_{+}\right)$, where
$y^{\alpha}B^{n}_{y}=y^{\alpha}\left(D_{yy}+\frac{c}{y}D_{y}\right)$. We start
with the $L^{2}$ theory.

###  6.1 The operator $\mathcal{L}=y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y}$
in $L^{2}_{c-\alpha}$

We use the Sobolev spaces of Section 4 and also $H^{1}_{\alpha,c}:=\\{u\in
L^{2}_{c-\alpha}:y^{\frac{\alpha}{2}}\nabla u\in L^{2}_{c-\alpha}$ equipped
with the inner product

$\displaystyle\left\langle u,v\right\rangle_{H^{1}_{\alpha,c}}:=\left\langle
u,v\right\rangle_{L^{2}_{c-\alpha}}+\left\langle y^{\frac{\alpha}{2}}\nabla
u,y^{\frac{\alpha}{2}}\nabla v\right\rangle_{L^{2}_{c-\alpha}}.$

Let $\mathcal{L}$ be the operator defined on
$C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ by

$\displaystyle\mathcal{L}$
$\displaystyle=y^{\alpha}\Delta+cy^{\alpha-1}D_{y}=y^{-c+\alpha}{\mbox{
div}}\Big{(}y^{c}\,\nabla u\Big{)}.$

Note that $c=\alpha$ if and only if $\mathcal{L}$ is formally self-adjoint
with respect to the Lebesgue measure. $\mathcal{L}$ is associated to the non-
negative, symmetric and closed form in
$L^{2}_{c-\alpha}(\mathbb{R}^{N+1}_{+})$

$\displaystyle\mathfrak{a}(u,v)$
$\displaystyle:=\int_{\mathbb{R}^{N+1}_{+}}\langle y^{\alpha}\nabla
u,\nabla\overline{v}\rangle\,y^{c-\alpha}dx\,dy=\int_{\mathbb{R}^{N+1}_{+}}\langle\nabla
u,\nabla\overline{v}\rangle\,y^{c}dx\,dy=\int_{\mathbb{R}^{N+1}_{+}}(-\mathcal{L}u)\,\overline{v}\,y^{c-\alpha}dx\,dy,$
$\displaystyle D(\mathfrak{a})$ $\displaystyle=H^{1}_{\alpha,c}.$

Accordingly we define the operator with Neumann boundary conditions by

$\displaystyle D(\mathcal{L})$ $\displaystyle=\\{u\in H^{1}_{\alpha,c}:\exists
f\in L^{2}_{c-\alpha}\ {\rm such\ that}\
\mathfrak{a}(u,v)=\int_{\mathbb{R}^{N+1}_{+}}f\overline{v}y^{c-\alpha}\,dz\
{\rm for\ every}\ v\in H^{1}_{\alpha,c}\\},$ $\displaystyle\mathcal{L}u$
$\displaystyle=-f.$ (12)

By construction $\mathcal{L}$ is a non-positive self-adjoint operator and, if
$u\in D(\mathcal{L})$, then $u\in H^{2}_{loc}(\mathbb{R}^{N+1}_{+})$ and
$\mathcal{L}u=y^{\alpha}\Delta u+cy^{\alpha-1}D_{y}u$ by standard arguments.
$\mathcal{L}$ generates a contractive analytic semigroup
$\left\\{e^{zy^{\alpha}\mathcal{L}}:\ z\in\mathbb{C}_{+}\right\\}$ in
$L^{2}_{c-\alpha}(\mathbb{R}^{N+1}_{+})$ and our aim is to characterize its
domain.

###### Proposition 6.1

If $c+1>|\alpha|$ then the set
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$, see (11), is a core for
$\mathcal{L}$ in $L^{2}_{c-\alpha}(\mathbb{R}^{N+1}_{+})$.

Proof. We observe, preliminarily, that under the given assumptions on
$\alpha,c$, the set $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is
contained in $H^{1}_{\alpha,c}$. Moreover, integrating by parts one sees that
any $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ satisfies (6.1)
with $\mathcal{L}u=y^{\alpha}\Delta_{x}u+y^{\alpha}B_{y}u\in
L^{2}_{c-\alpha}$. This yields
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subseteq D(\mathcal{L})$.

Since $I-\mathcal{L}$ is invertible we have to show that
$(I-\mathcal{L})\left(C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\right)$
is dense in $L^{2}_{c-\alpha}$ or, equivalently, that
$\left((I-\mathcal{L})\left(C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\right)\right)^{\perp}=\left\\{0\right\\}$.
Let $v\in L^{2}_{c-\alpha}(\mathbb{R}^{N+1}_{+})$ be such that

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}\left(I-\mathcal{L}\right)f\,\bar{v}\
dx\ y^{c-\alpha}dy=0,\quad\forall f\in
C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}.$

Let us choose $f=a(x)u(y)\in\mathcal{D}$ with $a\in
C_{c}^{\infty}(\mathbb{R}^{N})$ and $u\in\mathcal{D}$. Taking the Fourier
transform with respect to $x$ we get $\hat{f}(\xi,y)=\hat{a}(\xi)u(y)$ and

$\displaystyle\int_{\mathbb{R}^{N+1}_{+}}\Big{[}u(y)+y^{\alpha}|\xi|^{2}u(y)-y^{\alpha}B_{y}u(y)\Big{]}\,\hat{a}(\xi)\
\bar{\hat{v}}(\xi,y)\ d\xi\,y^{c-\alpha}dy=0.$ (13)

Fix $\xi_{0}\in\mathbb{R}^{N}$, $r>0$ and let
$w(\xi)=\frac{1}{|B(\xi_{0},r)|}\chi_{B(\xi_{0},r)}\in L^{2}(\mathbb{R}^{N})$.
Let $(a_{n})_{n}\in C_{c}^{\infty}(\mathbb{R}^{N})$ a sequence of test
functions such that $a_{n}\to\check{w}$ in $L^{2}(\mathbb{R}^{N})$; then
$\hat{a}_{n}\to w$ in $L^{2}(\mathbb{R}^{N})$ and writing (13) with $\hat{a}$
replaced by $\hat{a}_{n}$ and letting $n\to\infty$ we obtain

$\displaystyle\frac{1}{|B(\xi_{0},r)|}\int_{B(\xi_{0},r)}d\xi\int_{0}^{\infty}\Big{[}u(y)+y^{\alpha}|\xi|^{2}u(y)-y^{\alpha}B_{y}u(y)\Big{]}\,\bar{\hat{v}}(\xi,y)\
y^{c-\alpha}dy=0.$

Letting $r\to 0$ and using the Lebesgue Differentiation theorem, we have for
a.e. $\xi_{0}\in\mathbb{R}^{N}$

$\displaystyle\int_{0}^{\infty}\Big{[}u(y)+y^{\alpha}|\xi_{0}|^{2}u(y)-y^{\alpha}B_{y}u(y)\Big{]}\,\bar{\hat{v}}(\xi_{0},y)\
y^{c-\alpha}dy=0,$

which is valid for every $u\in\mathcal{D}$. Under the given hypotheses on $c$
and $\alpha$, Theorem 5.1 implies that $\mathcal{D}$ is a core for the
operator $y^{\alpha}B_{y}^{n}-y^{\alpha}|\xi_{0}|^{2}$ in
$L^{2}_{c-\alpha}(\mathbb{R}_{+})$. The last equation then implies
$\hat{v}(\xi_{0},\cdot)=0$ for a.e. $\xi_{0}\in\mathbb{R}^{N}$ and the proof
is complete.

###### Theorem 6.2

If $c+1>|\alpha|$ then

$\displaystyle D(\mathcal{L})$
$\displaystyle=W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)$

Proof.  Observe that

$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subset
W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)\cap D(\mathcal{L})$

and that it is core for $\mathcal{L}$ by Proposition 6.1 and is dense in
$W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)$ by 4.8.

We have to show that the graph norm and that of
$W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)$ are equivalent on
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$. Since the second is
obviously stronger, we have to show the converse.

We use Proposition 4.6 and endow
$W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)$ with the equivalent norm

$\|u\|_{W}=\|u\|_{L^{2}_{c-\alpha}}+\|y^{\alpha}\Delta_{x}u\|_{L^{2}_{c-\alpha}}+\|y^{\alpha}B_{y}u\|_{L^{2}_{c-\alpha}}.$

Let $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ and
$f=u-\mathcal{L}u$, so that
$\|u\|_{L^{2}_{c-\alpha}}\leq\|f\|_{L^{2}_{c-\alpha}}$. By taking the Fourier
transform with respect to $x$ (with co-variable $\xi$) we obtain

$\displaystyle(1+|\xi|^{2}y^{\alpha}-y^{\alpha}B^{n}_{y})\hat{u}(\xi,\cdot)=\hat{f}(\xi,\cdot),\qquad
y^{\alpha}|\xi|^{2}\hat{u}(\xi,\cdot)=y^{\alpha}|\xi|^{2}(1+|\xi|^{2}y^{\alpha}-y^{\alpha}B^{n}_{y})^{-1}\hat{f}(\xi,\cdot).$
(14)

This means $y^{\alpha}\Delta_{x}u=-{\cal F}^{-1}M(\xi){\cal F}f$, where ${\cal
F}$ denotes the Fourier transform and
$M(\xi)=y^{\alpha}|\xi|^{2}(1+|\xi|^{2}y^{\alpha}-y^{\alpha}B^{n}_{y})^{-1}$.

The estimate $\|y^{\alpha}\Delta_{x}u\|_{L^{2}_{c-\alpha}}\leq
C\|f\|_{L^{2}_{c-\alpha}}$ then follows from the boundedness of the multiplier
$M$ in $L^{2}(\mathbb{R}^{N};L^{2}_{c-\alpha}(\mathbb{R}_{+}))$ which follows
from Proposition 5.2 and Theorem 2.4 and yields
$\|y^{\alpha}B_{y}u\|_{L^{2}_{c-\alpha}}\leq C\|f\|_{L^{2}_{c-\alpha}}$ by
difference.

This gives the equivalence of the graph norm and of the norm of
$W^{2,2}_{\mathcal{N}}(\alpha,\alpha,c-\alpha)$ on
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ and concludes the proof.

###  6.2 The operator $\mathcal{L}=y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y}$
in $L^{p}_{m}$

In this section we prove domain characterization and maximal regularity for
the degenerate operator

$\mathcal{L}=y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y},\quad\alpha<2$

in $L^{p}_{m}$. To avoid any misinterpretation, we often write
$\mathcal{L}_{m,p}$ to emphasize the underlying space on which the operator
acts.

We shall use extensively the set $\mathcal{D}$ defined in (11). In particular
$\mathcal{L}$ is well defined on
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ when $(m+1)/p>\alpha^{-}$.

###### Lemma 6.3

Let $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. Then for any
$\lambda\in\mathbb{C}^{+}$ the operators

$(\lambda-\mathcal{L}_{c-\alpha,2})^{-1},\quad
y^{\alpha}\Delta_{x}(\lambda-\mathcal{L}_{c-\alpha,2})^{-1},\quad
y^{\alpha}B^{n}_{y}(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}$

initially defined on $L^{p}_{m}\cap L^{2}_{c-\alpha}$ by Theorem 6.2, extend
to bounded operators on $L^{p}_{m}$ which we denote respectively by
$\mathcal{R}(\lambda)$, $y^{\alpha}\Delta_{x}\mathcal{R}(\lambda)$,
$y^{\alpha}B^{n}_{y}\mathcal{R}(\lambda)$. Moreover the family
$\left\\{\lambda\mathcal{R}(\lambda):\lambda\in\mathbb{C}^{+}\right\\}$ is
$\mathcal{R}$-bounded on $L^{p}_{m}$.

Proof. Let $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ and
$f=\lambda u-\mathcal{L}u$. By taking the Fourier transform with respect to
$x$ we obtain

$(\lambda+|\xi|^{2}y^{\alpha}-y^{\alpha}B^{n}_{y})\hat{u}(\xi,\cdot)=\hat{f}(\xi,\cdot),\qquad\hat{u}(\xi,\cdot)=(\lambda-y^{\alpha}B^{n}_{y}+|\xi|^{2}y^{\alpha})^{-1}\hat{f}(\xi,\cdot).$

This means $u={\cal F}^{-1}N_{\lambda}(\xi){\cal F}f$, where

$N_{\lambda}(\xi)=(\lambda-y^{\alpha}B^{n}_{y}+|\xi|^{2}y^{\alpha})^{-1}.$

Since $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is a core for
$\mathcal{L}_{c-\alpha,2}$ we have proved the equality

$(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}={\cal F}^{-1}N_{\lambda}(\xi){\cal
F}.$

Proposition 5.2 and Theorem 2.4 yield the boundedness of the Fourier
multiplier $N_{\lambda}$ in the space
$L^{p}\left(\mathbb{R}^{N},L^{p}_{m}(\mathbb{R}_{+})\right)=L^{p}_{m}$ and the
existence of a bounded operator $\mathcal{R}(\lambda)\in L^{p}_{m}$ which
extends $(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}$. Furthermore [29, Theorem
4.3.9] and the $\mathcal{R}$-boundedness with respect to $\lambda$ of
$N_{\lambda}(\xi)$ and its $\xi$-derivatives, see again Proposition 5.2, imply
that the family
$\left\\{\lambda\mathcal{R}(\lambda):\lambda\in\mathbb{C}^{+}\right\\}$ is
$\mathcal{R}$-bounded.

The proof for $y^{\alpha}\Delta_{x}\mathcal{R}(\lambda)$ is similar. As before
we show that, see (14) in Theorem 6.2,

$y^{\alpha}\Delta_{x}(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}=-{\cal
F}^{-1}M_{\lambda}(\xi){\cal F}$

where
$M_{\lambda}(\xi)=y^{\alpha}|\xi|^{2}(\lambda+|\xi|^{2}y^{\alpha}-y^{\alpha}B^{n}_{y})^{-1}$,
and use Proposition 5.2 for the boundedness of the multiplier $M_{\lambda}$ in
$L^{p}(\mathbb{R}^{N};L^{p}_{m}(\mathbb{R}_{+}))$.

The boundedness of $y^{\alpha}B^{n}_{y}\mathcal{R}(\lambda)$ follows then by
difference, since
$y^{\alpha}\Delta_{x}\mathcal{R}(\lambda)+y^{\alpha}B^{n}_{y}\mathcal{R}(\lambda)=\lambda\mathcal{R}(\lambda)-I$.

###### Proposition 6.4

If $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$, an extension $\mathcal{L}_{m,p}$ of
the operator $\mathcal{L}$, initially defined on
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$, generates a bounded
analytic semigroup in $L^{p}_{m}(\mathbb{R}_{+}^{N+1})$ which has maximal
regularity and it is consistent with the semigroup generated by
$\mathcal{L}_{c-\alpha,2}$ in $L^{2}_{c-\alpha}(\mathbb{R}^{N+1}_{+})$.

Proof. Let us consider the $\mathcal{R}$-bounded family of operators
$\left\\{\lambda\mathcal{R}(\lambda):\lambda\in\mathbb{C}^{+}\right\\}$
defined by Lemma 6.3. In particular it satisfies

$\displaystyle\|\lambda\mathcal{R}(\lambda)\|_{\mathcal{B}(L^{p}_{m}(\mathbb{R}_{+}^{N+1}))}\leq
C,\qquad\forall\lambda\in\mathbb{C}^{+}.$

By construction $\mathcal{R}({\lambda})$ coincides with
$(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}$ when restricted to $L^{p}_{m}\cap
L^{2}_{c-\alpha}$. Hence, by density, the family
$\left\\{\mathcal{R}(\lambda):\lambda\in\mathbb{C}^{+}\right\\}$ satisfies the
resolvent equation

$\displaystyle\mathcal{R}(\lambda)-\mathcal{R}(\mu)=(\mu-\lambda)\mathcal{R}(\lambda)\mathcal{R}(\mu),\quad\forall\lambda,\mu\in\mathbb{C}^{+}$

in $L^{p}_{m}$ and therefore it is a pseudoresolvent, see [6, Section 4.a].
Furthermore $\mbox{rg}(\mathcal{R}(\lambda))$ is dense in $L^{p}_{m}$ for
every $\lambda\in\mathbb{C}^{+}$, since it contains
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$.

Let us prove that $\mathcal{R}(\lambda)$ is injective for every
$\lambda\in\mathbb{C}^{+}$. Let $f\in L^{p}_{m}$ s.t.
$\mathcal{R}(\lambda)f=0$ for some $\lambda\in\mathbb{C}^{+}$. Since
$\mbox{Ker}(\mathcal{R}(\lambda))=\mbox{Ker}(\mathcal{R}(\mu))$ for any
$\lambda,\mu\in\mathbb{C}^{+}$, see [6, Lemma 4.5], we have
$\mathcal{R}(\lambda)f=0$ for every $\lambda>0$. Given $\epsilon>0$, let us
choose $g\in L^{p}_{m}\cap L^{2}_{c-\alpha}$ s.t.
$\|f-g\|_{L^{p}_{m}}<\epsilon$. Then

$\displaystyle\lambda R(\lambda)g=\lambda R(\lambda)(g-f),\qquad\|\lambda
R(\lambda)g\|_{L^{p}_{m}}\leq C\epsilon,\qquad\forall\lambda>0.$

Since $\lambda R(\lambda)g=\lambda(\lambda-\mathcal{L}_{c-\alpha,2})^{-1}g\to
g$ as $\lambda\to\infty$ we may suppose, up to a subsequence, that $\lambda
R(\lambda)g\to g$ a.e.. Then Fatou’s Lemma yields

$\displaystyle\|g\|_{L^{p}_{m}}\leq\liminf_{\lambda\to\infty}\|\lambda\mathcal{R}(\lambda)g\|_{L^{p}_{m}}\leq
C\epsilon$

which implies
$\|f\|_{L^{p}_{m}}\leq\|f-g\|_{L^{p}_{m}}+\|g\|_{L^{p}_{m}}\leq(1+C)\epsilon$,
hence $f=0$ which proves the injectivity of $\mathcal{R}(\lambda)$.

At this point, [6, Proposition 4.6] yields the existence of a densely defined
closed operator $\mathcal{L}_{m,p}$ such that
$\mathbb{C}^{+}\subseteq\rho(\mathcal{L}_{m,p})$ and
$\mathcal{R}(\lambda)=(\lambda-\mathcal{L}_{m,p})^{-1}$ for any
$\lambda\in\mathbb{C}^{+}$. By construction,
$(\mathcal{L}_{m,p};D(\mathcal{L}_{m,p}))$ extends
$\left(\mathcal{L},C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\right)$
and one has

$\displaystyle\|\lambda\left(\lambda-\mathcal{L}_{m,p}\right)^{-1}\|_{\mathcal{B}\left(L^{p}_{m}\right)}\leq
C,\qquad\lambda\in C^{+}.$

Then from standard results on semigroup theory, see for example [1, Section
AII, Theorem 1.14], $(\mathcal{L}_{m,p},D(\mathcal{L}_{m,p}))$ generates a
bounded analytic semigroup
$\left(e^{z\mathcal{L}_{m,p}}\right)_{z\in\Sigma_{\theta}}$ for some
$\theta>0$, in $L^{p}_{m}$.

The maximal regularity of the semigroup follows, using Theorem 2.3, from the
$\mathcal{R}$-boundedness of the resolvent family
$\\{\lambda\left(\lambda-\mathcal{L}_{m,p}\right)^{-1},\
\lambda\in\mathbb{C}^{+}\\}$. Finally, the semigroup is consistent with that
in $L^{2}_{c-\alpha}$ since the resolvents are consistent.

Finally we characterize the domain of $\mathcal{L}_{m,p}$.

###### Theorem 6.5

If $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$, then

$\displaystyle D(\mathcal{L}_{m,p})$
$\displaystyle=W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$

and in particular $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is a core
for $\mathcal{L}_{m,p}$.

Proof. With the notation of the above proposition,
$D(\mathcal{L}_{m,p})=R(1)\left(L^{p}_{m}\right)$. Let
$u=R(1)f=(I-\mathcal{L}_{c-\alpha,2})^{-1}f$ with $f\in L^{2}_{c-\alpha}\cap
L^{p}_{m}$. Then Lemma 6.3 yields

$\displaystyle\|y^{\alpha}\Delta_{x}u\|_{L^{p}_{m}}+\|y^{\alpha}B_{y}u\|_{L^{p}_{m}}\leq
C\left(\|\mathcal{L}u\|_{L^{p}_{m}}+\|u\|_{L^{p}_{m}}\right).$ (15)

Using Theorem 5.1 and Theorem 6.2, we deduce that $u(x,\cdot)\in
D(y^{\alpha}B^{n}_{c-\alpha,2})$ for a.e. $x\in\mathbb{R}^{n}$. Moreover,
$u(x,\cdot)$, $\ y^{\alpha}B_{y}u(x,\cdot)\in L^{p}_{m}(\mathbb{R}_{+})$, for
a.e. $x\in\mathbb{R}^{n}$.

Let us show that $u(x,\cdot)\in D(y^{\alpha}B^{n}_{m,p})$. In fact, setting
$f:=u(x,\cdot)-B_{y}u(x,\cdot)\in L^{p}_{m}(\mathbb{R}_{+})\cap
L^{2}_{c-\alpha}(\mathbb{R}_{+})$ we have
$u=\left(I-y^{\alpha}B^{n}\right)^{-1}f\in D(y^{\alpha}B^{n}_{m,p})\cap
D(y^{\alpha}B^{n}_{c-\alpha,2})$ by the consistency of the resolvent
$\left(I-y^{\alpha}B^{n}\right)^{-1}$ in $L^{p}_{m}(\mathbb{R}_{+})$ and in
$L^{2}_{c-\alpha}(\mathbb{R}_{+})$ .

Theorem 5.1 then implies

$\displaystyle\|y^{\alpha}D_{yy}u\|_{L^{p}_{m}(\mathbb{R}_{+})}+\|y^{\frac{\alpha}{2}}D_{y}u\|_{L^{p}_{m}(\mathbb{R}_{+})}+\|y^{\alpha_{2}}D_{y}u\|_{L^{p}_{m}(\mathbb{R}_{+})}\leq
C\|u-y^{\alpha}B_{y}u\|_{L^{p}_{m}(\mathbb{R}_{+})}.$

Then, raising to the power $p$, integrating over $\mathbb{R}^{N}$ and using
Lemma 6.3 for the last inequality

$\displaystyle\|y^{\alpha}D_{yy}u\|_{L^{p}_{m}}+\|y^{\frac{\alpha}{2}}D_{y}u\|_{L^{p}_{m}}+\|y^{\alpha-1}D_{y}u\|_{L^{p}_{m}}\leq
C\left(\|u\|_{L^{p}_{m}}+\|y^{\alpha}B_{y}u\|_{L^{p}_{m}}\right)\leq
C\left(\|u\|_{L^{p}_{m}}+\|\mathcal{L}u\|_{L^{p}_{m}}\right).$ (16)

By the density of $L^{2}_{c-\alpha}\cap L^{p}_{m}$ in $L^{p}_{m}$, (15), (16)
hold for every $u\in D(\mathcal{L}_{m,p})$ and this last is contained in
$W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$, by 4.6.

Moreover, since the graph norm is clearly weaker than the norm of
$W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$, (15), (16) again show that they are
equivalent on $D(\mathcal{L}_{m,p})$, in particular on
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ which is dense in
$W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$, by 4.8.

Therefore $D(\mathcal{L}_{m,p})=W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$ and in
particular $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is a core.

###### Corollary 6.6

Under the hypotheses of Theorem 6.5 we have for every $u\in
W^{2,p}_{\mathcal{N}}(\alpha,\alpha,m)$

$\|y^{\alpha}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\|y^{\alpha}D_{yy}u\|_{L^{p}_{m}}+\|y^{\alpha-1}D_{y}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}u\|_{L^{p}_{m}}.$

Proof. By Theorem 6.5 the above inequality holds if
$\|u\|_{L^{p}_{m}(\mathbb{R}^{N+1}_{+})}$ is added to the right hand side.
Applying it to $u_{\lambda}(x,y)=u(\lambda x,\lambda y)$, $\lambda>0$ we
obtain

$\displaystyle\|y^{\alpha}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\|y^{\alpha}D_{yy}u\|_{L^{p}_{m}}+\|y^{\alpha-1}D_{y}u\|_{L^{p}_{m}}\leq
C\left(\|\mathcal{L}u\|_{L^{p}_{m}}+\lambda^{\alpha-2}\|u\|_{L^{p}_{m}}\right)$

and the proof follows letting $\lambda\to\infty$.

###  6.3 Mixed derivatives

By using classical covering results, Rellich inequalities and Theorem 6.5, we
obtain $L^{p}$ estimates for the mixed second order derivatives.

###### Theorem 6.7

Let $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. Then there exists $C>0$ such that
for every $u\in D(\mathcal{L}_{m,p})$

$\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}u\|_{L^{p}_{m}}.$

We need a Rellich type inequality for smooth functions vanishing near
$\\{y=0\\}$.

###### Lemma 6.8

Let $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. Assume, in addition, $\alpha\neq
1-\frac{m+1}{p}$, $\alpha\neq 2-\frac{m+1}{p}$. Then there exists a positive
constant $C$ such that for $u\in
C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$ we have

$\left\|y^{\alpha-2}u\right\|_{L^{p}_{m}}\leq C\|\mathcal{L}u\|_{L^{p}_{m}}.$

Proof. Let $u\in C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$. Let
$\alpha\neq 1-\frac{m+1}{p}$, $\alpha\neq 2-\frac{m+1}{p}$. Then by [25,
Proposition 3.10] (see also [16])

$\int_{\mathbb{R}^{+}}\left|y^{\alpha-2}u\right|^{p}\,y^{m}dy\leq
C\int_{\mathbb{R}^{+}}|y^{\alpha}D_{yy}u|^{p}\,y^{m}dy.$

Integrating the previous inequality over $\mathbb{R}^{N}$ and using Corollary
6.6 we get

$\left\|y^{\alpha-2}u\right\|_{L^{p}_{m}}\leq
C\|y^{\alpha}D_{yy}u\|_{L^{p}_{m}}\leq C\|\mathcal{L}u\|_{L^{p}_{m}}.$

We first prove mixed derivatives estimates for functions with support far away
from $\\{y=0\\}$.

###### Lemma 6.9

Let $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$. Assume, in addition, $\alpha\neq
1-\frac{m+1}{p}$, $\alpha\neq 2-\frac{m+1}{p}$. Then for every $u\in
C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$

$\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}u\|_{L^{p}_{m}}.$

Proof. For every $n\in\mathbb{Z}$ let

$\displaystyle I_{n}=[2^{n},2^{n+1}[,\qquad J_{n}=[2^{n-1},2^{n+2}[$

We fix $\vartheta\in C_{c}^{\infty}(\mathbb{R})$ such that $0\leq\vartheta\leq
1$, $\vartheta(y)=1$ for $y\in[1,2]$ and $\vartheta(y)=0$ for
$y\notin[\frac{1}{2},4]$ and set
$\vartheta_{n}(y)=\vartheta\left(\frac{y}{\rho_{n}}\right)$, where
$\rho_{n}=2^{n}$.

We apply the classical $L^{p}$ estimates for elliptic operators with constant
coefficients to the function $\vartheta_{n}u$ and obtain

$\|\rho_{n}^{\alpha}D_{y}\nabla_{x}(\vartheta_{n}u)\|_{L^{p}(\mathbb{R}^{N+1}_{+})}\leq
C\|\rho_{n}^{\alpha}D_{yy}(\vartheta_{n}u)+\rho_{n}^{\alpha}\Delta_{x}(\vartheta_{n}u)\|_{L^{p}(\mathbb{R}^{N+1}_{+})}.$

Then we get

$\displaystyle\|\rho_{n}^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}(\mathbb{R}^{N}\times
I_{n})}$ $\displaystyle\quad\leq
C\Big{(}\|\rho_{n}^{\alpha}D_{yy}u+\rho_{n}^{\alpha}\Delta_{x}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n})}+\frac{1}{\rho_{n}}\|\rho_{n}^{\alpha}D_{y}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n})}$
$\displaystyle\quad+\frac{1}{\rho_{n}^{2}}\|\rho_{n}^{\alpha}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n}}\Big{)}$

Since $\frac{\rho_{n}}{2}\leq y\leq 4\rho_{n}$ if $y\in J_{n}$ then we get

$\displaystyle\|y^{\alpha+\frac{m}{p}}D_{y}\nabla_{x}u\|_{L^{p}(\mathbb{R}^{N}\times
I_{n})}$ $\displaystyle\leq
C\Big{(}\|y^{\alpha+\frac{m}{p}}D_{yy}u+y^{\alpha+\frac{m}{p}}\Delta_{x}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n})}$
$\displaystyle+\|y^{\alpha-1+\frac{m}{p}}D_{y}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n})}+\|y^{\alpha-2+\frac{m}{p}}u\|_{L^{p}(\mathbb{R}^{N}\times
J_{n})}\Big{)}.$

Summing over $n$, since at most three among the intervals $J_{n}$ overlap, it
follows that

$\displaystyle\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}$ $\displaystyle\leq
C\left(\|\mathcal{L}_{m,p}u\|_{L^{p}_{m}}+\|y^{\alpha-1}D_{y}u\|_{L^{p}_{m}}+\left\|y^{\alpha-2}u\right\|_{L^{p}_{m}}\right).$

Using Corollary 6.6 and Lemma 6.8 we conclude the proof.

Next we remove the assumption on the supports and work in
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ which is a core for
$\mathcal{L}_{m,p}$.

###### Lemma 6.10

Let $\alpha^{-}<\frac{m+1}{p}<c+1-\alpha$ and assume also that $\alpha\neq
1-\frac{m+1}{p}$, $\alpha\neq 2-\frac{m+1}{p}$. Then

$\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq C\|\mathcal{L}u\|_{L^{p}_{m}}$

for every $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$.

Proof. Given $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$, let
$v(x,y)=u(x,\lambda y)$. Then $v\in
C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ and $u(x,0)=v(x,0)$. It
follows that $w=u-v\in C_{c}^{\infty}(\mathbb{R}^{N}\times]0,\infty[)$.
Moreover

$\|y^{\alpha}D_{y}\nabla_{x}v\|_{L^{p}_{m}}=\lambda^{1-\alpha-\frac{m+1}{p}}\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}$

and, by Corollary 6.6,

$\|\mathcal{L}v\|_{L^{p}_{m}}\leq\lambda^{-\alpha-\frac{m+1}{p}}\|y^{\alpha}\Delta_{x}u\|_{L^{p}_{m}}+\lambda^{2-\alpha-\frac{m+1}{p}}\|y^{\alpha}B^{n}_{y}u\|_{L^{p}_{m}}\leq
C(\lambda)\|\mathcal{L}u\|_{L^{p}_{m}}.$

Hence by applying Lemma 6.9 to $w$, we have

$\displaystyle\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\left(\|y^{\alpha}D_{y}\nabla_{x}w\|_{L^{p}_{m}}+\|y^{\alpha}D_{y}\nabla_{x}v\|_{L^{p}_{m}}\right)\leq
C\left(\|\mathcal{L}w\|_{L^{p}_{m}}+\|y^{\alpha}D_{y}\nabla_{x}v\|_{L^{p}_{m}}\right)$
$\displaystyle\leq
C\left(\|\mathcal{L}u\|_{L^{p}_{m}}+\|\mathcal{L}v\|_{L^{p}_{m}}+\|y^{\alpha}D_{y}\nabla_{x}v\|_{L^{p}_{m}}\right)\leq
C^{\prime}(\lambda)\|\mathcal{L}u\|_{L^{p}_{m}}+C\lambda^{1-\alpha-\frac{m+1}{p}}\|y^{\alpha}D_{y}\nabla_{x}u\|_{L^{p}_{m}}.$

Choosing $\lambda$ large enough or small enough accordingly to
$1-\alpha-\frac{m+1}{p}>0$ or $1-\alpha-\frac{m+1}{p}<0$ we conclude the
proof.

Proof. (Theorem 6.7). Since $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$
is a core for $\mathcal{L}_{m,p}$, by Lemma 6.10 the claim holds for
$\alpha\neq 1-\frac{m+1}{p}$, $\alpha\neq 2-\frac{m+1}{p}$.

Suppose now $p=\frac{m+1}{1-\alpha}$ (in particular $\alpha<1$ and
$m+\alpha>0$). Observe that, by the previous part of the proof, the operator
$y^{\alpha}D_{y}\nabla_{x}(I-\mathcal{L}_{m,q})^{-1}$ is bounded in $L^{q}$
for $q<\frac{m+1}{1-\alpha}$ and for $q>\frac{m+1}{1-\alpha}$, $q$ close to
$p$ to satisfy the condition $\alpha^{-}<(m+1)/q<c+1-\alpha$. The Riesz-Thorin
interpolation theorem then yields the boundedness of
$y^{\alpha}D_{y}\nabla_{x}(I-\mathcal{L}_{m,p})^{-1}$ also for
$p=\frac{m+1}{1-\alpha}$. By arguing similarly for $p=\frac{m+1}{2-\alpha}$ we
conclude the proof.

##  7 The operator $y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B^{n}_{y}$

In this section we consider for $\alpha_{1}\in\mathbb{R}$, $\alpha_{2}<2$ the
operator

$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B^{n}_{y}$

in the space $L^{p}_{m}$. The generation and domain properties for
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ are deduced from the case
$\alpha_{1}=\alpha_{2}$ by using the isometry

$\displaystyle T_{k,\beta\,}u(x,y)$
$\displaystyle:=|\beta+1|^{\frac{1}{p}}y^{k}u(x,y^{\beta+1}),\quad(x,y)\in\mathbb{R}^{N+1}_{+}$

introduced in Section 3.

###### Theorem 7.1

Let $\alpha_{2}-\alpha_{1}<2$ and

$\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}.$

Then $\mathcal{L}^{\alpha_{1},\alpha_{2}}$ with domain
$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$
generates a bounded analytic semigroup in $L^{p}_{m}$ which has maximal
regularity. Moreover for every $u\in
W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$

$\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{y}\nabla_{x}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}u\|_{L^{p}_{m}}.$

Proof. We use the isometry

$T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}:L^{p}_{\tilde{m}}\to
L^{p}_{m},\qquad\tilde{m}=\frac{2m-\alpha_{1}+\alpha_{2}}{\alpha_{1}-\alpha_{2}+2}$

which, according to Proposition 3.2, transforms
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ into

$T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}^{-1}\mathcal{L}^{\alpha_{1},\alpha_{2}}T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}=y^{\alpha}\Delta_{x}+\left(\frac{\alpha_{1}-\alpha_{2}+2}{2}\right)^{2}y^{\alpha}\tilde{B}^{n}_{y}$

where

$\alpha=\frac{2\alpha_{1}}{\alpha_{1}-\alpha_{2}+2},\quad\tilde{B}^{n}_{y}=D_{yy}+\frac{\tilde{c}}{y}D_{y},\quad\tilde{c}=\frac{4c+(\alpha_{1}-\alpha_{2})(2c+2+\alpha_{1}-\alpha_{2})}{(\alpha_{1}-\alpha_{2}+2)^{2}}.$

Observe that $\alpha<2$ by assumption as well as
$\alpha^{-}<\frac{\tilde{m}+1}{p}<\tilde{c}+1-\alpha$. Generation properties
and maximal regularity for $\mathcal{L}^{\alpha_{1}\alpha_{2}}_{m,p}$ in
$L^{p}_{m}$ are then immediate consequence of the same properties of

$y^{\alpha}\Delta_{x}+\left(\frac{\alpha_{1}-\alpha_{2}+2}{2}\right)^{2}y^{\alpha}\tilde{B}^{n}_{y}$

in $L^{p}_{\tilde{m}}$ proved in Proposition 6.4 and Theorem 6.5. Concerning
the domain, we have

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}\left(W^{2,p}_{\mathcal{N}}\left(\alpha,\alpha,\tilde{m}\right)\right)$

which, by 4.3, coincides with
$W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m\right)$. The estimates for
the mixed derivatives follow from the equality

$y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{xy}u=\frac{2+\alpha_{1}-\alpha_{2}}{2}T_{0,\frac{\alpha_{1}-\alpha_{2}}{2}}\left(y^{\frac{2\alpha_{1}}{\alpha_{1}-\alpha_{2}+2}}D_{xy}\tilde{u}\right).$

and Theorem 6.7.

###### Remark 7.2

The operator $y^{\alpha}\Delta_{x}+ay^{\alpha}B^{n}_{y}$, $a>0$, has the same
domain and properties of $y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y}$. This
follows by using the map $Tu(x,y)=u(x,a^{-\frac{1}{2}}y)$ since
$T^{-1}\left(y^{\alpha}\Delta_{x}+ay^{\alpha}B^{n}_{y}\right)T=a^{\frac{\alpha}{2}}\left(y^{\alpha}\Delta_{x}+y^{\alpha}B^{n}_{y}\right)$.
We used this in the above proof.

##  8 Degenerate operators with Dirichlet boundary conditions

In this section we add a potential term to $B$ and study the operator

$\mathcal{L}=\mathcal{L}^{\alpha_{1},\alpha_{2}}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c}{y}D_{y}-\frac{b}{y^{2}}\right),\quad\alpha_{2}<2$

in $L^{p}_{m}$, under Dirichlet boundary conditions, in the sense specified
below.

We recall that the equation $L_{y}u=0$ has solutions $y^{-s_{1}}$,
$y^{-s_{2}}$ where $s_{1},s_{2}$ are the roots of the indicial equation
$f(s)=-s^{2}+(c-1)s+b=0$ given by

$s_{1}:=\frac{c-1}{2}-\sqrt{D},\quad s_{2}:=\frac{c-1}{2}+\sqrt{D}$

where

$D:=b+\left(\frac{c-1}{2}\right)^{2}$

is supposed to be nonnegative. When $b=0$, then $\sqrt{D}=|c-1|/2$ and
$s_{1}=0,s_{2}=c-1$ for $c\geq 1$ and $s_{1}=c-1,s_{2}=0$ for $c<1$.

###### Remark 8.1

All the results of this section will be valid, with obvious changes, also in
$\mathbb{R}_{+}$ for the $1$d operators $y^{\alpha_{2}}L_{y}$ changing (when
it appears in the various conditions on the parameters) $\alpha_{1}^{-}$ to
$0$ (see also Remark 4.2). We also refer to [2, 17, 18, 20, 27] for the
analogous results concerning the $Nd$ version of $L_{y}$.

A multiplication operator transforms $\mathcal{L}$ into an operator of the
form $y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B^{n}_{y}$ and allows to transfer
the results of the previous sections to this situation. Indeed, we use the map
defined in Section 3

$\displaystyle T_{k,0\,}u(x,y)$
$\displaystyle:=y^{k}u(x,y),\quad(x,y)\in\mathbb{R}^{N+1}_{+}$ (17)

for a suitable choice of $k$ and with $\beta=0$. We recall that $T_{k,0\,}$
maps isometrically $L^{p}_{\tilde{m}}$ onto $L^{p}_{m}$ where $\tilde{m}=m+kp$
and for every $u\in W^{2,1}_{loc}\left(\mathbb{R}^{N+1}_{+}\right)$ one has

$T_{k,0\,}^{-1}\Big{(}y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}\Big{)}T_{k,0\,}u=\Big{(}y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{L}_{y}\Big{)}u$

where $\tilde{L}$ is the operator defined as above with parameters $b,c$
replaced, respectively, by

$\displaystyle\tilde{b}$ $\displaystyle=b-k\left(c-1+k\right),$
$\displaystyle\tilde{c}$ $\displaystyle=c+2k.$ (18)

Moreover the discriminant $\tilde{D}$ and the parameters $\tilde{s}_{1,2}$ of
$\tilde{L}$ are given by

$\displaystyle\tilde{D}$ $\displaystyle=D,\quad\tilde{s}_{1,2}=s_{1,2}+k.$
(19)

Choosing $k=-s_{i}$, $i=1,2$, we get $\tilde{b}=0$, $\tilde{c}_{i}=c-2s_{i}$
and therefore

$T_{-s_{i},0}^{-1}\mathcal{L}T_{-s_{i},0}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{i}_{y}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\left(D_{yy}+\frac{c-2s_{i}}{y}D_{y}\right).$

###### Theorem 8.2

Let $\alpha_{2}-\alpha_{1}<2$ and

$s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}.$

Then $\mathcal{L}^{\alpha_{1},\alpha_{2}}$ generates a bounded analytic
semigroup in $L^{p}_{m}$ which has maximal regularity. Moreover,

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=y^{-s_{1}}W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right).$
(20)

Finally, the estimate

$\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\|y^{\alpha_{2}}L_{y}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}^{\alpha_{1},\alpha_{2}}u\|_{L^{p}_{m}}$ (21)

holds for every $u\in D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})$.

Proof. According to the discussion above the map
$T_{-s_{1},0}:L^{p}_{m-s_{1}p}\to L^{p}_{m}$ transforms
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ into
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{n}_{y}$ where
$\tilde{B}^{n}_{y}=D_{yy}+\frac{\tilde{c}}{y}D_{y},\quad\tilde{c}=c-2s_{1}$.

Since $s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}$ is equivalent to
$\alpha_{1}^{-}<\frac{m-ps_{1}+1}{p}<\tilde{c}+1-\alpha_{2}$, the statement on
generation and maximal regularity is therefore a translation to
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ and in $L^{p}_{m}$ of the results of
Section 7 for $y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{n}_{y}$ in
$L^{p}_{m-s_{1}p}$.

Also
$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=T_{-s_{1},0}\left(W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right)\right)$.
Finally, (21) holds since the similar statement holds for
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{n}_{y}$ in
$L^{p}_{m-s_{1}p}$ and

$T_{-s_{1},0}^{-1}\left(y^{\alpha_{1}}D_{x_{i}x_{j}}\right)T_{-s_{1},0}=y^{\alpha_{1}}D_{x_{i}x_{j}},\qquad
T_{-s_{1},0}^{-1}\left(y^{\alpha_{2}}L_{y}\right)T_{-s_{1},0}=y^{\alpha_{2}}\tilde{B}_{y}.$

The following corollary explains why we use the term Dirichlet boundary
conditions.

###### Corollary 8.3

Let $\alpha_{2}-\alpha_{1}<2$ and
$s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}.$

  * (i)

If $D>0$ then

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}L_{y}u\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}y^{s_{2}}u(x,y)=0\right\\}.$

  * (ii)

If $D=0$ then $s_{1}=s_{2}$ and

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}L_{y}u\in L^{p}_{m}\
\text{\;and\;}\lim_{y\to 0}y^{s_{2}}u(x,y)\in\mathbb{C}\right\\}.$

Proof. Since $\tilde{c}=c-2s_{1}=1+2\sqrt{D}\geq 1$, both points follow by the
previous theorem and 4.7.

###### Remark 8.4

Equality (20) says that $u\in D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})$ if
and only for every $i,j=1,\dots N$ all functions

$u,\ y^{\frac{\alpha_{1}}{2}}D_{x_{i}}u,\ y^{\alpha_{1}}D_{x_{i}x_{j}}u,\
y^{\alpha_{2}-1}\left(D_{y}u+s_{1}\frac{u}{y}\right),\ y^{\alpha_{2}}L_{y}u$

belong to $L^{p}_{m}$ but one cannot deduce, in general, that
$y^{\alpha_{2}-1}D_{y}u$ and $y^{\alpha_{2}}D_{yy}u$ belong to $L^{p}_{m}$, as
one can check on functions like $y^{-s_{1}}u(x)$, $u\in
C_{c}^{\infty}(\mathbb{R}^{N})$, near $y=0$. This is however possible in the
special case below.

###### Corollary 8.5

Let $\alpha_{2}-\alpha_{1}<2$ and
$s_{1}+2-\alpha_{2}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}$. Then
$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$.

Proof. Observe that $s_{1}+2-\alpha_{2}>s_{1}+\alpha_{1}^{-}$, since
$\alpha_{2}<2$, $\alpha_{2}-\alpha_{1}<2$. By Theorem 8.2 and Proposition 4.10

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=y^{-s_{1}}\left(W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right)\right)=W^{2,p}_{\mathcal{R}}\left(\alpha_{1},\alpha_{2},m\right)$

under the assumption $\frac{m-ps_{1}+1}{p}>2-\alpha_{2}$ which is equivalent
to $s_{1}+2-\alpha_{2}<\frac{m+1}{p}$.

Concerning the mixed derivatives, we have the following result.

###### Corollary 8.6

Let $\alpha_{2}-\alpha_{1}<2$ and

$s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2},\quad\frac{m+1}{p}>s_{1}+1-\frac{\alpha_{1}+\alpha_{2}}{2}.$

Then

$\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}-1}D_{x_{i}}u\|_{L^{p}_{m}}+\|y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{x_{i}y}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}^{\alpha_{1},\alpha_{2}}u\|_{L^{p}_{m}}$

for every $u\in D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})$.

Proof. Let us write $u=y^{-s_{1}}v$ with $v\in
W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right)$. Then

$y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{x_{i}y}u=y^{\frac{\alpha_{1}+\alpha_{2}}{2}}\left(y^{-s_{1}}D_{x_{i}y}v-s_{1}y^{-s_{1}-1}D_{x_{i}}v\right).$

The first term on the right hand side belongs to
$L^{p}_{m}(\mathbb{R}^{N+1}_{+})$ by Theorem 7.1 and the second by Proposition
4.9, provided $\frac{m+1}{p}>s_{1}+1-\frac{\alpha_{1}+\alpha_{2}}{2}$. This
gives the estimate for $y^{\frac{\alpha_{1}+\alpha_{2}}{2}}D_{x_{i}y}u$. That
for $y^{\frac{\alpha_{1}+\alpha_{2}}{2}-1}D_{x_{i}}u$ follows similarly, using
Proposition 4.9 again.

Observe that the condition
$\frac{m+1}{p}>s_{1}+1-\frac{\alpha_{1}+\alpha_{2}}{2}$ in the previous
corollary is necessary for the integrability of the mixed derivatives of
functions like $y^{-s_{1}}u(x)$, $u\in C_{c}^{\infty}(\mathbb{R}^{N})$, near
$y=0$.

###### Corollary 8.7

Let $\alpha_{2}-\alpha_{1}<2$ and

$s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2},\quad\frac{m+1}{p}>s_{1}+1-\frac{\alpha_{2}}{2}.$

Then

$\|y^{\frac{\alpha_{2}}{2}}D_{y}u\|_{L^{p}_{m}}\leq
C\left(\|u\|_{L^{p}_{m}}+\|\mathcal{L}^{\alpha_{1},\alpha_{2}}u\|_{L^{p}_{m}}\right).$

Proof.  Let us write $u=y^{-s_{1}}v$ with $v\in
W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right)$. Then

$y^{\frac{\alpha_{2}}{2}}D_{y}u=y^{\frac{\alpha_{2}}{2}}\left(y^{-s_{1}}D_{y}v-s_{1}y^{-s_{1}-1}v\right)$

and the thesis follows from Proposition 4.9 (i).

The above results apply also to the operator
$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}$,
$B_{y}=D_{yy}+\frac{c}{y}D_{y}$, when $c<1$, so that $s_{1}=c-1\neq 0$, and
allow to construct a realization of $\mathcal{L}$ different from that of
Theorem 7.1.

###### Corollary 8.8

Let $\alpha_{2}-\alpha_{1}<2$, $c<1$ and and
$c-1+\alpha_{1}^{-}<\frac{m+1}{p}<2-\alpha_{2}.$ Then
$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}$ with domain

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=\left\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}B_{y}u\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}u(x,y)=0\right\\}.$

generates a bounded analytic semigroup in $L^{p}_{m}$ which has maximal
regularity.

Proof.  This follows from Corollary 8.3 (i), since $s_{1}=c-1$ and $s_{2}=0$.

Note that the generation interval
$c-1+\alpha_{1}^{-}<\frac{m+1}{p}<2-\alpha_{2}$ under Dirichlet boundary
conditions, is larger than $\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}$ given
by Theorem 7.1 for Neumann boundary conditions.

Let us explain what happens in Theorem 8.2 if we choose the second root
$s_{2}$ instead of $s_{1}$. Proceeding similarly, one proves an identical
result under the condition

$s_{2}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{1}+2-\alpha_{2}.$ (22)

However this requires the assumption $s_{2}<s_{1}+2-\alpha_{2}$ which is not
always satisfied. When (22) holds this procedure leads to a different
operator, as we explain in more detail in Section 9.2.

##  9 Further results, examples and applications

###  9.1 The range of contractivity

Here we investigate when the semigroups generated by our operators are
contractive on the positive real axis.

Let $\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}^{n}$ with
$\alpha_{2}<2$, $\alpha_{2}-\alpha_{1}<2$ and
$\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}$ so that the generation
conditions are satisfied and
$C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$ is a core.

If $I_{s}u(x,y)=u(s^{1-\frac{(\alpha_{2}-\alpha_{1})}{2}}x,sy)$, then
$I_{s}^{-1}\mathcal{L}I_{s}=s^{2-{\alpha_{2}}}\mathcal{L}$ and an estimate
$\|e^{t\mathcal{L}}\|\leq e^{\omega t}$ implies $\|e^{t\mathcal{L}}\|\leq 1$
(operator norms in $L^{p}_{m}$). Therefore quasi-contractivity is equivalent
to contractivity.

###### Lemma 9.1

The operator $\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}^{n}$ is
dissipative on $C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}\subset
L^{p}_{m}$ if and only if $y^{\alpha_{2}}B$ is dissipative on
$\mathcal{D}\subset L^{p}_{m}(\mathbb{R}_{+})$.

Proof.  For $u\in C_{c}^{\infty}(\mathbb{R}^{N})\otimes\mathcal{D}$

$\displaystyle-\int_{\mathbb{R}^{N+1}_{+}}(\mathcal{L}u)u|u|^{p-2}y^{m}\,dx\,dy$
$\displaystyle=(p-1)\int_{\mathbb{R}^{N+1}_{+}}|\nabla_{x}u|^{2}|u|^{p-2}y^{\alpha_{1}+m}\,dx\,dy$
$\displaystyle-\int_{\mathbb{R}^{N+1}_{+}}(y^{\alpha_{2}}B_{y}u)u|u|^{p-2}y^{m}\,dx\,dy$

and the dissipativity of $\mathcal{L}$ follows from that of
$y^{\alpha_{2}}B^{n}$. Conversely, assuming the dissipativity of
$\mathcal{L}$, we fix $v\in\mathcal{D}$, $0\neq\phi\in
C_{c}^{\infty}(\mathbb{R}^{N})$ and consider $u_{n}(x,y)=\phi(x/n)v(y)$.
Inserting in the above identity and letting $n\to\infty$ it follows that
$-\int_{0}^{\infty}(y^{\alpha_{2}}B_{y}u)u|u|^{p-2}y^{m}\,dy\geq 0$.

The dissipativity of $y^{\alpha_{2}}B^{n}$ will be deduced from the case
$\alpha=0$, via a change of variable.

###### Lemma 9.2

The best constant in the inequality

$\int_{0}^{\infty}u_{y}^{2}|u|^{p-2}y^{m}\ dy\geq
C\int_{0}^{\infty}|u|^{p}y^{m-2}\,dy,\quad u\in
C_{c}^{\infty}(\mathbb{R}_{+})$ (23)

is $C=\left(\frac{m-1}{p}\right)^{2}$. When $m>1$ the inequality above holds
also for every $u\in\mathcal{D}$.

Proof.  A proof that the best constant is that indicated above can be found in
[25, Proposition 8.3].

When $m>1$ and $u\in\mathcal{D}$, let $\phi$ be a smooth cut-off functions
which is equal to $0$ in $[0,1]$ and to $1$ in $[2,\infty[$. We apply the
inequality above to $u_{n}(y)=u(y)\phi(ny)$ and get

$C\int_{0}^{\infty}|u_{n}|^{p}y^{m}\,dy\leq\int_{0}^{\infty}u_{y}^{2}\phi(ny)u(y)|\phi(ny)u(y)|^{p-2}y^{m}\,dy+n^{2}\int_{\frac{1}{n}}^{\frac{2}{n}}|u|^{p}{\phi}^{2}_{y}(ny)|\phi(ny)|^{p-2}y^{m}\,dy$

and the last term tends to 0 as $n\to\infty$, since $m>1$ and
$u,\phi,\phi_{y}$ are bounded. One concludes by dominate convergence.

###### Proposition 9.3

Assume $0<\frac{m+1}{p}<c+1$. The operator $B_{p}^{n}$ is dissipative in
$L^{p}_{m}(\mathbb{R}_{+})$ if and only if

  * (i)

$m=c$ or

  * (ii)

$m\geq 1$ and $\frac{m-1}{p}\leq c-1$.

Proof.  For $u\in\mathcal{D}$, $u$ constant in $[0,a]$ we have integrating by
parts

$\displaystyle-\int_{0}^{\infty}(Bu)u|u|^{p-2}y^{m}\,dy$
$\displaystyle=(p-1)\int_{a}^{\infty}u_{y}^{2}|u|^{p-2}y^{m}\,dy+\frac{(1-m)(m-c)}{p}\int_{a}^{\infty}|u|^{p}y^{m-2}\,dy$
$\displaystyle-\frac{m-c}{p}|u(a)|^{p}a^{m-1}$

and (i) is immediate.

The inequality

$-\int_{0}^{\infty}(Bu)u|u|^{p-2}y^{m}\,dy=(p-1)\int_{a}^{\infty}u_{y}^{2}|u|^{p-2}y^{m}\,dy+\frac{(1-m)(m-c)}{p}\int_{a}^{\infty}|u|^{p}y^{m-2}\,dy\geq
0$

holds in $C_{c}^{\infty}(\mathbb{R}_{+})$ if and only if
$\frac{(m-1)(m-c)}{p(p-1)}\leq\left(\frac{m-1}{p}\right)^{2}$ by the above
lemma, which means

$\frac{m-1}{p}\left(c-1-\frac{m-1}{p}\right)\geq 0.$ (25)

Therefore, dissipativity can hold when $m\geq 1$ only if (ii) holds. On the
other hand, if $m>1$ and (ii) holds, then letting $a\to 0$ in (9.1) we obtain

$-\int_{0}^{\infty}(Bu)u|u|^{p-2}y^{m}\,dy=(p-1)\int_{0}^{\infty}u_{y}^{2}|u|^{p-2}y^{m}\,dy+\frac{(1-m)(m-c)}{p}\int_{0}^{\infty}|u|^{p}y^{m-2}\,dy$

which is nonnegative since (23) holds in $\mathcal{D}$, by Lemma 9.2.
Therefore (ii) is proved for $m>1$. If $m=1$ let us observe that (9.1)
trivially holds when $c\geq 1$.

Finally we consider the case $m\leq 1$ and show that $B^{n}$ is never
dissipative for $m<1$ and $c\neq m$ or for $m=1$ and $c\leq 1$, even though
(23) can hold on $C_{c}^{\infty}(\mathbb{R}_{+})$.

Let assume that (25) holds, or $c-1\leq(m-1)/p$, otherwise dissipativity fails
already on $C_{c}^{\infty}(\mathbb{R}_{+})$, and let $u(y)=y^{-\beta}$ for
$y\geq 1$ and constant in $[0,1]$. The function $u$ is not properly in
$\mathcal{D}$ but smoothing and cutting at infinity do not make any problem.

Assuming $(m-1)/p<\beta$ all integrals in the right hand side of (9.1)
converge and a straightforward computation shows that positivity is equivalent
to

$\beta\left((p-1)\beta-(m-c)\right)\geq 0$

for every $\beta>(m-1)/p$. However this is false for $m<1$ since the
expression above is negative between 0 and $(m-c)/(p-1)$ and
$(m-1)/p<0,(m-1)/p\leq(m-c)/(p-1)$.

When $m=1$ the inequality (25) is always verified and the positivity of (9.1)
on $y^{-\beta}$ is equivalent to

$\beta\left((p-1)\beta-(1-c)\right)\geq 0$

for $\beta>0$ which is false for small $\beta>0$ when $c<1$.

We can now state the final contractivity result.

###### Theorem 9.4

  * (i)

Assume that $\alpha_{2}-\alpha_{1}<2$ and

$\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}.$

Then the semigroup generated by
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B^{n}_{y}$ is contractive in
$L^{p}_{m}$ if and only if

$\displaystyle
m=c-\alpha_{2}\qquad\text{or}\qquad\frac{2-\alpha_{2}}{p}\leq\frac{m+1}{p}\leq
c-1+\frac{2-\alpha_{2}}{p}.$

  * (ii)

Assume that $\alpha_{2}-\alpha_{1}<2$ and

$s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}.$

Then the semigroup generated by
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}$, under Dirichlet boundary
conditions, is contractive in $L^{p}_{m}$ if and only if

$s_{1}+\frac{2-\alpha_{2}}{p}\leq\frac{m+1}{p}\leq
s_{2}+\frac{2-\alpha_{2}}{p}.$

Proof. Concerning (i), observe that by Lemma 9.1 it is enough to consider
$y^{\alpha_{2}}B^{n}_{y}$. According to Proposition 3.2, we use the isometry

$T_{0,-\frac{\alpha_{2}}{2}}:L^{p}_{\tilde{m}}\left(\mathbb{R}_{+}\right)\to
L^{p}_{m}\left(\mathbb{R}_{+}\right),\quad
T_{0,-\frac{\alpha_{2}}{2}}u(y)=\left|1-\frac{\alpha_{2}}{2}\right|^{\frac{1}{p}}u(y^{1-\frac{\alpha_{2}}{2}}),$

$\tilde{m}=\frac{m+\frac{\alpha_{2}}{2}}{1-\frac{\alpha_{2}}{2}}$, under whose
action $y^{\alpha}_{2}B_{y}^{n}$ becomes isometrically equivalent to
$\left(1-\frac{\alpha_{2}}{2}\right)^{2}\tilde{B}$ where
$\tilde{B}=D_{yy}+\frac{\tilde{c}}{y}D_{y}$ and
$\tilde{c}=\frac{c-\frac{\alpha_{2}}{2}}{1-\frac{\alpha_{2}}{2}}$.

The dissipativity for $y^{\alpha_{2}}B$ in $L^{p}_{m}$ is then immediate
consequence of that of $\tilde{B}$ in $L^{p}_{\tilde{m}}$ already proved in
Proposition 9.3.

Concerning (ii), observe that, as in the previous Section, the map
$T_{-s_{1},0}:L^{p}_{m-s_{1}p}\to L^{p}_{m}$ transforms
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ into
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{n}_{y}$ where
$\tilde{B}^{n}_{y}=D_{yy}+\frac{\tilde{c}}{y}D_{y},\quad\tilde{c}=c-2s_{1}$.
Therefore the dissipativity of $\mathcal{L}^{\alpha_{1},\alpha_{2}}$ in
$L^{p}_{m}$ follows from that of
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}^{n}_{y}$ in
$L^{p}_{m-s_{1}p}$ proved in (i). We have that
$\mathcal{L}^{\alpha_{1},\alpha_{2}}$ is dissipative in $L^{p}_{m}$ if and
only if $m-ps_{1}=c-2s_{1}-\alpha_{2}$ or $m-ps_{1}\geq 1-\alpha_{2}$ and
$\frac{m-ps_{1}-1+\alpha_{2}}{p}\leq c-2s_{1}-1$. The claim follows since
$m-ps_{1}\geq 1-\alpha_{2}$ and $\frac{m-ps_{1}-1+\alpha_{2}}{p}\leq
c-2s_{1}-1$ are equivalent respectively to $\frac{m+1}{p}\geq
s_{1}+\frac{2-\alpha_{2}}{p}$ and $\frac{m+1}{p}\leq
s_{2}+\frac{2-\alpha_{2}}{p}$ and after observing that
$m-ps_{1}=c-2s_{1}-\alpha_{2}$ is equivalent to
$\frac{m+1}{p}=s_{1}+\frac{2-\alpha_{2}}{p}+\frac{s_{2}-s_{1}}{p}$ and
obviously
$s_{1}+\frac{2-\alpha_{2}}{p}<s_{1}+\frac{2-\alpha_{2}}{p}+\frac{s_{2}-s_{1}}{p}<s_{2}+\frac{2-\alpha_{2}}{p}$.

###  9.2 Further generation results and uniqueness

Let $\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}$,
$\alpha_{2}<2$, and keep the notation of Section 8, in particular
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}$ is the operator constructed
therein. Let us define the maximal operator $\mathcal{L}^{max}_{m,p}$ as
$\mathcal{L}$ on the maximal domain

$D(\mathcal{L}^{max}_{m,p})=\\{u\in L^{p}_{m}\cap
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):\mathcal{L}u\in L^{p}_{m}\\}$ (26)

and the minimal operator $\mathcal{L}^{min}_{m,p}$ as the closure of
$\mathcal{L}$ initially defined on $C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$. By
local elliptic regularity $\mathcal{L}^{max}_{m,p}$ is closed and then, since
$(\mathcal{L},C_{c}^{\infty}(\mathbb{R}^{N+1}_{+}))$ admits the closed
extension $\mathcal{L}^{max}_{m,p}$, its closure is well defined. Clearly
$\mathcal{L}^{min}_{m,p}\subset\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}\subset\mathcal{L}^{max}_{m,p}$.

Integrating by parts one sees that the formal adjoint of $\mathcal{L}$ is the
operator $\mathcal{L}^{*}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L^{*}_{y}$ in
$L^{p^{\prime}}_{m}$ where

$L^{*}_{y}=D_{yy}+\frac{\tilde{c}}{y}D_{y}-\frac{\tilde{b}}{y^{2}},\quad\tilde{c}=2\alpha_{2}+2m-c,\quad\tilde{b}=b-(\alpha_{2}+m-c)(\alpha_{2}+m-1).$

Moreover, the characteristic numbers of $\mathcal{L}^{*}$ are given by

$D^{*}=D,\quad s_{1}^{*}=\alpha_{2}+m-1-s_{2},\quad
s_{2}^{*}=\alpha_{2}+m-1-s_{1}.$

###### Lemma 9.5

The dual of $\mathcal{L}^{min}_{m,p}$ is $\mathcal{L}^{*,max}_{m,p^{\prime}}$
and the dual of $\mathcal{L}^{max}_{m,p}$ is
$\mathcal{L}^{*,min}_{m,p^{\prime}}$.

Proof. Since $L^{p}_{m}$ is reflexive, it is sufficient to prove the first
equality. The second follows by duality from the first, changing $p$ with
$p^{\prime}$. If $u\in C_{c}^{\infty}(\mathbb{R}^{N+1}_{+})$ and $v\in
D(\mathcal{L}^{*,max}_{m,p^{\prime}})$ one can integrate by parts and get

$\int_{\mathbb{R}^{N+1}_{+}}v(\mathcal{L}u)\,y^{m}\,dx\,dy=\int_{\mathbb{R}^{N+1}_{+}}u(\mathcal{L}^{*}v)\,y^{m}\,dx\,dy$

and hence $\mathcal{L}^{*,max}_{m,p^{\prime}}$ is a restriction of the dual of
$\mathcal{L}^{min}_{m,p}$. Conversely, if $v\in L^{p^{\prime}}_{m}$ and

$\int_{\mathbb{R}^{N+1}_{+}}v(\mathcal{L}u)\,y^{m}\,dx\,dy=\int_{\mathbb{R}^{N+1}_{+}}ufy^{m}\,dx\,dy$

for some $f\in L^{p^{\prime}}_{m}$, then by local elliptic regularity (the
coefficients of $\mathcal{L}$ are smooth in the interior of
$\mathbb{R}^{N+1}_{+}$), $v\in W^{2,p^{\prime}}_{loc}(\mathbb{R}^{N+1}_{+})$
and ${\mathcal{L}}^{*}v=f\in L^{p^{\prime}}_{m}$.

###### Proposition 9.6

Let $\alpha_{2}-\alpha_{1}<2$ and
$s_{1}+2-\alpha_{2}\leq\frac{m+1}{p}<s_{2}+2-\alpha_{2}$. Then
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}=\mathcal{L}^{min}_{m,p}$.

Proof. Observe that $s_{1}+2-\alpha_{2}>s_{1}+\alpha_{1}^{-}$, since
$\alpha_{2}<2$, $\alpha_{2}-\alpha_{1}<2$. By Theorem 8.2

$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=y^{-s_{1}}\left(W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{1}p\right)\right).$

The assumption $s_{1}+2-\alpha_{2}\leq\frac{m+1}{p}$ is equivalent to
$\frac{m-ps_{1}+1}{p}\geq 2-\alpha_{2}$ and one concludes by
LABEL:Core_C_c_infty.

Note that when $s_{1}+2-\alpha_{2}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}$, then we
have also
$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})=W^{2,p}_{\mathcal{R}}(\alpha_{1},\alpha_{2},m)$,
by Corollary 8.5.

###### Proposition 9.7

Let $\alpha_{2}-\alpha_{1}<2$ and $s_{1}<\frac{m+1}{p}\leq s_{2}$. Then
$\mathcal{L}^{max}_{m,p}$ generates an analytic semigroup.

Proof. Let us consider the adjoint $\mathcal{L}^{*}$. Then
$s_{1}^{*}+2-\alpha_{2}\leq\frac{m+1}{p^{\prime}}<s_{2}^{*}+2-\alpha_{2}$ and
then, by the proposition above, $\mathcal{L}^{*,min}_{m,p^{\prime}}$ generates
a semigroup in $L^{p^{\prime}}_{m}$. By standard semigroup duality in
reflexive spaces, the dual operator $\mathcal{L}^{max}_{m,p}$, see Lemma 9.5,
generates a semigroup in $L^{p}_{m}$.

Observe that when $\alpha_{1}<0$, $\mathcal{L}^{max}_{m,p}$ generates a
semigroup also when the condition $s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}$ is
violated. However, if this last holds, then
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}=\mathcal{L}^{max}_{m,p}$.

###### Proposition 9.8

Let $\alpha_{2}-\alpha_{1}<2$ and $s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}\leq
s_{2}$. Then
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}=\mathcal{L}^{max}_{m,p}$.

Proof. In fact $\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}$ is well defined and
generates a semigroup. Since $\mathcal{L}^{max}_{m,p}$ extends
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}$ and both are generators, they
coincide.

By duality, we can extend the generation interval.

###### Proposition 9.9

If $\alpha_{2}-\alpha_{1}<2$ and
$\frac{m+1}{p}\in(s_{1},s_{2}+2-\alpha_{2}-\alpha_{1}^{-})\cup(s_{1}+\alpha_{1}^{-},s_{2}+2-\alpha_{2})$
a realization
$\mathcal{L}^{min}_{m,p}\subset\mathcal{L}_{D}\subset\mathcal{L}^{max}_{m,p}$
generates a semigroup.

Proof. In fact, if $s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}$ we
can take $\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}$ and if
$s_{1}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}-\alpha_{1}^{-}$ we can take the
adjoint of $\mathcal{L}^{*,\alpha_{1},\alpha_{2}}_{m,p^{\prime}}$ since the
condition is equivalent to
$s^{*}_{1}+\alpha_{1}^{-}<\frac{m+1}{p^{\prime}}<s_{2}^{*}+2-\alpha_{2}$.

For the $1d$ operator $y^{\alpha_{2}}L_{y}$ it is known that a such a
realization exists if and only if $s_{1}<\frac{m+1}{p}<s_{2}+2-\alpha_{2}$,
see [24, Theorem 1.1, Theorem 1.2] for the case $m=0$. For general $m$ and
$\alpha_{2}=0$ see [20, Propositions 2.4, 2.5] from which, by the
transformation $T_{0,-\frac{\alpha_{2}}{2}}$, it is possible deduce the
general case. However the above proposition yields a semigroup in this
interval only when $s_{1}+\alpha_{1}^{-}<s_{2}+2-\alpha_{2}-\alpha_{1}^{-}$.
For example, when $\alpha_{1}=\alpha_{2}=\alpha<0$ this requires
$|\alpha|<s_{2}-s_{1}+2$.

Let us show that when the condition $s_{1}+\alpha_{1}^{-}<\frac{m+1}{p}$ is
violated, the regularity estimate

$\|y^{\alpha_{1}}D_{x_{i}x_{j}}u\|_{L^{p}_{m}}+\|y^{\alpha_{2}}L_{y}u\|_{L^{p}_{m}}\leq
C\|\mathcal{L}u\|_{L^{p}_{m}}$ (27)

may fail for $u$ in the domain of the operator.

###### Example 9.10

Let $\mathcal{L}=y^{-\beta}(\Delta_{x}+D_{yy})$, $\beta>0$. Then $s_{1}=-1$,
$s_{2}=0$, $\mathcal{L}$ generates under Neumann boundary conditions when
$\beta<\frac{m+1}{p}<1+\beta$ and under Dirichlet boundary conditions when
$-1+\beta<\frac{m+1}{p}<2+\beta$ and both operators satisfy (27).

However, when $-1<\frac{m+1}{p}\leq(-1+\beta)\wedge 0$,
$\mathcal{L}^{max}_{m,p}$ is a generator for which (27) fails. Indeed, let
$\eta$ be a smooth function equal to $1$ in $[0,\frac{1}{2}]$ and to $0$ in
$[1,\infty[$ and

$u(x,y)=\eta(y)\left(\frac{y+1}{\left(|x|^{2}+(y+1)^{2}\right)^{\frac{N+1}{2}}}+\frac{y-1}{\left(|x|^{2}+(y-1)^{2}\right)^{\frac{N+1}{2}}}\right).$

Note that $u$ is, for small $y$, the difference of the Poisson kernels on the
hyperplanes $y=\pm 1$. Then $u\in L^{p}_{m}$, since $(m+1)/p+1>0$, and
$\mathcal{L}u\in L^{p}_{m}$, since $\Delta u=0$ for $0\leq y\leq\frac{1}{2}$,
so that $u\in D(\mathcal{L}^{max}_{m,p})$. However, $y^{-\beta}\Delta_{x}u$
and $y^{-\beta}D_{yy}u$ do not belong to $L^{p}_{m}$, since
$(m+1)/p\leq-1+\beta$.

A natural question arises if different boundary conditions can be imposed to
produce different semigroups in $L^{p}_{m}$. This is the case, for example,
for the operator $\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}B_{y}$ in
Theorem 7.1 and Corollary 8.8, in the range $c<1$,
$\alpha_{1}^{-}<\frac{m+1}{p}<c+1-\alpha_{2}$, where both boundary conditions
$\lim_{y\to 0}u=0$ and $\lim_{y\to 0}y^{c}D_{y}u=0$ can be imposed and produce
different semigroups.

As in [21, Section 5] we look for realizations $\mathcal{L}_{D}$ such that
$\mathcal{L}^{min}_{m,p}\subset\mathcal{L}_{D}\subset\mathcal{L}^{min}_{m,p}$
. From Propositions 9.6 and 9.7 it follows that $\mathcal{L}_{D}$ is unique in
$L^{p}_{m}$ if $s_{1}<\frac{m+1}{p}\leq s_{2}$ or
$s_{1}+2-\alpha-2\leq\frac{m+1}{p}<s_{2}+2-\alpha_{2}$. Uniqueness then holds
in the generation range of $L_{y}$, namely $(s_{1},s_{2}+2-\alpha_{2})$, when
these two intervals overlap, that is when $s_{1}+2-\alpha_{2}\leq s_{2}$ or
equivalently $D\geq\left(1-\frac{\alpha_{2}}{2}\right)^{2}$. In this case,
uniqueness does not depend on $p$ and $m$.

Uniqueness may fail if $s_{2}<s_{1}+2-\alpha_{2}$ and
$(m+1)/p\in(s_{2},s_{1}+2-\alpha_{2})$, as we show under the stronger
assumptions $s_{1}\neq s_{2}$ and
$s_{2}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{1}+2-\alpha_{2}$.

###### Proposition 9.11

If $0<D<\left(1-\frac{\alpha_{2}}{2}\right)^{2}$ and
$s_{2}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{1}+2-\alpha_{2}$, the operator
$\mathcal{L}=y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}L_{y}$ with domain

$\displaystyle D(\mathcal{L})$
$\displaystyle=y^{-s_{2}}W^{2,p}_{\mathcal{N}}\left(\alpha_{1},\alpha_{2},m-s_{2}p\right)$
(28) $\displaystyle=\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
y^{\alpha_{1}}\Delta_{x}u,y^{\alpha_{2}}L_{y}u\in
L^{p}_{m}\text{\;and\;}\lim_{y\to
0}y^{s_{1}+1}\left(D_{y}u+s_{2}\frac{u}{y}\right)=0\right\\}.$

generates a semigroup in $L^{p}_{m}$.

Proof. We proceed as in the proof of Theorem 8.2 but in place of the isometry
$T_{-s_{1},0}$ we use $T_{-s_{2},0}:L^{p}_{m-s_{2}p}\to L^{p}_{m}$ which
transform $\mathcal{L}$ into
$y^{\alpha_{1}}\Delta_{x}+y^{\alpha_{2}}\tilde{B}_{y}^{n}$ where
$\tilde{B}_{y}^{n}=D_{yy}+\frac{\tilde{c}}{y}D_{y}$, $\tilde{c}=c-2s_{2}$.
Observe that, under the given hypotheses,
$\tilde{c}=1-2\sqrt{D}>-1+\alpha_{2}$ and the claim follows by Theorem 7.1.

We point out that in the range
$s_{2}+\alpha_{1}^{-}<\frac{m+1}{p}<s_{1}+2-\alpha_{2}$ the operators
$\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p}$ of Theorem 8.2 and
$(\mathcal{L},D(\mathcal{L}))$ just constructed are different. In fact let
$f=a(x)b(y)\in C_{c}^{\infty}(\mathbb{R}^{N})\times\mathcal{D}$ a function in
the core defined in (11). Then $u=y^{-s_{1}}f$ belongs to
$D(\mathcal{L}^{\alpha_{1},\alpha_{2}}_{m,p})$ but not to $D(\mathcal{L})$
since $\lim_{y\to
0}y^{s_{1}+1}\left(D_{y}u+s_{2}\frac{u}{y}\right)=s_{2}-s_{1}>0$.

###  9.3 Baouendi-Grushin operator

Our results apply to generalized Baouendi-Grushin operators

$\mathcal{L}=y^{\alpha}\Delta_{x}+L_{y},\quad\alpha>-2$

in the half space $\mathbb{R}^{N+1}_{+}$ both with Neumann and Dirichlet
boundary conditions, but we restrict ourselves to the classical case
$\mathcal{L}=y^{\alpha}\Delta_{x}+D_{yy}$ in the whole space
$\mathbb{R}^{N+1}$ with the Lebesgue measure. Our results improve those from
[19], allowing negative $\alpha$ and showing maximal regularity, besides
domain characterization.

###### Proposition 9.12

Let $\alpha>-\frac{1}{p}$. Then $\mathcal{L}=|y|^{\alpha}\Delta_{x}+D_{yy}$
with domain

$D(\mathcal{L})=\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}):\ u,\
y^{\alpha}D_{x_{i}x_{j}}u,\ y^{\frac{\alpha}{2}}D_{x_{i}}u,\ D_{yy}u,\
D_{y}u,\ y^{\frac{\alpha}{2}}D_{x_{i}y}u\in L^{p}(\mathbb{R}^{N+1})\right\\}$

generates a bounded analytic semigroup in $L^{p}(\mathbb{R}^{N+1})$ which has
maximal regularity.

Proof. By Theorems 7.1, 8.2 the operator $\mathcal{L}$ generates an analytic
semigroup in $L^{p}(\mathbb{R}^{N+1}_{+})$ both with Dirichlet and Neumann
boundary conditions. We can therefore consider the operators
$\mathcal{L}_{i}$, $i=1,2$, that is $\mathcal{L}$ with domains

$\displaystyle D_{1}=$ $\displaystyle\Bigl{\\{}u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\ y^{\alpha}D_{x_{i}x_{j}}u,\
y^{\frac{\alpha}{2}}D_{x_{i}}u,D_{yy}u,\
D_{y}u,y^{\frac{\alpha}{2}}D_{x_{i}y}u\in L^{p}(\mathbb{R}^{N+1}_{+}),$
$\displaystyle\hskip 301.38794pt\lim_{y\to 0}u(x,y)=0\Bigr{\\}},$
$\displaystyle D_{2}$
$\displaystyle=W^{2,p}_{\mathcal{N}}\left(\alpha,0,0\right)$
$\displaystyle=\Bigl{\\{}u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}):\ u,\
y^{\alpha}D_{x_{i}x_{j}}u,\ y^{\frac{\alpha}{2}}D_{x_{i}}u,D_{yy}u,\
D_{y}u,y^{\frac{\alpha}{2}}D_{x_{i}y}u\in L^{p}(\mathbb{R}^{N+1}_{+}),$
$\displaystyle\hskip 301.38794pt\lim_{y\to 0}D_{y}u(x,y)=0\Bigr{\\}}.$

The mixed derivatives estimates follows from Theorem 7.1 and Corollary 8.6.

Let $P_{1},P_{2}:L^{p}(\mathbb{R}^{N+1})\to L^{p}(\mathbb{R}^{N+1}_{+})$ be
the even and odd projections

$(P_{1}f)(x,y)=\frac{f(x,y)+f(x,-y)}{2},\quad(P_{2}f)(x,y)=\frac{f(x,y)-f(x,-y)}{2}$

and $E_{1},E_{2}:L^{p}(\mathbb{R}^{N+1}_{+})\to:L^{p}(\mathbb{R}^{N+1})$ the
even and odd extensions

$\displaystyle E_{1}u(x,y)=\begin{cases}u(x,y),\qquad&\text{if}\quad
y>0;\\\\[4.30554pt] u(x,-y),\qquad&\text{if}\quad y<0;\\\\[4.30554pt]
\end{cases}$ $\displaystyle
E_{2}u(x,y)=\begin{cases}u(x,y),\qquad&\text{if}\quad y>0;\\\\[4.30554pt]
-u(x,-y),\qquad&\text{if}\quad y<0.\\\\[4.30554pt] \end{cases}$

Note that $E_{1}P_{1}+E_{2}P_{2}=I_{L^{p}(\mathbb{R}^{N+1})}$,
$P_{i}(D(\mathcal{L}))\subset D_{i}$, $E_{i}(D_{i})\subset D(\mathcal{L})$,
$i=1,2$ and that $D(\mathcal{L})=D_{1}\oplus D_{2}$ algebraically and
topologically with respect to the Sobolev norm. Then
$\mathcal{L}=E_{1}\mathcal{L}_{1}P_{1}+E_{2}\mathcal{L}_{2}P_{2}$ and
everything follows from the properties of $\mathcal{L}_{1},\mathcal{L}_{2}$.

###  9.4 The operator $y\Delta_{x}+yB_{y}$

We specialize and comment here the results obtained in the special case
$\alpha=1$, that is for
$\mathcal{L}=y\Delta_{x}+yB_{y}=y\Delta_{x}+yD_{yy}+cD_{y}$ where
$B_{y}=D_{yy}+\frac{c}{y}D_{y}$.

Theorem 7.1 applies when $0<\frac{m+1}{p}<c$ and yields generation and all
other properties listed therein for $y\Delta_{x}+yB^{n}_{y}$ with domain

$W^{2,p}_{\mathcal{N}}(1,1,m)=\\{u\in
W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\,\nabla_{x}u,\,D_{y}u,\,yD_{x_{i}x_{j}}u,\,yD_{yy}u,\,yD_{x_{i}y}u\in
L^{p}_{m}\\},$

using also Proposition 4.9 for $\nabla_{x}u$.

This result has been already proved in [14] but also in [28] when $c\geq 1$
and $m=0$ and in [10] when $p=2,m=0$ (and $c>\frac{1}{2}$).

Note that, when $m=0$, then

$W^{2,p}_{\mathcal{N}}(1,1,0)=\\{u\in
W^{1,p}(\mathbb{R}^{N+1}_{+}):yD_{x_{i}x_{j}}u,\,yD_{yy}u,\,yD_{x_{i}y}u\in
L^{p}(\mathbb{R}^{N+1}_{+})\\}$

and the associated elliptic and parabolic problems seem to have no boundary
condition. In our approach, the Neumann boundary condition is indeed imposed
to $yD_{y}u$, by requiring that $\frac{1}{y}(yD_{y}u)\in
L^{p}(\mathbb{R}^{N+1}_{+})$.

Theorem 8.2 says nothing new when $c\geq 1$, since then $s_{1}=0$ and the
transformation $T_{-s_{1},0}$ is the identity. However, when $c<1$ then
$s_{1}=c-1,s_{2}=0$ and Theorem 8.2 yields a different operator
$\mathcal{L}^{1,1}_{m,p}$ in the range $c-1<\frac{m+1}{p}<1$. Its domain is

$\left\\{u\in W^{2,p}_{loc}(\mathbb{R}^{N+1}_{+}):u,\
yD_{x_{i}x_{j}}u,\,yD_{x_{i}y}u,\,yD_{yy}u+cD_{y}u\in
L^{p}_{m}\text{\;and\;}\lim_{y\to 0}u(x,y)=0\right\\}$

by Corollary 8.3(i) and Corollary 8.6 for the mixed derivative. However, it is
not true that $yD_{yy}u$ and $D_{y}u$ belong to $L^{p}_{m}$ separately, even
when $m=0$, see also [11]. On the other hand, when $c<\frac{m+1}{p}<1$, then
Corollary 8.5 applies and gives

$D(\mathcal{L}^{1,1}_{m,p})=W^{2,p}_{\mathcal{R}}(1,1,m).$

In particular, if $m=0$, it follows that for $c<\frac{1}{p}$

$D(\mathcal{L}^{1,1}_{0,p})=\\{u\in
W^{1,p}_{0}(\mathbb{R}^{N+1}_{+}):yD_{x_{i}x_{j}}u,\,yD_{yy}u,\,yD_{x_{i}y}u\in
L^{p}(\mathbb{R}^{N+1}_{+})\\},$

a result already proved in [8].

Finally, let us specialize the results of Section 9.2, see also [9]. If $c<1$

  * (i)

$\mathcal{L}^{1,1}_{m,p}=\mathcal{L}^{min}_{m,p}$ when $c\leq(m+1)/p<1$;
$\mathcal{L}^{1,1}_{m,p}=\mathcal{L}^{max}_{m,p}$ when $c-1<(m+1)/p\leq 0$;

  * (ii)

uniqueness fails if and only if $0<(m+1)/p<c$.

Instead, if $c\geq 1$

  * (i)

$\mathcal{L}^{1,1}_{m,p}=\mathcal{L}^{min}_{m,p}$ when $1\leq(m+1)/p<c$;
$\mathcal{L}^{1,1}_{m,p}=\mathcal{L}^{max}_{m,p}$ when $0<(m+1)/p\leq c-1$;

  * (ii)

uniqueness fails if and only if $c-1<(m+1)/p<1$.

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