# The Regularity Problem in Domains with Lower Dimensional Boundaries

Zanbing Dai  Zanbing Dai. School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA<EMAIL_ADDRESS>,  Joseph Feneuil  Joseph
Feneuil. Mathematical Sciences Institute, Australian National University,
Acton, ACT, Australia<EMAIL_ADDRESS>and  Svitlana Mayboroda
Svitlana Mayboroda. School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA<EMAIL_ADDRESS>
###### Abstract.

In the present paper we establish the solvability of the Regularity boundary
value problem in domains with lower dimensional boundaries (flat and
Lipschitz) for operators whose coefficients exhibit small oscillations
analogous to the Dahlberg-Kenig-Pipher condition.

The proof follows the classical strategy of showing bounds on the square
function and the non-tangential maximal function. The key novelty and
difficulty of this setting is the presence of multiple non-tangential
derivatives. To solve it, we consider a cylindrical system of derivatives and
establish new estimates on the “angular derivatives”.

S. Mayboroda was partly supported by the NSF RAISE-TAQS grant DMS-1839077 and
the Simons foundation grant 563916, SM. J. Feneuil was partially supported by
the Simons foundation grant 601941, GD and by the European Research Council
via the project ERC-2019-StG 853404 VAREG

###### Contents

  1. 1 Introduction
    1. 1.1 Remarks on the proof of Theorem 1.1.
      1. 1.1.1 Cylindrical Coordinate Derivatives
      2. 1.1.2 Commutators
      3. 1.1.3 Local bounds
      4. 1.1.4 Approximation Results
      5. 1.1.5 Self-improvement
  2. 2 Equation in Cylindrical Coordinates
  3. 3 $N\leq S$ Local Estimates, Part 1: Integration by Parts
  4. 4 $N\leq S$ Local Estimates, Part 2: the Good Lambda Argument
  5. 5 $S\leq N$ Local Estimates
  6. 6 Global Estimates for Energy Solutions
  7. 7 Approximation by operators with Lipschitz coefficients.
  8. 8 Proof of Theorem 1.5: The Regularity Problem for a Reduced Class of Operators.
  9. 9 Proof of Theorem 1.1
  10. 10 A complement of a Lipschitz graph

##  1\. Introduction

There are three principal types of boundary value problems for elliptic
operators with rough ($L^{p}$) data: Dirichlet, Neumann, and Regularity. The
Dirichlet problem consists of establishing the existence and uniqueness of
solutions with a given trace on the boundary, the Neumann problem corresponds
to prescribing the flux, that is, the normal derivative on the boundary,
again, in $L^{p}$. The Regularity problem postulates that the tangential
derivative of the trace of the solution is known, once again, in some $L^{p}$
space. As such, it can be seen as a companion of the Neumann problem in which
the tangential rather than the normal derivative of the solution is given, or
as a version of the Dirichlet problem corresponding to the smoother boundary
data.

The Dirichlet problem has received a lot of attention in the past 30-40 years
and we will not be able to even briefly mention all the references in the
subject. Its well-posedness was established, in particular, for
$t$-independent operators on all Lipschitz domains [JK81, KKPT00, HKMP15], for
the the Laplacian on all uniformly rectifiable sets with mild topological
conditions [Dah77, HM14, Azz21, AHM+20], which was then extended to the sharp
class of the so-called Dahlberg-Kenig-Pipher (DKP) operators [KP01, DPP07,
HMM+21] and for their analogues in domains with lower-dimensional boundaries
[DFM19, FMZ21].

The Neumann and Regularity problems in $L^{p}$ proved to be much more
challenging. In particular, concerning the latter, up until recently the only
known results pertained to either $t$-independent scenario [KP93] or a “small
constant” DKP case [DPR17]. The breakthrough article [MT21] by Mourgoglou and
Tolsa was the first one to consider the regularity problem on domains beyond
Lipschitz graphs: they proved the solvability of the regularity problem for
the Laplacian on domains with uniformly rectifiable boundaries and some mild
topology. Just in the past few months the first “big constant” DKP result was
announced, by two different arguments, by Dindoš, Hofmann, Pipher [DHP22] in
the half plane and Lipschitz domains, and simultaneously, by Mourgoglou,
Poggi, Tolsa [MPT22] on domains with uniformly rectifiable boundaries.

The present paper is devoted to the setting of domains with lower dimensional
boundaries. It establishes the solvability of the regularity problem in the
complement of $\mathbb{R}^{d}$, or more generally, of a Lipschitz graph, for
an appropriate analogue of the “small constant” DKP coefficients. The higher
co-dimensional setting presented numerous new challenges, particularly, due to
the presence of “torsion”, the derivatives which roughly speaking turn the
solution around a thin boundary which are not present in the traditional
$(n-1)$-dimensional case. Respectively, we had to invent new structural
properties of the operators which on one hand, are amenable to the analysis in
desired geometric scenarios, and on the other, still allow for a control of
the second derivatives of a solution in a square function. All this will be
discussed in detail below.

Let us also mention that in the setting of the domains with lower dimensional
boundaries we are bound to work with degenerate elliptic operators, whose
coefficients grow as powers of the distance to the boundary. This provides a
curious new motivation point. Our operators, as explained below, essentially
look like $-{\rm div}\operatorname{dist}(\cdot,\partial\Omega)^{\beta}\nabla$
with a suitable power $\beta$ depending on the dimension of the set and of the
boundary. This is reminiscent of the Caffarelli-Silvestre extension operator
which allows one to view the fractional Laplacian $(-\Delta)^{\gamma}$,
$\gamma\in(0,1)$, on $\mathbb{R}^{d}$ as a Dirichlet-to-Neumann map for the
operator $-{\rm div}\,{\rm dist}(\cdot,\mathbb{R}^{d})^{\beta}\nabla$ on
$\mathbb{R}^{d+1}$, where $\beta=1-2\gamma$ (see [CS07] and also an extension
to higher powers by A. Chang and co-authors in [CY17]). Respectively, the
mapping properties of the Dirichlet-to-Neumann map become the mapping
properties of the fractional Laplacian. By the same token, one could view the
Dirichlet-to-Neumann map of our operators as an embodiment of a new concept of
differentiation or integration on rough lower-dimensional sets, and in this
vein the appropriate estimates correspond exactly to the solution of the
Regularity and Neumann problems. This paper is the first step in the
direction.

Let us now turn to definitions and statements of the main results. Let $0<d<n$
be two integers. If $d=n-1$, the domain $\Omega$ is the half-space
$\mathbb{R}^{n}_{+}:=\\{(x,t)\in\mathbb{R}^{d}\times(0,\infty)\\}$ and if
$d<n-1$, then
$\Omega:=\mathbb{R}^{n}\setminus\mathbb{R}^{d}:=\\{(x,t)\in\mathbb{R}^{d}\times(\mathbb{R}^{n-d}\setminus\\{0\\})\\}$.
In the rest of the article, $t$ will be seen as a horizontal vector, and
thence $t^{T}$ will correspond to the vertical vector. It is technically
simpler and more transparent to work in $\mathbb{R}^{n}_{+}$ and
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}$ rather than a more general graph
domain, but the goal is to treat the class of coefficients which would
automatically cover the setting of Lipschitz domains via a change of variables
– see Corollary 1.2.

We take an operator $L:=-{\operatorname{div}}|t|^{d+1-n}\mathcal{A}\nabla$ and
the first condition that we impose is of course the ellipticity and
boundedness of $\mathcal{A}$: there exists $\lambda>0$ such that for
$\xi,\zeta\in\mathbb{R}^{n}$, and $(x,t)\in\Omega$,

(1.1) $\displaystyle\lambda|\xi|^{2}\leq\mathcal{A}(x,t)\xi\cdot\xi\ \
\text{and}\ \ |\mathcal{A}(x,t)\xi\cdot\zeta|\leq\lambda^{-1}|\xi||\zeta|.$

We write (1.1)λ when we want to refer to the constant in (1.1). Then, we say
that $u\in W^{1,2}_{loc}(\Omega)$ is a weak solution to $Lu=0$ if for any
$\varphi\in C^{\infty}_{0}(\Omega)$, we have

(1.2) $\displaystyle\iint_{\Omega}\mathcal{A}\nabla
u\cdot\nabla\varphi\frac{dt}{|t|^{n-d-1}}dx=0.$

When $d=n-1$ these are the classical elliptic operators and when $d<n-1$ the
weight given by the power of distance to the boundary is necessary and
natural: if the coefficients are not degenerate, the solutions do not see the
lower dimensional sets. For instance, a harmonic function in
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}$ is the same as a harmonic function in
$\mathbb{R}^{n}$ for sufficiently small $d$. All this is discussed in detail
in [DFM21b] where we develop the elliptic theory for the operators at hand. In
particular, in the aforementioned work we construct the elliptic measure
$\omega_{L}^{X}$ associated to $L$ so that for any continuous and compactly
supported boundary data $g$, the function

(1.3) $u(X):=\int_{\mathbb{R}^{d}}g(y)\,d\omega^{X}(y)$

is a weak solution to $Lu=0$, which continuously extends to
$\overline{\Omega}$ by taking the values $u=g$ on
$\partial\Omega=\mathbb{R}^{d}$.

With this at hand, we turn to the definition of the Regularity problem. The
averaged non-tangential maximal function $\widetilde{N}$ is defined for any
function $u\in L^{2}_{loc}(\Omega)$ as

(1.4) $\displaystyle\widetilde{N}(u)(x)=\sup_{(z,r)\in\Gamma(x)}u_{W}(z,r),$

where $\Gamma(x)$ is the cone $\\{(z,r)\in\mathbb{R}^{d+1}_{+},\,|z-x|<r\\}$,
and $u_{W}(z,r)$ is the $L^{2}$-average

$\displaystyle
u_{W}(z,r):=\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{W(z,r)}|u(y,s)|^{2}dy\,ds\bigg{)}^{\frac{1}{2}},$

over the Whitney box

(1.5) $\displaystyle W(z,r):=\\{(y,s)\in\Omega,\,|y-z|<r/2,\,r/2\leq|s|\leq
2r\\}.$

Observe that when $d<n-1$, a Whitney cube is a bounded, annular region, so in
particular, the higher co-dimensional Whitney cubes $W(z,r)$ are invariant
under rotation around the boundary. We say that the Regularity problem is
solvable in $\mathbf{L^{p}}$ if for any $g\in C^{\infty}_{0}(\mathbb{R}^{d})$,
the solution given by (1.3) verifies

(1.6) $\|\widetilde{N}(\nabla u)\|_{L^{p}(\mathbb{R}^{d})}\leq C\|\nabla
g\|_{L^{p}(\mathbb{R}^{d})}$

with a constant $C>0$ that is independent of $g$. If the Regularity problem is
solvable in $L^{p}$, then we deduce by density that for any $g\in
L^{1}_{loc}(\mathbb{R}^{d})$ such that $\|\nabla
g\|_{L^{p}(\mathbb{R}^{d})}<\infty$, there exists a solution to $Lu=0$ subject
to (1.6) which converges non-tangentially to $g$. The proof of this fact is
non-trivial, but classical. See for instance Theorem 3.2 of [KP93] for the
proof of the non-tangential convergence from the bound (1.6), and since the
space $\\{g\in L^{1}_{loc}(\mathbb{R}^{d}),\,\|\nabla
g\|_{L^{p}(\mathbb{R}^{d})}<\infty\\}$ is homogenous and only equipped of a
semi-norm, we need density results analogous to Lemma 5.7, Remark 5.10, Lemma
5.11 in [DFM21b].

Going further, we say that a function $f$ satisfies the Carleson measure
condition if $\sup_{W(z,s)}|f|^{2}\frac{dsdz}{s}$ is a Carleson measure on
$\Omega$, that is, there exists a constant $M\geq 0$ such that

(1.7) $\sup_{x\in\mathbb{R}^{d},r>0}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{z\in
B(x,r)}\int_{0}^{r}\sup_{W(z,s)}|f|^{2}\frac{dsdz}{s}\leq M.$

We write $f\in CM$, or $f\in CM(M)$ when we want to refer to the constant in
(1.7). It is fairly easy to check that $f\in L^{\infty}(\Omega)$, and we even
have

(1.8) $\|f\|_{L^{\infty}(\Omega)}\leq CM^{1/2}\qquad\text{ whenever }f\in
CM(M),$

with a constant that depends only on $d$ and $n$.

The main result of the present paper is as follows.

######  Theorem 1.1.

Let $0\leq d<n$ be two integers. For any $\lambda>0$, there exists a small
parameter $\kappa>0$ and a large constant $C$, both depending only on
$\lambda$, $d$, and $n$, with the following property. Consider an elliptic
operator $L:=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$ that satisfies
(1.1)λ and such that $\mathcal{A}$ can be decomposed as
$\mathcal{A}=\mathcal{B}+\mathcal{C}$, $\mathcal{B}$ is a block matrix

(1.9) $\mathcal{B}=\begin{pmatrix}B_{1}&B_{2}\frac{t}{|t|}\\\
\frac{t^{T}}{|t|}B_{3}&b_{4}I\end{pmatrix},$

where $B_{1}$, $B_{2}$, $B_{3}$, and $b_{4}$ are respectively a $d\times d$
matrix, a $d$-dimensional vertical vector111Since $t$ is a horizontal vector,
$B_{2}\frac{t}{|t|}$ is seen as a matrix product giving a $d\times(n-d)$
matrix., a $d$-dimensional horizontal vector222That is
$\frac{t^{T}}{|t|}B_{3}$ is a $(n-d)\times d$ matrix., a scalar function, and

(1.10) $|t||\nabla B_{1}|+|t||\nabla B_{2}|+|t||\nabla B_{3}|+|t||\nabla
b_{4}|+|\mathcal{C}|\in CM(\kappa).$

Then the Regularity problem is solvable in $L^{2}(\mathbb{R}^{d})$, that is

(1.11) $\|\widetilde{N}(\nabla u_{g})\|_{L^{2}(\mathbb{R}^{d})}\leq C\|\nabla
g\|_{L^{2}(\mathbb{R}^{d})}$

whenever $g\in C^{\infty}_{0}(\mathbb{R}^{d})$ and $u_{g}$ is a solution to
$Lu=0$ given by in (1.3).

Note that when $d=n-1$ our result corresponds to the main result in [DPR17] by
Dindoš, Pipher, and Rule. In this case, the coefficients of $\mathcal{B}$
satisfy the so-called Dahlberg-Kenig-Pipher (DKP) condition with a small
constant and the addition of $\mathcal{C}$ is made possible by the
perturbation results [KP95, DFM21]. The DKP condition is sharp, that is, its
failure could result in the failure of solvability of the Dirichlet problem
[FKP91] and hence, a failure of solvability of the Regularity problem by
[FKP91].

In the setting of the domains with lower dimensional boundaries the special
structure (1.9) is new. It is dictated by the aforementioned need to control
the “torsion” of the coefficients, that is, not only to control the
oscillations of the coefficients in the transversal direction to the boundary,
but also to make sure that they are well-behaved, in a very peculiar sense, in
the angular coordinate in cylindrical coordinates naturally induced by
$\mathbb{R}^{n}\setminus\mathbb{R}^{d}$. Roughly speaking, we want to have an
almost isometry to some constant coefficient matrix as far as the $t$
direction is concerned.

One good test for whether our class of coefficients is sound structure-wise is
whether it allows for a change of variables that would yield the results on
rougher, e.g., Lipschitz, domains. After all, this was an initial motivation
for the DKP Carleson conditions on the coefficients in half-space back when
Dahlberg suggested them. To this end, consider $d<n-1$ and take a Lipschitz
function $\varphi:\,\mathbb{R}^{d}\mapsto\mathbb{R}^{n-d}$. Let
$\Omega_{\varphi}:=\\{(x,t)\in\mathbb{R}^{n},\,t\neq\varphi(x)\\}$. We set
$\sigma:=\mathcal{H}^{d}|_{\partial\Omega_{\varphi}}$ to be the
$d$-dimensional Hausdorff measure on the graph of $\varphi$, which is the
boundary of $\Omega_{\varphi}$, and we construct the “smooth distance”

$D_{\varphi}(X):=\left(\int_{\partial\Omega_{\varphi}}|X-y|^{-d-\alpha}\,d\sigma(y)\right)^{-\frac{1}{\alpha}},\quad\alpha>0.$

The quantity $D_{\varphi}(X)$ is equivalent to
$\operatorname{dist}(X,\partial\Omega_{\varphi})$, see Lemma 5.1 in [DFM19],
so the operator $L_{\varphi}:=-\operatorname{div}[D_{\varphi}^{d+1-n}\nabla]$
falls under the elliptic theory developed in [DFM21b]. Moreover, it was proved
that the Dirichlet problem for such an operator $L_{\varphi}$ is solvable in
$L^{p}$ in a complement of a small Lipschitz graph [FMZ21] and much more
generally, in a complement of a uniformly rectifiable set [DM20, Fen20]. It is
also explained in the aforementioned works why $D_{\varphi}$ as opposed to the
Euclidean distance has to be used in this context. Using the results from
[FMZ21], one can prove solvability of the Dirichlet problem in $L^{2}$. Here
we establish solvability of Regularity problem.

######  Corollary 1.2.

Let $\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{n-d}$ be a Lipschitz
function, and set $\Omega_{\varphi}$ and $L_{\varphi}$ as above. There exists
$\kappa>0$ such that if
$\|\nabla\varphi\|_{L^{\infty}(\mathbb{R}^{d})}\leq\kappa$, then the
Regularity problem is solvable in $L^{2}(\partial\Omega_{\varphi})$.

The reader can consult Section 10 for the proof and the detailed definitions.

###  1.1. Remarks on the proof of Theorem 1.1.

At this point let us return to the Main result, Theorem 1.1, and discuss some
highlights of the proof along with the particular challenges of the higher co-
dimensional setting.

Similarly to the strategy used in codimension 1, we want to prove that for any
$g$ smooth enough and $u_{g}$ constructed as in (1.3), we have

(1.12) $\|S(\nabla u_{g})\|_{L^{2}(\mathbb{R}^{d})}\leq
C\|g\|_{L^{2}(\mathbb{R}^{d})}+C\kappa\|\widetilde{N}(\nabla
u_{g})\|_{L^{2}(\mathbb{R}^{d})}$

and

(1.13) $\|\tilde{N}(\nabla u_{g})\|_{L^{2}(\mathbb{R}^{d})}\leq C\|S(\nabla
u_{g})\|_{L^{2}(\mathbb{R}^{d})},$

for some $C>0$. Here, $S$ is a square function that will be defined in (1.19)
below. We can see that when $\kappa$ is small the two estimates above would
formally imply the bound (1.11). They are the crux of the matter and the core
of the argument. However, even in this passage there are considerable
additional difficulties. Nothing guarantees that $\|\widetilde{N}(\nabla
u)\|_{L^{2}(\mathbb{R}^{d})}$ is finite, and if we do not know a priori
whether $\|\widetilde{N}(\nabla u)\|_{L^{2}(\mathbb{R}^{d})}$ is finite, we
cannot use (1.12)–(1.13) to deduce that $\|\tilde{N}(\nabla
u)\|_{L^{2}(\mathbb{R}^{d})}\leq C\|g\|_{L^{2}(\mathbb{R}^{d})}$. For that
reason we cannot simply concentrate on (1.12)–(1.13), but rather have to prove
local versions of those estimates, where all the terms are guaranteed to be
finite, and we then carefully take a limit to directly establish

(1.14) $\|\tilde{N}(\nabla u)\|_{L^{2}(\mathbb{R}^{d})}\leq C\|\nabla
g\|_{L^{2}(\mathbb{R}^{d})}<+\infty.$

Unfortunately, taking the limit is already far from trivial, because the term
$\|\nabla g\|_{2}$ is obtained roughly by taking the limit of $\|\nabla
u(x,\epsilon)\|_{2}$, and to ensure convergence, we had to assume that
$\|\nabla\mathcal{A}\|_{\infty}<+\infty$ as in [KP93], and then obtain (1.14)
for all $\mathcal{A}$ by interchanging two limits. In the classical case of
codimension 1, the situation is considerably easier because more tools are
available to us (for instance layer potential representations).

The principal issue though are still the estimates on the quantity $S(\nabla
u)$, (1.19). Clearly, it involves two derivatives, and in principle we do not
have enough regularity of the coefficients ($\mathcal{C}$ is not necessarily
continuous) to be able to directly bound the second derivatives of the
solution, not to mention the actual refined estimates that we are targeting.
This led us to a separate paper devoted to the Carleson perturbation theory
for the Regularity problem [DFM21] (cf. [KP95] when $d=n-1$). However, even
with that and even for $\mathcal{A}=\mathcal{B}$ we could not follow the route
paved for $d=n-1$ in [DPR17]. We finally realized that these arguments are not
well adapted to the cylindrical structure of our space and the additional,
quite involved, structural considerations are necessary. Let us try to give
some ideas here.

####  1.1.1. Cylindrical Coordinate Derivatives

As we mentioned, we shall use $S(\nabla u)$ as an intermediate quantity in our
computations, and so we will need to estimate second derivatives. However,
taking the second derivatives in the cartesian system of coordinates will not
be adapted to our context, and we prefer to consider “cylindrical derivatives”
defined below.

We notice that there are three difference types of directions. One is the
tangential direction, which goes alone the boundary
$\mathbb{R}^{d}\times\\{t=0\\}$. The second one is the angular direction,
which rotates around the boundary, and the last one is the radial direction
that moves away from the boundary. We write
$\nabla_{x}=(\partial_{1},\partial_{2},...,\partial_{d})$ and
$\nabla_{t}=(\partial_{d+1},\partial_{d+2},...,\partial_{n})$, where
$\partial_{i}=\vec{e}_{i}\cdot\nabla$ and $\vec{e}_{i}\in\mathbb{R}^{n}$
denotes the vector with a $1$ in the $i$-th coordinate and $0$’s elsewhere.

######  Definition 1.3.

The radial directional derivative $\partial_{r}$ is defined as:

(1.15)
$\displaystyle\partial_{r}:=\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|}\partial_{\alpha}.$

For each $d+1\leq i,j\leq n$, the directional derivative
$\partial_{\varphi_{ij}}$ is defined as:

(1.16)
$\displaystyle\partial_{\varphi_{ij}}:=-\frac{t_{i}}{|t|}\partial_{j}+\frac{t_{j}}{|t|}\partial_{i}.$

The important property of $\partial_{\varphi}$ is that

(1.17) $\partial_{\varphi}|t|=0$

To lighten the notation, we write $\partial_{\varphi}$ for any angular
directional derivative. We will mention $i,j$ explicitly when it is necessary.
Furthermore we define the angular gradient $\nabla_{\varphi}$ as a vector
derivative whose components are all angular directional derivatives
$(\partial_{\varphi_{ij}})_{d+1\leq i,j\leq n}$ and

$|\nabla_{\varphi}u|^{2}=\frac{1}{2}\sum_{i,j=d+1}^{n}|\partial_{\varphi_{ij}}u|^{2}.$

Note that $\partial_{\varphi_{ii}}=0$ for all $d+1\leq i\leq n$ and
$\partial_{\varphi_{ij}}=\partial_{\varphi_{ij}}$ for all $d+1\leq i,j\leq n$.
Also, we can easily check that the tangential, angular, and radial directions
are perpendicular to each other. More importantly, for any $u\in
W^{1,2}_{loc}$, we have the identity that
$|\nabla_{t}u|^{2}=|\partial_{r}u|^{2}+|\nabla_{\varphi}u|^{2}$ almost
everywhere (see Proposition 2.1). Consequently, it suffices to establish
estimates for the average non-tangential maximal functions of $\nabla_{x}$,
$\nabla_{\varphi}$ and $\partial_{r}$. In the rest of the article, we will
write

(1.18)
$\displaystyle\overline{\nabla}=(\nabla_{x},\nabla_{\varphi},\partial_{r}).$

One of the main reasons for using the cylindrical coordinate system is that
the operator $L=-\operatorname{div}[|t|^{d+1-n}\mathcal{A}\nabla]$ can be
written in terms of $\partial_{x},\partial_{\varphi},$ and $\partial_{r}$ (see
Proposition 2.3) when the coefficient matrix $\mathcal{A}$ is in the form of
(1.26). The expression (2.1) not only simplifies the computations, but also
helps us to better understand the geometric structure of the operator $L$.

######  Remark 1.4.

The notation $\partial_{r}$, $\partial_{\varphi}$, … might be confusing at
first, as these are not derivatives in a new system of coordinates. We will
not use a change of variable to turn our system of coordinates from a
cartesian to a cylindrical one. Instead, $\partial_{r}$ and
$\partial_{\varphi}$ denote linear combinations of derivatives in cartesian
coordinates, or derivatives along some curves (i.e., $r$ and $\varphi$ are not
“new variables”). They are used for properly grouping the derivatives. In
particular, we do not need to properly define a bijection
$(x,t)\mapsto(x,r,\varphi)$ or its Jacobian.

####  1.1.2. Commutators

The common point between $\partial_{x}$ and $\partial_{\varphi}$ is that they
both cancel out the weight $|t|^{d+1-n}$, so they will be handled in a similar
manner by commuting them with the operator $L$; the estimates on the last
derivative $\partial_{r}$ will then be obtained by using the equation
(Proposition 2.3). The difference between the two differential operators
$\partial_{x}$ and $\partial_{\varphi}$ is that $\partial_{x}$ commute with
$\nabla$ and $\overline{\nabla}$, and $\partial_{\varphi}$ do not commute with
the radial and other angular derivatives, but fortunately, everything will
work out at the end because the commutators have zero average on $W(z,r)$. The
computations pertaining to commutators are performed in Section 2, for
instance Proposition 2.4 gives that

$[\partial_{r},\partial_{\varphi}]:=\partial_{r}\partial_{\varphi}-\partial_{\varphi}\partial_{r}=-\frac{\partial_{\varphi}}{|t|}.$

####  1.1.3. Local bounds

We want to prove local versions of (1.12)–(1.13). Before introducing the
notation, let us mention that a weak solution is in $W^{2,2}_{loc}$ whenever
$\nabla A\in L^{\infty}_{loc}$, this is a well known fact which we proved
again in Proposition 7.1.

We have already defined the non-tangential maximal function in (1.4), and the
square function of $v\in W^{1,2}_{loc}(\Omega)$ is defined as:

(1.19) $\displaystyle S(v)(x):=\bigg{(}\iint_{\widehat{\Gamma}_{a}(x)}|\nabla
v(y,s)|^{2}\,\frac{dyds}{|s|^{n-2}}\bigg{)}^{\frac{1}{2}},$

where

$\widehat{\Gamma}(x)=\\{(y,s)\in\mathbb{R}^{n}\setminus\mathbb{R}^{d}:|y-x|\leq|s|\\}$

is a higher-codimension cone with vertex $x\in\mathbb{R}^{d}$. We write

(1.20) $\displaystyle
S(\overline{\nabla}u)^{2}:=\sum_{i=1}^{d}S(\partial_{x_{i}}u)^{2}+\sum_{d<i,j\leq
n}S(\partial_{\varphi_{ij}}u)^{2}+S(\partial_{r}u)^{2},$

and the square functions of $\nabla_{x}u$ and $\nabla_{\varphi}u$ are defined
in a similar manner.

For a function $0\leq\Psi\leq 1$, the definitions of the localized square
functions and the non-tangential maximal functions are

(1.21) $\displaystyle S(v|\Psi)(x):=\bigg{(}\iint_{\widehat{\Gamma}(x)}|\nabla
v|^{2}\Psi\frac{dsdy}{|s|^{n-d}}\bigg{)}^{1/2}$

and

$\displaystyle\widetilde{N}(v|\Psi)(x)=\sup_{(z,r)\in\Gamma(x)}(v|\Psi)_{W,a}(z,r)$

where $(v|\Psi)_{W,a}$ is defined on $\mathbb{R}^{d+1}_{+}$ by

$\displaystyle(v|\Psi)_{W,a}(z,r):=\bigg{(}\frac{1}{|W_{a}(z,r)|}\iint_{W_{a}(z,r)}|v|^{2}\Psi
dyds\bigg{)}^{1/2}.$

“Good” cut-off functions will satisfy the following hypothesis.

######  Hypothesis ($\mathcal{COF}$).

We say that a function $\Psi$ satisfies ($\mathcal{COF}$) if $\Psi$ is a cut-
off function, that is if $\Psi\in C^{\infty}(\overline{\Omega})$,
$0\leq\Psi\leq 1$, $\Psi$ is radial - i.e. there exists $\psi\in
C^{\infty}(\mathbb{R}^{d+1}_{+})$ such that $\Psi(x,t)=\psi(x,|t|)$ \- and we
have the bound

$|t||\nabla\Psi|\leq K\quad\text{ and
}\quad{\mathds{1}}_{\operatorname{supp}\nabla\Psi}\in CM(K).$

We write ($\mathcal{COF}$)K when we want to refer to a constant for which
$|t|\nabla\Psi|\leq K$ and ${\mathds{1}}_{\operatorname{supp}\nabla\Psi}\in
CM(K)$, and $K$ will always be chosen $\geq 1$.

We show that if $\Psi$ is a “good” cut-off function, then for any weak
solution $u\in W^{2,2}_{loc}(\Omega)$ to the equation $Lu=0$, we have

$\|S(\overline{\nabla}u|\Psi)\|^{2}_{2}\leq C_{1}\kappa\|\widetilde{N}(\nabla
u|\Psi)\|^{2}_{2}+\|\operatorname{Tr}_{\Psi}(\nabla_{x}u)\|^{2}_{2}+\text{``error
terms''},$

where $\operatorname{Tr}_{\Psi}(\nabla_{x}u)$ is an approximation of trace of
$\nabla_{x}u$ that depends on how far is $\operatorname{supp}\Psi$ to
$\partial\Omega$. The precise statement can be found in Lemma 5.5. In
addition, for a reduced class of “good” cut-off function we will obtain the
local $N\leq S$

$\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}\lesssim\|S(\overline{\nabla}u|\Psi)\|^{2}_{2}+\text{``error
terms''},$

where an exact estimate is given in Lemma 4.6. The “error terms” that we
mentioned above go to zero once we extend local estimates to global ones. The
careful definitions of the “good” cutoffs, a delicate splitting of the
derivatives, and an enhanced structure of the operator are all important for
the algebra of the computations. Afterwards, when $\kappa$ is small, by taking
$\Psi\uparrow 1$, we are able obtain the estimate

(1.22) $\|\widetilde{N}(\nabla u)\|_{2}\lesssim\lim_{\epsilon\to
0}\|\operatorname{Tr}_{\epsilon}(\nabla_{x}u)\|_{2}$

whenever $u$ is an energy solution (see Theorem 6.4). Finally, with this at
hand, two natural questions now arise. Does the limit
$\lim_{\epsilon}\|\operatorname{Tr}_{\epsilon}(\nabla_{x}u_{g})\|_{2}$ exists
and does it converge to $\|\nabla g\|_{2}$?

####  1.1.4. Approximation Results

We want to follow the strategy that Kenig and Pipher used in [KP93]. The idea
is to construct a sequence of coefficients
$\\{\mathcal{A}^{j}\\}_{j\in\mathbb{N}}$ such that
$\mathcal{A}^{j}\equiv\mathcal{A}$ on $\\{|t|>1/j\\}$ and $\mathcal{A}^{j}$ is
Lipschitz up to the boundary. In particular $\mathcal{A}_{j}$ converges
pointwise to $\mathcal{A}$, which guarantees the convergence of the solution
$u^{j}_{g}$ to $u_{g}$ (see Theorem 8.1). Meanwhile, since $\mathcal{A}^{j}$
is continuous up to the boundary,
$\|\operatorname{Tr}_{\epsilon}(\nabla_{x}u^{j}_{g})\|_{2}$ converges indeed
to $\|\nabla g\|_{2}$ because $\nabla_{x}u_{j}$ is continuous/smooth up to the
boundary. We can swap the two limits (in $\epsilon$ and in $j$), because
(1.22) entails a uniform convergence of the traces in $j$.

However, the construction of the $\mathcal{A}^{j}$ used by Kenig and Pipher
does not immediately transfer to our higher codimensional setting. In
addition, we only succeeded to obtain global bounds on $\nabla\nabla_{x}u$
(and not on all the second derivatives, like we could do in the codimension 1
setting), and this forced us to prove Theorem 6.4 before doing the
approximation. For that reason, even if we globally follow the spirit of Kenig
and Pipher’s method, we cannot say that our argument is a simple adaptation of
[KP93].

####  1.1.5. Self-improvement

All the arguments that we presented will allow us to prove the
$L^{2}$-solvability of the Regularity problem for a reduced class of
operators, and then we will “self improve” it to Theorem 1.1. The reduced
class of operators on which most of intermediate results will be written is
given as follows.

######  Hypothesis ($\mathcal{H}$).

We say that the operator
$L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ satisfies the
assumption ($\mathcal{H}$) if

  * •

$L$ is uniformly elliptic, that is there exists $\lambda\in(0,1)$ such that

(1.23) $\lambda|\xi|^{2}\leq\mathcal{A}(x,t)\xi\cdot\xi\ \ \text{and}\ \
|\mathcal{A}(x,t)\xi\cdot\zeta|\leq\lambda^{-1}|\xi||\zeta|\qquad\text{ for
}(x,t)\in\Omega,\,\xi,\zeta\in\mathbb{R}^{n};$

  * •

the matrix $\mathcal{A}$ can be written as

(1.26)
$\displaystyle\mathcal{A}(x,t)=\left({\begin{array}[]{cc}\mathcal{A}_{1}(x,t)&\mathcal{A}_{2}(x,t)\frac{t}{|t|}\\\
0&Id_{(n-d)\times(n-d)}\\\ \end{array}}\right),$

where $\mathcal{A}_{1}$ is a $d\times d$-matrix function, and
$\mathcal{A}_{2}$ is vertical vector of length $d$333That is,
$\mathcal{A}_{2}(x,t)\frac{t}{|t|}$ is a matrix operation which gives a
$d\times(n-d)$ matrix;

  * •

There exists $\kappa>0$ such that

(1.27) $|t||\nabla\mathcal{A}_{1}|+|t||\nabla\mathcal{A}_{2}|\in CM(\kappa).$

We write ($\mathcal{H}$)λ,κ when we want to refer to the constants in (1.23),
and (1.27). The constant $\kappa$ will ultimately be small.

Keep in mind that we consider the operators satisfying ($\mathcal{H}$) at
first, partially because some of our intermediate results can not be stated
with the assumptions from Theorem 1.1 (for instance we need $u\in
W^{2,2}_{loc}$ for Lemma 5.9, and so cannot consider Carleson perturbation
$\mathcal{C}$ for this result), but also because we want to simplify the
proofs (for instance, our proofs would work with $\mathcal{A}$ in the form
(1.9) instead of (1.26), but many extra computations would be needed in
Sections 3 and 5). That is, we sacrificed the optimality of the intermediate
results in order to shorten our proof.

We prove in Section 8 the following result, which seems at first glance weaker
than Theorem 1.1.

######  Theorem 1.5.

Take $\lambda,M>0$. There exists $\kappa\in(0,1)$ small enough (depending only
on $\lambda$, $d$, and $n$) such that if
$L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ is an elliptic operator
satisfying ($\mathcal{H}$)λ,κ, then for any boundary data $g\in
C^{\infty}_{0}(\mathbb{R}^{d})$, the solution $u$ to $Lu=0$ constructed as in
(1.3) or equivalently by using Lax-Milgram theorem (see Lemma 6.1) verifies

(1.28) $\displaystyle\|\widetilde{N}(\nabla u)\|_{L^{2}(\mathbb{R}^{d})}\leq
C\|\nabla g\|_{L^{2}(\mathbb{R}^{d})},$

where $C>0$ depends only on $\lambda$, $d$, and $n$.

Then, using the theory of Carleson perturbations for the Regularity problem
[DFM21, KP95] we improve the above result in Section 9 and we get Theorem 1.1,
as desired.

##  2\. Equation in Cylindrical Coordinates

In Subsection 1.1.1, we introduced a set of directional derivatives adapted to
the cylindrical structure of $\Omega$ (when $d<n-1$). The gradient
$\overline{\nabla}=(\nabla_{x},\nabla_{\varphi},\partial_{r})$ in cylindrical
coordinate has a norm equivalent to the one of the classical gradient (see
Proposition 2.1), which makes $\overline{\nabla}$ equivalent to $\nabla$ for
estimates on first order derivatives. We compute the expression of our
elliptic operator in the cylindrical system of derivatives (see Proposition
2.3).

For the second order derivatives in cylindrical coordinates, we will need to
know the commutators between $\nabla_{x}$, $\nabla_{\varphi}$, and
$\partial_{r}$, which we compute in Proposition 2.4 and Proposition 2.6. We
observe that the non trivial commutators will always involve the angular
derivative $\nabla_{\varphi}$. In order to deal with them, we shall crucially
rely on Proposition 2.7, which uses the fact that the angular directional
derivative $\partial_{\varphi}u(x,r\theta)$ has zero mean on the unit sphere
for almost every $(x,r)\in\mathbb{R}^{d+1}_{+}$. From there, we will be able
to use the Poincaré inequality and recover second order derivatives (that will
eventually be controlled).

Recall that, as mentioned in Remark 1.4, $r$ and $\varphi$ are not “new
variables in a cylindrical system”, and $\partial_{r}$ and
$\partial_{\varphi_{ij}}$ are just a linear combination of Euclidean
derivatives.

######  Proposition 2.1.

Let $\partial_{\varphi_{ij}}$ and $\partial_{r}$ be directional derivatives
defined in Definition 1.3, and let $\overline{\nabla}u$ be the cylindrical
gradient defined in (1.18). We have

$\nabla u\cdot\nabla
v=\nabla_{x}u\cdot\nabla_{x}v+(\partial_{r}u)(\partial_{r}v)+\frac{1}{2}\sum_{i,j=d+1}^{n}(\partial_{\varphi_{ij}}u)(\partial_{\varphi_{ij}}v)=\overline{\nabla}u\cdot\overline{\nabla}v$

whenever it makes sense (for instance for $u\in W^{1,2}_{loc}(\Omega)$ and
almost every $(x,t)\in\Omega$). In particular, we have $|\nabla
u|^{2}=|\overline{\nabla}u|^{2}$.

###### Proof.

We just need to prove

$\nabla_{t}u\cdot\nabla_{t}v:=\sum_{\alpha=d+1}^{n}(\partial_{t_{\alpha}}u)(\partial_{t_{\alpha}}v)=(\partial_{r}u)(\partial_{r}v)+\frac{1}{2}\sum_{i,j=d+1}^{n}(\partial_{\varphi_{ij}}u)(\partial_{\varphi_{ij}}v).$

According to the definition of $\partial_{\varphi_{ij}}$ in (1.16), we have

$\displaystyle\sum_{i,j=d+1}^{n}(\partial_{\varphi_{ij}}u)(\partial_{\varphi_{ij}}v)=\sum_{i,j=d+1}^{n}\Big{\\{}\frac{t_{i}^{2}}{|t|^{2}}(\partial_{t_{j}}u)(\partial_{t_{j}}v)-2\frac{t_{i}t_{j}}{|t|^{2}}(\partial_{t_{i}}u)(\partial_{t_{j}}v)+\frac{t^{2}_{j}}{|t|^{2}}(\partial_{t_{i}}u)(\partial_{t_{i}}v)\Big{\\}}.$

The first term on the righthand side equals $\nabla_{t}u\cdot\nabla_{t}v$
since $\sum_{i=d+1}^{n}t_{i}^{2}/|t|^{2}=1$. For the same reason, the last
term is also $\nabla_{t}u\cdot\nabla_{t}v$. We can factorize the second term
of the righthand side into the product of a sum in $i$ and a sum in $j$, and
we easily observe from the definition (1.15) that the middle term is indeed
$-2(\partial_{r}u)(\partial_{r}v)$. The proposition follows. ∎

The second proposition establishes an integration by parts for the angular and
radial derivatives.

######  Proposition 2.2.

Let $u,v\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$ be such that either $u$ or
$v$ is compactly supported in $\Omega$. We have the identities

$\iint_{\Omega}(\partial_{r}u)\,v\,|t|^{d+1-n}\,dt\,dx=-\iint_{\Omega}u\,(\partial_{r}v)\,|t|^{d+1-n}\,dt\,dx$

and

$\iint_{\Omega}(\partial_{\varphi}u)\,v\,|t|^{d+1-n}\,dt\,dx=-\iint_{\Omega}u\,(\partial_{\varphi}v)\,|t|^{d+1-n}\,dt\,dx$

where $\partial_{\varphi}$ stands for any of the $\partial_{\varphi_{ij}}$,
$d+1\leq i,j\leq n$.

###### Proof.

If one writes the integrals in cylindrical coordinates, the integration by
parts for $\partial_{r}$ is immediate once you notice that we imposed the
boundary condition $uv=0$ when $r=0$.

The second identity is also expected, but let us write is formally. Take
$d+1\leq i,j\leq n$ and we have by definition of $\partial_{\varphi_{ij}}$
that

$I:=\iint_{\Omega}(\partial_{\varphi_{ij}}u)\,v\,|t|^{d+1-n}\,dt\,dx=\iint_{\Omega}v\Big{[}(\partial_{i}u)\,t_{j}|t|^{d-n}-(\partial_{j}u)\,t_{i}|t|^{d-n}\Big{]}\,dt\,dx$

We use the integration by part to remove $\partial_{i}$ and $\partial_{j}$
from $u$, and we get

$\begin{split}I=-\iint_{\Omega}u(\partial_{\varphi_{ij}}v)\,|t|^{d+1-n}\,dt\,dx-\iint_{\Omega}uv\Big{[}\,\partial_{i}(t_{j}|t|^{d-n})-\,\partial_{j}(t_{i}|t|^{d-n})\Big{]}\,dt\,dx.\end{split}$

It is easy to check that
$\partial_{i}(t_{j}|t|^{d-n})-\,\partial_{j}(t_{i}|t|^{d-n})=0$ in $\Omega$,
thus the proposition follows. ∎

The following proposition rearranges the derivatives, in order to use
$\partial_{r}$ and $\partial_{\varphi}$ instead of the $t$-derivatives in the
expression of $L$.

######  Proposition 2.3.

Let $L=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be such that

$\displaystyle\mathcal{A}(x,t)=\left({\begin{array}[]{cc}\mathcal{A}_{1}(x,t)&\mathcal{A}_{2}(x,t)\frac{t}{|t|}\\\
\frac{t^{T}}{|t|}\mathcal{A}_{3}(x,t)&b(x,t)Id_{(n-d)\times(n-d)}\\\
\end{array}}\right),$

where $t\in\mathbb{R}^{n-d}$ is seen as a $d$-dimensional horizontal vector,
and where $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$, and $b$ are
respectively a $d\times d$ matrix, a $d$-dimensional vertical vector, a
$d$-dimensional horizontal vector, and a scalar function.

Then:

$L=-|t|^{d+1-n}\Big{[}\operatorname{div}_{x}(\mathcal{A}_{1}\nabla_{x})+\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{r})+\partial_{r}(\mathcal{A}_{3}\nabla_{x})+\partial_{r}(b\partial_{r})+\frac{1}{2}\sum_{i,j=d+1}^{n}\partial_{\varphi_{ij}}(b\partial_{\varphi_{ij}})\Big{]}.$

In particular, if $L$ satisfies ($\mathcal{H}$), then

(2.1)
$L=-|t|^{d+1-n}\Big{[}\operatorname{div}_{x}(\mathcal{A}_{1}\nabla_{x})+\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{r})+\partial^{2}_{r}+\frac{1}{2}\sum_{i,j=d+1}^{n}\partial^{2}_{\varphi_{ij}}\Big{]}.$

###### Proof.

We first decompose as

(2.2)
$L=-\operatorname{div}_{x}(|t|^{d+1-n}\mathcal{A}_{1}\nabla_{x})-\operatorname{div}_{x}(|t|^{d-n-1}\mathcal{A}_{2}\frac{t}{|t|}\nabla_{t})-\operatorname{div}_{t}(|t|^{d-n-1}\frac{t^{T}}{|t|}\mathcal{A}_{3}\nabla_{x})\\\
-\operatorname{div}_{t}(|t|^{d+1-n}b\nabla_{t})=:L_{1}+L_{2}+L_{3}+L_{4}.$

Since the weight $|t|^{d+1-n}$ is independent of $x$, one has
$L_{1}=-|t|^{d+1-n}\operatorname{div}_{x}(\mathcal{A}_{1}\nabla_{x})$ and
$L_{2}=-|t|^{d+1-n}\operatorname{div}_{x}(\mathcal{A}_{2}\frac{t}{|t|}\nabla_{t})=-|t|^{d+1-n}\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{r})$,
since by definition $\partial_{r}$ is $\frac{t}{|t|}\nabla_{t}$.

Recall that $\mathcal{A}_{3}$ is a horizontal vector and $\nabla_{x}$ is a
vertical vector differential operator, so $\mathcal{A}_{3}\nabla_{x}$ is a
scalar (differential operator). In conclusion,

$L_{3}=-\operatorname{div}_{t}(|t|^{d-n}t^{T})\mathcal{A}_{3}\nabla_{x}-|t|^{d-n-1}\frac{t}{|t|}\cdot\nabla_{t}(\mathcal{A}_{3}\nabla_{x})\\\
=0-|t|^{d-n-1}\partial_{r}(\mathcal{A}_{3}\nabla_{x}).$

At this point, it remains to treat $L_{4}$. The integration by parts entails
that, for $u,v\in C^{\infty}_{0}(\Omega)$,

$\begin{split}\iint_{\Omega}(L_{4}u)v\,dt\,dx&=\iint_{\Omega}b\nabla_{t}u\cdot\nabla_{t}v\,|t|^{d+1-n}\,dt\,dx\\\
&=\iint_{\Omega}b(\partial_{r}u)(\partial_{r}v)\,|t|^{d+1-n}\,dt\,dx+\frac{1}{2}\sum_{i,j=d+1}^{n}\iint_{\Omega}b(\partial_{\varphi_{ij}}u)(\partial_{\varphi_{ij}}v)\,|t|^{d+1-n}\,dt\,dx\end{split}$

by Proposition 2.1. Using the integration by part for $\partial_{r}$ and
$\partial_{\varphi_{ij}}$ given by Proposition 2.2, we deduce

$\iint_{\Omega}(L_{4}u)v\,dt\,dx=-\iint_{\Omega}\partial_{r}(b\partial_{r}u)\,v\,|t|^{d+1-n}\,dt\,dx-\frac{1}{2}\sum_{i,j=d+1}^{n}\iint_{\Omega}b\partial_{\varphi_{ij}}(b\partial_{\varphi_{ij}}u)v\,|t|^{d+1-n}\,dt\,dx$

Since the above equality is true for all $u,v\in C^{\infty}_{0}(\Omega)$, we
conclude

$L_{4}=-|t|^{d+1-n}\Big{[}\partial_{r}(b\partial_{r})+\frac{1}{2}\sum_{i,j=d+1}^{n}\partial_{\varphi_{ij}}(b\partial_{\varphi_{ij}})\Big{]}.$

The proposition follows. ∎

In the next results, we want to compute commutators. We immediately have that
$[\partial_{x},\partial_{r}]=0$ and $[\partial_{x},\partial_{\varphi}]=0$. The
normal derivative $\partial_{r}$ and the angular directional derivative
$\partial_{\varphi}$ do not commute, therefore we want to compute their
commutator.

######  Proposition 2.4.

Let $\partial_{\varphi}$ and $\partial_{r}$ be the derivatives defined in
Definition 1.3. Then we have

$[\partial_{r},\partial_{\varphi}]:=\partial_{r}\partial_{\varphi}-\partial_{\varphi}\partial_{r}=-\frac{\partial_{\varphi}}{|t|}.$

###### Proof.

Fix a angular directional derivative $\partial_{\varphi_{ij}}$. We use the
expressions of $\partial_{\varphi_{ij}}$ and $\partial_{r}$ given in
Definition 1.3 to write

(2.3)
$\partial_{r}\partial_{\varphi_{ij}}=\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|}\partial_{\alpha}\partial_{\varphi_{ij}}=\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|}\Big{[}\partial_{\varphi_{ij}}\partial_{\alpha}-\partial_{\alpha}\Big{(}\frac{t_{i}}{|t|}\Big{)}\partial_{j}+\partial_{\alpha}\Big{(}\frac{t_{j}}{|t|}\Big{)}\partial_{i}\Big{]}\\\
=\sum_{\alpha=d+1}^{n}\Big{[}\partial_{\varphi_{ij}}\Big{(}\frac{t_{\alpha}}{|t|}\partial_{\alpha}\Big{)}-\partial_{\varphi_{ij}}\Big{(}\frac{t_{\alpha}}{|t|}\Big{)}\partial_{\alpha}-\frac{t_{\alpha}}{|t|}\partial_{\alpha}\Big{(}\frac{t_{i}}{|t|}\Big{)}\partial_{j}+\frac{t_{\alpha}}{|t|}\partial_{\alpha}\Big{(}\frac{t_{j}}{|t|}\Big{)}\partial_{i}\Big{]}.$

We notice that the first term on the last line of (2.3) is exactly
$\partial_{\varphi_{ij}}\partial_{r}$ after summing over all
$d+1\leq\alpha\leq n$. The third and forth terms of (2.3) are similar, and are
both zero. Indeed,

$-\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|}\partial_{\alpha}\Big{(}\frac{t_{i}}{|t|}\Big{)}\partial_{j}=-\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|}\Big{(}\frac{\delta_{i\alpha}}{|t|}-\frac{t_{i}t_{\alpha}}{|t|^{3}}\Big{)}\partial_{j}=-\frac{t_{i}}{|t|}\partial_{j}+\frac{t_{i}}{|t|}\sum_{\alpha=d+1}^{n}\frac{t_{\alpha}}{|t|^{2}}\partial_{j}=0.$

The second term on the last line of (2.3) can be handled as follows:

(2.4)
$-\sum_{\alpha=d+1}^{n}\partial_{\varphi_{ij}}\Big{(}\frac{t_{\alpha}}{|t|}\Big{)}\partial_{\alpha}=-\sum_{\alpha=d+1}^{n}\Big{[}-\frac{t_{i}}{|t|}\partial_{j}\Big{(}\frac{t_{\alpha}}{|t|}\Big{)}+\frac{t_{j}}{|t|}\partial_{i}\Big{(}\frac{t_{\alpha}}{|t|}\Big{)}\Big{]}\partial_{\alpha}\\\
=\sum_{\alpha=d+1}^{n}\Big{[}\frac{t_{i}}{|t|}\Big{(}\frac{\delta_{j\alpha}}{|t|}-\frac{t_{j}t_{\alpha}}{|t|^{3}}\Big{)}-\frac{t_{j}}{|t|}\Big{(}\frac{\delta_{i\alpha}}{|t|}-\frac{t_{i}t_{\alpha}}{|t|^{3}}\Big{)}\Big{]}\partial_{\alpha}=\frac{t_{i}}{|t|^{2}}\partial_{j}-\frac{t_{j}}{|t|^{2}}\partial_{i}=-\frac{\partial_{\varphi_{ij}}}{|t|}.$

By combining our observations all together, the proposition follows. ∎

Different angular derivatives do not commute either, and we give their
commutator below.

######  Proposition 2.5.

We trivially have
$[\partial_{\varphi_{ij}},\partial_{\varphi_{\alpha\beta}}]=0$ when
$i,j,\alpha,\beta$ are all different. If $i,j,k$ are all different, we have

$[\partial_{\varphi_{ij}},\partial_{\varphi_{ik}}]=-[\partial_{\varphi_{ji}},\partial_{\varphi_{ik}}]=\frac{1}{|t|}\partial_{\varphi_{jk}}.$

###### Proof.

The identity
$[\partial_{\varphi_{ij}},\partial_{\varphi_{ik}}]=-[\partial_{\varphi_{ji}},\partial_{\varphi_{ik}}]$
comes from the fact that $\partial_{\varphi_{ij}}=-\partial_{\varphi_{ji}}$.
For the second identity, we brutally compute. We use the definitions of the
angular derivatives, and develop the expressions to obtain 8 terms that we
pair as follows:

$[\partial_{\varphi_{ij}},\partial_{\varphi_{ik}}]=\left[\frac{t_{i}}{|t|}\partial_{j}\Big{(}\frac{t_{i}}{|t|}\partial_{k}\Big{)}-\frac{t_{i}}{|t|}\partial_{k}\Big{(}\frac{t_{i}}{|t|}\partial_{j}\Big{)}\right]-\left[\frac{t_{i}}{|t|}\partial_{j}\Big{(}\frac{t_{k}}{|t|}\partial_{i}\Big{)}-\frac{t_{k}}{|t|}\partial_{i}\Big{(}\frac{t_{i}}{|t|}\partial_{j}\Big{)}\right]\\\
-\left[\frac{t_{j}}{|t|}\partial_{i}\Big{(}\frac{t_{i}}{|t|}\partial_{k}\Big{)}-\frac{t_{i}}{|t|}\partial_{k}\Big{(}\frac{t_{j}}{|t|}\partial_{i}\Big{)}\right]+\left[\frac{t_{j}}{|t|}\partial_{i}\Big{(}\frac{t_{k}}{|t|}\partial_{i}\Big{)}-\frac{t_{k}}{|t|}\partial_{i}\Big{(}\frac{t_{j}}{|t|}\partial_{i}\Big{)}\right]\\\
:=T_{1}+T_{2}+T_{3}+T_{4}.$

By using the product rule for every term and the fact that $i,j,k$ are
pairwise different, we easily get that $T_{1}=T_{4}=0$ and

$T_{2}=\frac{t_{k}}{|t|^{2}}\partial_{j}\quad\text{ and
}T_{3}=-\frac{t_{j}}{|t|^{2}}\partial_{k}.$

We conclude that

$[\partial_{\varphi_{ij}},\partial_{\varphi_{ik}}]=-\frac{t_{j}}{|t|^{2}}\partial_{k}+\frac{t_{k}}{|t|^{2}}\partial_{j}=\frac{1}{|t|}\partial_{\varphi_{jk}}$

as desired. ∎

Now it is time to compute the commutator $[L,\partial_{\varphi}]$, which is a
crucial step for establishing local bounds between the square functions and
the non-tangential maximal functions. We will explain more when we start
building up these estimates. We compute the commutator when $L$ satisfies
($\mathcal{H}$); we could compute the commutator for general elliptic operator
$L$, but we do not need it, so we spare ourselves the extra complications.

######  Proposition 2.6.

Let $\mathcal{A}$ be a $n\times n$ matrix in the form of (1.26), then for any
$v\in W^{2,2}_{loc}(\Omega)$

$[L,\partial_{\varphi}](v)=|t|^{d-n}\Big{[}\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{\varphi})+2\partial_{r}\partial_{\varphi}v\Big{]}+\operatorname{div}_{x}(|t|^{d+1-n}(\partial_{\varphi}\mathcal{A})\nabla
v).$

Here we identity $\partial_{\varphi}\mathcal{A}$ with its non-trivial
submatrix, that is the first $d$ rows.

###### Proof.

Fix an angular directional derivative $\partial_{\varphi}$. We rearrange the
derivatives to avoid using any $t$-derivatives, and Proposition 2.3 entails
that

$L=-|t|^{d+1-n}\Big{[}\operatorname{div}_{x}(\mathcal{A}_{1}\nabla_{x})+\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{r})+\partial_{r}^{2}+\frac{1}{2}\sum_{i,j=d+1}^{n}\partial_{\varphi_{ij}}^{2}\Big{]}=:L_{1}+L_{2}+L_{3}+L_{4}.$

We note that
$[L,\partial_{\varphi}]=\sum_{\alpha=1}^{4}[L_{\alpha},\partial_{\varphi}].$
So we will compute each $[L_{\alpha},\partial_{\varphi}]$ individually. Let us
start from the easiest one $[L_{1},\partial_{\varphi}]$. Since $\nabla_{x}$
and $\partial_{\varphi}$ commute and $\partial_{\varphi}|t|=0$, we have

(2.5)
$\displaystyle[L_{1},\partial_{\varphi}]=\operatorname{div}_{x}(|t|^{d+1-n}(\partial_{\varphi}\mathcal{A}_{1})\nabla_{x}).$

We turn to the operator $L_{2}$. By product rule, one has

(2.6)
$L_{2}\partial_{\varphi}=-\operatorname{div}_{x}(|t|^{d+1-n}\mathcal{A}_{2}\partial_{r}\partial_{\varphi})=-\operatorname{div}_{x}(|t|^{d+1-n}\mathcal{A}_{2}\partial_{\varphi}\partial_{r})-\operatorname{div}_{x}(|t|^{d+1-n}\mathcal{A}_{2}[\partial_{r},\partial_{\varphi}])\\\
=-\operatorname{div}_{x}(|t|^{d+1-n}\partial_{\varphi}(\mathcal{A}_{2}\partial_{r}))+\operatorname{div}_{x}(|t|^{d+1-n}(\partial_{\varphi}\mathcal{A}_{2})\partial_{r})+|t|^{d-n}\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{\varphi}),$

where we used Proposition 2.4 to compute the commutator. The first term on the
last line of (2.6) is exactly $\partial_{\varphi}L_{2}$ because
$\partial_{\varphi}|t|^{d+1-n}\equiv 0$ and $\partial_{x}$ and
$\partial_{\varphi}$ commute. Thus, (2.6) becomes

(2.7)
$\displaystyle[L_{2},\partial_{\varphi}]=\operatorname{div}_{x}(|t|^{d+1-n}(\partial_{\varphi}\mathcal{A}_{2})\partial_{r})-|t|^{d-n}\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{\varphi}).$

For simplicity, we group $[L_{1},\partial_{\varphi}]$ and
$[L_{2},\partial_{\varphi}]$. We have for any $v\in W^{2,2}_{loc}(\Omega)$
that

$[L_{1},\partial_{\varphi}](v)+[L_{2},\partial_{\varphi}](v)=\operatorname{div}_{x}\Big{(}|t|^{d+1-n}(\partial_{\varphi}\mathcal{A})\nabla
v\Big{)}+|t|^{d-n}|\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{\varphi})|.$

As for the commutator between $L_{3}$ and $\partial_{\varphi}$, we use
Proposition 2.4 multiple times to write

$\partial_{r}^{2}\partial_{\varphi}=\partial_{r}\partial_{\varphi}\partial_{r}-\partial_{r}\Big{(}\frac{\partial_{\varphi}}{|t|}\Big{)}=\partial_{\varphi}\partial_{r}^{2}-\frac{1}{|t|}\partial_{\varphi}\partial_{r}-\frac{1}{|t|}\partial_{r}\partial_{\varphi}+\frac{1}{|t|^{2}}\partial_{\varphi}=\partial_{\varphi}\partial_{r}^{2}-2\partial_{r}\partial_{\varphi}$

and thus deduce

$[L_{3},\partial_{\varphi}]=2|t|^{d+1-n}\partial_{r}\partial_{\varphi}.$

It remains to establish that $[L_{4},\partial_{\varphi}]=0$. We take
$d+1\leq\alpha,\beta\leq n$ so that
$\partial_{\varphi}=\partial_{\varphi_{\alpha\beta}}$. We invoke the fact that
$\partial_{\varphi_{ij}}=-\partial_{\varphi_{ij}}$ and then Proposition 2.5 to
obtain

$\begin{split}-2|t|^{n-d-1}[L_{4},\partial_{\varphi}]&=-\sum_{i\neq\alpha}\Big{(}\partial_{\varphi_{\beta
i}}^{2}\partial_{\varphi_{\beta\alpha}}-\partial_{\varphi_{\beta\alpha}}\partial_{\varphi_{\beta
i}}^{2}\Big{)}+\sum_{j\neq\beta}\Big{(}\partial_{\varphi_{\alpha
j}}^{2}\partial_{\varphi_{\alpha\beta}}-\partial_{\varphi_{\alpha\beta}}\partial_{\varphi_{\alpha
j}}^{2}\Big{)}\\\ &=-\sum_{i\neq\alpha}\Big{(}\partial_{\varphi_{\beta
i}}\partial_{\varphi_{i\alpha}}-\partial_{\varphi_{\alpha
i}}\partial_{\varphi_{\beta
i}}\Big{)}+\sum_{j\neq\beta}\Big{(}\partial_{\varphi_{\alpha
j}}\partial_{\varphi_{j\beta}}-\partial_{\varphi_{\beta
j}}\partial_{\varphi_{\alpha j}}\Big{)}\\\
&=-\sum_{i\neq\alpha,\beta}\Big{(}\partial_{\varphi_{\beta
i}}\partial_{\varphi_{i\alpha}}-\partial_{\varphi_{\alpha
i}}\partial_{\varphi_{\beta
i}}\Big{)}+\sum_{j\neq\alpha,\beta}\Big{(}\partial_{\varphi_{\alpha
j}}\partial_{\varphi_{j\beta}}-\partial_{\varphi_{\beta
j}}\partial_{\varphi_{\alpha j}}\Big{)}.\end{split}$

because $\partial_{\varphi_{kk}}=0$. We can freely change $j$ in $i$ in the
second sum, and after recalling again that
$\partial_{\varphi_{ij}}=-\partial_{\varphi_{ij}}$, we observe that two sums
in the right-hand side above cancel with each other. We conclude that
$[L_{4},\partial_{\varphi}]=0$, which finishes the proof of the proposition. ∎

Finally, we will need the following version of the Poincaré inequality.

######  Proposition 2.7.

Let $u\in W^{1,2}_{loc}(\Omega)$ and let $\Phi\in
C^{\infty}_{0}(\Omega,\mathbb{R}^{+})$ be a radial function. Then
$\partial_{\varphi}u$ has zero mean on sphere, that is, for almost every
$(x,r)\in\mathbb{R}^{d+1}_{+}$, we have

$\displaystyle(\partial_{\varphi}u)_{\mathbb{S}^{n-d}}(x,r):=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{S}^{n-d}}\partial_{\varphi}u(x,r\theta)d\sigma(\theta)=0,$

where $\sigma$ is the surface measure on the unit sphere $\mathbb{S}^{n-d}$.

As a result, we have

$\iint_{\Omega}|\partial_{\varphi}u|^{2}\Phi\,dt\,dx\leq
C\iint_{\Omega}|t|^{2}|\partial_{\varphi}^{2}u|^{2}\Phi\,dt\,dx,$

where $C>0$ is a universal constant.

###### Proof.

Let $\phi(x,r)\in C_{0}^{\infty}(\mathbb{R}^{d+1}_{+})$ and set
$\Phi(x,t):=\phi(x,|t|)$. Observe that

$\iint_{\mathbb{R}^{d+1}_{+}}\left|\int_{\mathbb{S}^{n-d}}\partial_{\varphi}u\,d\theta\right||\phi|\,r^{n-d-1}\,dr\,dx\leq\iint_{\mathbb{R}^{n}}|\partial_{\varphi}u||\Phi|\,dx\,dt\leq
C_{\phi}$

which proves by Fubini’s theorem that
$\int_{\mathbb{S}^{n-d}}\partial_{\varphi}u\,d\theta$ and thus
$(\partial_{\varphi}u)_{\mathbb{S}^{n-d}}$ exists for almost every
$(x,r)\in\mathbb{R}^{d+1}_{+}$. Notice now that
$\partial_{\varphi}\Phi=(\partial_{\varphi}|t|)(\partial_{r}\phi)\equiv 0$
because $\partial_{\varphi}|t|\equiv 0$ and $|\partial_{r}\phi|<\infty$.
Therefore the integration by parts (see Proposition 2.2) entails that

(2.8)
$\iint_{\mathbb{R}^{d+1}_{+}}(\partial_{\varphi}u)_{\mathbb{S}^{n-d-1}}\,\phi\,r^{n-d-1}\,dr\,dx=\frac{1}{\sigma(\mathbb{S}^{n-d-1})}\iint_{\Omega}(\partial_{\varphi}u)\Phi\,dt\,dx\\\
=-\frac{1}{\sigma(\mathbb{S}^{n-d-1})}\iint_{\Omega}u\partial_{\varphi}\Phi\,\,dt\,dx=0.$

Since the identity (2.8) holds for every $\phi\in
C_{0}^{\infty}(\mathbb{R}^{d+1}_{+})$, it is enough to conclude that
$(\partial_{\varphi}u)_{\mathbb{S}^{n-d-1}}(x,r)=0$ for almost every
$(x,r)\in\mathbb{R}^{d+1}_{+}$.

Let us turn to the second part of the Proposition. Without loss of generality,
we can assume that $d\leq n-2$ (because otherwise angular derivatives do not
exist) and $\partial_{\varphi}=\partial_{\varphi_{ij}}$ with $i=n$ and
$j=n-1$. Write a running point of $\mathbb{R}^{n}$ as
$(x,t^{\prime},t_{n-1},t_{n})\in\mathbb{R}^{d}\times\mathbb{R}^{n-d-2}\times\mathbb{R}\times\mathbb{R}$.
We consider a function
$\psi\in\mathbb{R}^{n-1}_{+}:=\\{(x,t^{\prime},r)\in\mathbb{R}^{n-2}\times(0,\infty)\\}$,
and then $\Psi(x,t):=\psi(x,t^{\prime},|(t_{n-1},t_{n})|)$. The same argument
as before shows that for almost every
$(x,t^{\prime},r)\in\mathbb{R}^{n-1}_{+}$, the function
$\theta\to\partial_{\varphi}u(x,t^{\prime},r\theta)$ lies in
$L^{2}(\mathbb{S}^{1},d\sigma)$ and

$(\partial_{\varphi}u)_{\mathbb{S}^{1}}(x,t^{\prime},r):=\fint_{\mathbb{S}_{1}}\partial_{\varphi}u(x,t^{\prime},r\theta)\,d\sigma(\theta)=0.$

However, $\mathbb{S}^{1}$ is just the unit circle, so we have the bijection

$\rho:\,z\in[0,2\pi)\mapsto\theta=(\cos(z),\sin(z))\in\mathbb{S}^{1}$

and we even have $d\sigma(\theta)=dz$. Moreover,

$\begin{split}\frac{\partial}{\partial
z}[u(x,t^{\prime},r\theta)]&=-r\sin(\theta)\frac{\partial}{\partial
t_{n-1}}u(x,t^{\prime},r\theta)+r\cos(\theta)\frac{\partial}{\partial
t_{n}}u(x,t^{\prime},r\theta)=r\partial_{\varphi}u(x,t^{\prime},r\theta)\end{split}$

and similarly

$\frac{\partial^{2}}{\partial
z^{2}}[u(x,t^{\prime},r\theta)]=r^{2}\partial_{\varphi}^{2}u(x,t^{\prime},r\theta).$

We deduce that, for almost every $(x,t^{\prime},r)\in\mathbb{R}^{n-1}_{+}$,

$\fint_{0}^{2\pi}\frac{\partial}{\partial
z}[u(x,t^{\prime},r\rho(z))]\,dz=(\partial_{\varphi}u)_{\mathbb{S}^{1}}(x,t^{\prime},r)=0$

and then, by the Poincaré inequality on $[0,2\pi]$,

$\begin{split}\int_{\mathbb{S}^{1}}|\partial_{\varphi}u(x,t^{\prime},r\theta)|^{2}\,d\sigma(\theta)&=r^{-2}\int_{0}^{2\pi}\Big{|}\frac{\partial}{\partial
z}[u(x,t^{\prime},r\rho(z))]\Big{|}^{2}dz\\\ &\leq
Cr^{2}\int_{0}^{2\pi}\Big{|}\frac{\partial^{2}}{\partial
z^{2}}[u(x,t^{\prime},r\rho(z))]\Big{|}^{2}dz=Cr^{2}\int_{\mathbb{S}^{1}}|\partial_{\varphi}^{2}u(x,t^{\prime},r\theta)|^{2}\,d\sigma(\theta).\end{split}$

We conclude by integrating over $(x,t^{\prime},r)\in\mathbb{R}^{n-1}_{+}$.
Since a radial function $\Phi$ depends on $t_{n-1}$ and $t_{n}$ only via the
norm $|(t_{n-1},t_{n})|$, we get

$\begin{split}\iint_{\Omega}|\partial_{\varphi}u|^{2}\Phi\,dt\,dx&=\int_{\mathbb{R}^{n-2}}\int_{0}^{\infty}\Phi\left(\int_{\mathbb{S}^{1}}|\partial_{\varphi}u(x,t^{\prime},r\theta)|^{2}\,d\sigma(\theta)\right)\,r\,dr\,dt^{\prime}\,dx\\\
&\lesssim\int_{\mathbb{R}^{n-2}}\int_{0}^{\infty}\Phi\left(\int_{\mathbb{S}^{1}}|\partial_{\varphi}^{2}u(x,t^{\prime},r\theta)|^{2}\,d\sigma(\theta)\right)\,r^{3}dr\,dt^{\prime}\,dx\\\
&\quad=\iint_{\Omega}|t|^{2}|\partial_{\varphi}^{2}u|^{2}\Phi\,dt\,dx.\end{split}$

The lemma follows. ∎

##  3\. $N\leq S$ Local Estimates, Part 1: Integration by Parts

We want to bound of the non-tangential maximal function by the square
functional. In this section, we prove preliminary estimates that will be
improved to the desired $N<S$ estimate in the next section by using a “good
$\lambda$” argument.

We observe first that if $v\in L^{2}_{loc}(\Omega)$ and $\Psi$ is a cut-off
function, we have by a simple application of Fubini’s theorem that

(3.1)
$\iint_{\Omega}|v|^{2}\Psi\,\frac{dt}{|t|^{n-d-2}}\,dx\approx\int_{\mathbb{R}^{d}}\left(\iint_{(y,t)\in\widehat{\Gamma}(x)}|v(y,t)|^{2}\Psi(y,t)\,\frac{dt}{|t|^{n-2}}\,dy\right)\,dx$

so in particular, for any $v\in W^{1,2}_{loc}(\Omega)$

(3.2) $\iint_{\Omega}|\nabla
v|^{2}\Psi\,\frac{dt}{|t|^{n-d-2}}\,dx\approx\|S(v|\Psi)\|^{2}_{L^{2}(\mathbb{R}^{d})}.$

The constants in (3.1) and (3.2) depends only on $d$ and $n$.

Moreover, the Carleson measure condition is well adapted to the averaged non-
tangential maximal function, in that we have

(3.3) $\iint_{\Omega}|v|^{2}|f|^{2}\Psi\,\frac{dt}{|t|^{n-d}}\,dx\leq
CM\|\widetilde{N}(u|\Psi)\|^{2}_{L^{2}(\mathbb{R}^{d})}$

whenever $f\in CM(M)$. The statement in this particular context can be found
as Proposition 4.3 in [FMZ21], but the proof is an easy consequence of the
classical Carleson inequality.

######  Lemma 3.1.

In this lemma, $\partial_{v}$ stands for either a tangential derivative
$\partial_{x}$ or an angular derivative $\partial_{\varphi}$. For any function
$u\in W^{2,2}_{loc}(\Omega)$, any cut-off function $\Psi\in
C^{\infty}_{0}(\Omega,[0,1])$ satisfying ($\mathcal{COF}$)K, any real constant
$\alpha$, and any $\delta\in(0,1)$, we have

(3.4)
$\left|\iint_{\Omega}|\partial_{v}u-\alpha|^{2}\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}\right|\leq\delta\|\widetilde{N}(\partial_{v}u-\alpha|\Psi^{3})\|^{2}_{2}+C(1+\delta^{-1}K)\|S(\overline{\nabla}u|\Psi)\|^{2}_{2},$

where $C>0$ depends only on $n$.

###### Proof.

To lighten the notation, we write $V$ for $\partial_{v}u$. First, by the
integration by parts (Proposition 2.2), we have

$\mathcal{T}:=\iint_{\Omega}|V-\alpha|^{2}\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}=-2\iint_{\Omega}(V-\alpha)(\partial_{r}V)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}$

We introduce $1=\partial_{r}|t|$, and we proceed to another integration by
parts in order to write

$\mathcal{T}=-2\iint_{\Omega}(V-\alpha)(\partial_{r}V)\Psi^{3}(\partial_{r}|t|)\frac{dtdx}{|t|^{n-d-1}}=2\iint_{\Omega}|\partial_{r}V|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\\\
+2\iint_{\Omega}(V-\alpha)(\partial_{r}V)\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-2}}+2\iint_{\Omega}(V-\alpha)(\partial_{r}^{2}V)\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}:=\textup{I}+\textup{II}+\textup{III}.$

Thanks to (3.2), the term I is bounded by the square function
$\|S(V|\Psi^{3})\|^{2}_{2}$. Since $\Psi$ satisfies ($\mathcal{COF}$)K, the
cut-off function $|t|\nabla\Psi\in CM(K)$ and so the Cauchy-Schwarz inequality
and the Carleson inequality (3.3) imply

$\displaystyle\textup{II}\leq
CK^{\frac{1}{2}}\|\widetilde{N}(V-\alpha|\Psi^{3})\|_{2}\|S(V|\Psi)\|_{2}\leq\frac{\delta}{3}\|\widetilde{N}(V-\alpha|\Psi^{3})\|^{2}_{2}+C\delta^{-1}K\|S(\overline{\nabla}u|\Psi)\|^{2}_{2}$

for any $\delta\in(0,1)$. As for the term III, we have

$\textup{III}=2\iint_{\Omega}(V-\alpha)\Big{(}[\partial_{r}^{2},\partial_{v}]u\Big{)}\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}+2\iint_{\Omega}(V-\alpha)(\partial_{v}\partial^{2}_{r}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}:=\textup{III}_{1}+\textup{III}_{2}.$

Since $\partial_{v}|t|=0$ whenever $\partial_{v}=\partial_{x}$ or
$\partial_{v}=\partial_{\varphi}$, an integration by parts yields that

$\textup{III}_{2}=-2\iint_{\Omega}(\partial_{v}^{2}u)(\partial^{2}_{r}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}-2\iint_{\Omega}(V-\alpha)(\partial^{2}_{r}u)\partial_{v}(\Psi^{3})\frac{dtdx}{|t|^{n-d-2}}:=\textup{III}_{21}+\textup{III}_{22}.$

The term $\textup{III}_{21}$ is easily bounded by
$C\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}$, and similarly to II, since
$|t||\partial_{v}\Psi|\in CM(K)$, we have that

$|\textup{III}_{22}|\leq\frac{\delta}{3}\|\widetilde{N}(V-\alpha|\Psi^{3})\|^{2}_{2}+C\delta^{-1}K\|S(\overline{\nabla}u|\Psi)\|_{2}^{2}.$

It remains to bound $\textup{III}_{1}$. Since $\partial_{x}$ and
$\partial_{r}$ commute, the commutator $[\partial^{2}_{r},\partial_{x}]$ is
zero, and hence - when $\partial_{v}=\partial_{x}$ \- we have
$\textup{III}_{1}=0$. Using Proposition 2.4 multiple times gives that

$[\partial_{r}^{2},\partial_{\varphi}]=\partial_{r}[\partial_{r},\partial_{\varphi}]+[\partial_{r},\partial_{\varphi}]\partial_{r}=-\frac{2}{|t|}\partial_{r}\partial_{\varphi}.$

So when $\partial_{v}=\partial_{\varphi}$, we have

$\textup{III}_{1}=-4\iint_{\Omega}(\partial_{\varphi}u-\alpha)(\partial_{r}\partial_{\varphi}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}\\\
=4\iint_{\Omega}(\partial_{r}\partial_{\varphi}u)(\partial_{\varphi}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+4\iint_{\Omega}(\partial_{\varphi}u-\alpha)(\partial_{\varphi}u)\partial_{\varphi}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}:=\textup{III}_{11}+\textup{III}_{12}$

by the integration by part given in Proposition 2.2. Observe that Proposition
2.7 and (3.2) infer that

(3.5) $\iint_{\Omega}|\partial_{\varphi}u|^{2}\Phi\frac{dt\,dx}{|t|^{n-d}}\leq
C\|S(\partial_{\varphi}u|\Phi)\|_{2}^{2}$

So by the Cauchy-Schwarz inequality, we have

$\textup{III}_{11}\leq\|S(\partial_{\varphi}u|\Psi^{3})\|_{2}\left(\iint_{\Omega}|\partial_{\varphi}u|^{2}\Psi^{3}\frac{dt\,dx}{|t|^{n-d}}\right)^{\frac{1}{2}}\lesssim\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}$

and similarly to II and $\textup{III}_{22}$,

$\textup{III}_{12}\leq\delta\|\widetilde{N}(\partial_{\varphi}u-\alpha|\Psi^{3})\|^{2}_{2}+C(\delta
K)^{-1}\left(\iint_{\Omega}|\partial_{\varphi}u|^{2}\Psi\frac{dt\,dx}{|t|^{n-d}}\right)^{\frac{1}{2}}\\\
\leq\frac{\delta}{3}\|\widetilde{N}(\partial_{\varphi}u-\alpha|\Psi^{3})\|^{2}_{2}+C\delta^{-1}K\|S(\overline{\nabla}u|\Psi)\|_{2}^{2}.$

The lemma follows. ∎

Now, we prove the analogue of the previous lemma for the radial derivative,
and we shall use that $u$ is solution to $Lu=0$.

######  Lemma 3.2.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, any cut-off function $\Psi\in
C^{\infty}_{0}(\Omega,[0,1])$ satisfying ($\mathcal{COF}$)K, any real constant
$\alpha$, and any $\delta\in(0,1)$, we have

$\Big{|}\iint_{\Omega}|\partial_{r}u-\alpha|^{2}\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}\Big{|}\leq\delta\|\widetilde{N}(\partial_{r}u-\alpha|\Psi^{3})\|^{2}_{2}+C(1+\delta^{-1}K^{2})\kappa\|\widetilde{N}(\nabla
u|\Psi)\|^{2}_{2}\\\
+C(1+\delta^{-1}K^{2})\|S(\overline{\nabla}u|\Psi)\|^{2}_{2},$

and

(3.6)
$\Big{|}\iint_{\Omega}|\partial_{r}u|^{2}\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}\Big{|}\leq(\delta+\delta^{-1}K^{2}\kappa)\|\widetilde{N}(\partial_{r}u|\Psi)\|^{2}_{2}+C(1+\delta^{-1}K^{2})\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2},$

$C$ depends only on $\lambda$, $d$, and $n$.

###### Proof.

We only prove the first bound, since (3.6) is established with the same
computations, by simply shifting switching the place of $\Psi$ and $\Psi^{3}$
when we bound $|\textup{I}_{3}|+|\textup{I}_{5}|$ below. By integration by
parts (see Proposition 2.2), we have

$\mathcal{T}:=\iint_{\Omega}|\partial_{r}u-\alpha|^{2}\partial_{r}(\Psi^{3})\frac{dtdx}{|t|^{n-d-1}}=-2\iint_{\Omega}(\partial_{r}u-\alpha)(\partial_{r}^{2}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}$

But now, we can use the equation in cylindrical coordinate, that is (2.1), to
obtain

$\partial^{2}_{r}u=-\operatorname{div}_{x}\mathcal{A}_{1}\nabla_{x}u+\operatorname{div}_{x}\mathcal{A}_{2}\partial_{r}u-\frac{1}{2}\sum_{i,j=d+1}^{n}\partial_{\varphi_{ij}}^{2}u$

and then

$\mathcal{T}=2\iint_{\Omega}(\partial_{r}u-\alpha)(\operatorname{div}_{x}\mathcal{A}_{1}\nabla_{x}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+2\iint_{\Omega}(\partial_{r}u-\alpha)(\operatorname{div}_{x}\mathcal{A}_{2}\partial_{r}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}\\\
+\sum_{i,j=d+1}^{n}\iint_{\Omega}(\partial_{r}u-\alpha)(\partial_{\varphi_{ij}}^{2}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}:=\textup{I}+\textup{II}+\textup{III}.$

We first deal with III, which is easier. Since $\Psi$ and $|t|$ are radial,
$\partial_{\varphi_{ij}}(|t|^{d+1-n}\Psi^{3})=0$ and thus, thanks to
integration by parts, III becomes

$\textup{III}=-\sum_{i,j=d+1}^{n}\iint_{\Omega}(\partial_{\varphi_{ij}}\partial_{r}u)(\partial_{\varphi_{ij}}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}\lesssim\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}$

by the Cauchy-Schwarz inequality and then (3.5). The terms I and II are
similar. We write $\mathcal{A}_{1,2}\nabla_{x,r}u$ for
$\mathcal{A}_{1}\nabla_{x}u+\mathcal{A}_{2}\partial_{r}u$, and by using the
fact that $\partial_{r}|t|=1$, we get

$\textup{I}+\textup{II}=2\iint_{\Omega}(\partial_{r}u-\alpha)(\operatorname{div}_{x}\mathcal{A}_{1,2}\nabla_{x,r}u)\,\Psi^{3}\,\partial_{r}(|t|)\frac{dtdx}{|t|^{n-d-1}}.$

So with an integration by parts to move the derivative $\partial_{r}$ away
from $|t|$, we have

$\textup{I}+\textup{II}=-2\iint_{\Omega}(\partial_{r}u-\alpha)(\partial_{r}\operatorname{div}_{x}\mathcal{A}_{1,2}\nabla_{x,r}u)\,\Psi^{3}\,\frac{dtdx}{|t|^{n-d-2}}\\\
-2\iint_{\Omega}(\partial^{2}_{r}u)(\operatorname{div}_{x}\mathcal{A}_{1,2}\nabla_{x,r}u)\,\Psi^{3}\,\frac{dtdx}{|t|^{n-d-2}}\\\
-2\iint_{\Omega}(\partial_{r}u-\alpha)(\operatorname{div}_{x}\mathcal{A}_{1,2}\nabla_{x,r}u)\,\partial_{r}(\Psi^{3})\,\frac{dtdx}{|t|^{n-d-2}}=\textup{I}_{1}+\textup{I}_{2}+\textup{I}_{3}.$

The integrate further by parts in $\textup{I}_{1}$ to move the
$\operatorname{div}_{x}$ away from
$\partial_{r}\mathcal{A}_{1,2}\nabla_{x,r}u$ (note beforehand that
$\partial_{r}$ and $\operatorname{div}_{x}$ commute), and we obtain

$\textup{I}_{1}=2\iint_{\Omega}(\nabla_{x}\partial_{r}u)\cdot(\partial_{r}\mathcal{A}_{1,2}\nabla_{x,r}u)\,\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\\\
+2\iint_{\Omega}(\partial_{r}u-\alpha)\,(\partial_{r}\mathcal{A}_{1,2}\nabla_{x,r}u)\cdot\nabla_{x}(\Psi^{3})\,\frac{dtdx}{|t|^{n-d-2}}:=\textup{I}_{4}+\textup{I}_{5}.$

So it remains to bounds $\textup{I}_{2}$, $\textup{I}_{3}$, $\textup{I}_{4}$,
and $\textup{I}_{5}$. The terms $\textup{I}_{2}$ and $\textup{I}_{4}$ are
similar, in that

$\begin{split}|\textup{I}_{2}|+|\textup{I}_{4}|&\lesssim\iint_{\Omega}|\nabla\overline{\nabla}u||\nabla\mathcal{A}_{1,2}\nabla_{x,r}u|\,\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\\\
&\lesssim\iint_{\Omega}|\nabla\overline{\nabla}u||\nabla\mathcal{A}_{1,2}||\nabla_{x,r}u|\,\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+\iint_{\Omega}|\mathcal{A}_{1,2}||\nabla\overline{\nabla}u|^{2}\,\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\\\
\end{split}$

We use the boundedness of $\mathcal{A}_{1,2}$ and (3.2) to get that the last
term in the right-hand side is bounded by
$\|S(\overline{\nabla}u|\Psi)\|_{2}^{2}$. As the first term in the right-hand
side above, we use the inequality $ab\leq a^{2}+b^{2}/4$, the fact that
$\nabla\mathcal{A}_{1,2}\in CM(\kappa)$, and the Carleson inequality (3.3) to
bound it by $C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|_{2}^{2}+C\|S(\overline{\nabla}u|\Psi)\|_{2}^{2}$. Altogether,

$|\textup{I}_{2}|+|\textup{I}_{4}|\lesssim\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|_{2}^{2}+\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}.$

The terms $\textup{I}_{3}$ and $\textup{I}_{5}$ are also similar, in that they
are bounded as follows

$\begin{split}|\textup{I}_{3}|+|\textup{I}_{5}|&\lesssim\iint_{\Omega}|\partial_{r}u-\alpha||\nabla\mathcal{A}_{1,2}\nabla_{x,r}u|\,|\nabla\Psi^{3}|\frac{dtdx}{|t|^{n-d-2}}\\\
&\lesssim\iint_{\Omega}|\partial_{r}u-\alpha||\nabla\mathcal{A}_{1,2}||\nabla_{x,r}u|\,|\nabla\Psi^{3}|\frac{dtdx}{|t|^{n-d-2}}+\iint_{\Omega}|\mathcal{A}_{1,2}||\partial_{r}u-\alpha||\nabla\overline{\nabla}u|\,|\nabla\Psi^{3}|\frac{dtdx}{|t|^{n-d-1}}\\\
&\lesssim\delta\|\widetilde{N}(\partial_{r}u-\alpha|\Psi^{3})\|_{2}^{2}+\delta^{-1}K^{2}\kappa\|\widetilde{N}(\nabla
u|\Psi)\|_{2}^{2}+\delta^{-1}K^{2}\|S(\overline{\nabla}u|\Psi)\|_{2}^{2}\end{split}$

by using the inequality $ab\leq\delta a^{2}+b^{2}/4\delta$, the Carleson
inequality (3.3), the fact that $\Psi$ satisfies $|\nabla\Psi|\leq K/|t|$ and
${\mathds{1}}_{\operatorname{supp}\nabla\Psi}\in CM(K)$, and the fact that
$|\nabla\mathcal{A}_{1,2}|^{2}\leq\kappa$ by (1.8). The lemma follows. ∎

In the following, we summarize the results from Lemma 3.1 and Lemma 3.2.
Before stating the precise result, we should introduce a notation first. We
write $|\nabla u-\vec{\alpha}|^{2}$ for a sum of
$|\nabla_{x}u-\vec{\alpha}_{\parallel}|^{2}$,
$|\nabla_{\varphi}u-\vec{\alpha}_{\varphi}|^{2}$, and
$|\partial_{r}u-\vec{\alpha}_{r}|^{2}$, where $\vec{\alpha}_{x}$,
$\vec{\alpha}_{\varphi}$, and $\vec{\alpha}_{r}$ are different components of
constant vector $\vec{\alpha}$ corresponding to $\nabla_{x}$,
$\nabla_{\varphi}$, and $\partial_{r}$ respectively.

######  Lemma 3.3.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, any cut-off function $\Psi\in
C^{\infty}_{0}(\Omega,[0,1])$ satisfying ($\mathcal{COF}$)K, any real constant
$\alpha$, and any $\delta\in(0,1)$, we have

(3.7) $\Big{|}\iint_{\Omega}|\nabla
u-\vec{\alpha}|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}\Big{|}\leq\delta\|\widetilde{N}(\partial_{r}u-\alpha|\Psi^{3})\|^{2}_{2}+C(1+\delta^{-1}K^{2})\kappa\|\widetilde{N}(\nabla
u|\Psi)\|^{2}_{2}\\\
+C(1+\delta^{-1}K^{2})\|S(\overline{\nabla}u|\Psi)\|^{2}_{2},$

$C$ depends only on $\lambda$, $d$, and $n$.

###### Proof.

Immediate from Lemma 3.1 and Lemma 3.2. ∎

##  4\. $N\leq S$ Local Estimates, Part 2: the Good Lambda Argument

The main goal of this section is to establish the “good-lambda” distributional
inequality, that will give the desired $N<S$ estimate.

In this section, a boundary ball (a ball in $\mathbb{R}^{d}$) with center $x$
and radius $l$ will be written $B_{l}(x)$. First, we recall several results
from [FMZ21]. Let $h_{\beta}:\,\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a
function such that for any compactly supported and continuous function $v$,

$\displaystyle
h_{\beta}(v)(x):=\inf\Big{\\{}r>0,\sup_{(y,s)\in\Gamma(x,r)}|v(y,s)|<\beta\Big{\\}},$

where $\Gamma(x,r)\subset\mathbb{R}^{d+1}_{+}$ is defined as the translation
of the cone $\Gamma(0)$ with vertex at $(x,r)$.

######  Lemma 4.1 (Lemma 6.1 in [FMZ21]).

For any $v$ such that $h_{\beta}(v)<\infty$, the map $h_{\beta}(v)$ is a
$1$-Lipschitz function.

######  Lemma 4.2 (Lemma 6.2 in [FMZ21]).

Let $v\in L^{2}_{loc}(\Omega)$ and $\Psi$ be a smooth function which satisfies
$0\leq\Psi\leq 1$. Set $h_{\beta}:=h_{\beta}((v|\Psi^{3})_{W})$. There exists
a small constant $c>0$ depending only on $d$ and $n$ such that for any
$\beta>0$ and $\widetilde{N}(v|\Psi^{3})(x)>\beta$, we have:

$\displaystyle\mathcal{M}\Big{[}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y\in
B_{h_{\beta}(.)/2}(.)}\int_{s\in\mathbb{R}^{n-d}}|v|^{2}\Psi^{3}\partial_{r}[\chi_{\beta}^{3}]\frac{ds}{|s|^{n-d-1}}dy\Big{)}^{\frac{1}{2}}\Big{]}(x)\geq
c\beta,$

where $\chi_{\beta}$ is a cut-off function defined as $\chi_{\beta}(y,.)\equiv
0$ if $h_{\beta}(y)=0$ and

$\displaystyle\chi_{\beta}(y,s):=\phi\Big{(}\frac{|s|}{h_{\beta}(y)}\Big{)},\
\ \text{ with }\ \ \phi(r)$ $\displaystyle:=\begin{cases}0&\text{if }0\leq
r<1/5,\\\ (25-5r)/24&\text{ if }1/5\leq r\leq 5,\\\ 1&\text{if
}r>5\end{cases}$

otherwise.

The two above lemmas are analogues to results from [KKPT00] and [DP19] adapted
to our setting and to the use of cut-off functions $\Psi$. Let us first
introduce some specific cut-off functions.

######  Definition 4.3.

Let $\phi\in C_{0}^{\infty}(\mathbb{R})$ be a non-increasing function such
that $\phi\equiv 1$ on $[0,1]$ and $\phi\equiv 0$ on $[2,\infty)$. We define
the cut-off functions on $\Omega$ as

$\Psi_{e}(y,t):=\phi\Big{(}\frac{e(x)}{|t|}\Big{)}{\mathds{1}}_{\Omega}(x,t)$

if $x\to e(x)\geq 0$ is a 1-Lipschitz function, in particular,

$\Psi_{\epsilon}(y,t):=\phi\Big{(}\frac{\epsilon}{|t|}\Big{)}{\mathds{1}}_{\Omega}(x,t)$

if $\epsilon>0$. Also, let us denote

$\Psi_{B}(y,t):=\phi\Big{(}\frac{\operatorname{dist}(y,B)}{100|t|}\Big{)}{\mathds{1}}_{\Omega}(x,t)$

if $B$ is a boundary ball. Moreover, we write $\Psi_{B,l,\epsilon}$ for the
product $\Psi_{B}(1-\Psi_{2l})\Psi_{\epsilon}$.

Note that from the fact that $\phi$ is non-increasing, for any (non-negative)
$1$-Lipschitz function $e$, we have

(4.1) $\partial_{r}\Psi_{e}\geq 0.$

The proof of next lemma is easy but can nevertheless be found after Lemma 4.5
in [FMZ21].

######  Lemma 4.4 ([FMZ21]).

There exists a uniform $K$ that depends only on $d$ and $n$ such that the
functions $\Psi_{e}$ and their “complements” $1-\Psi_{e}$ satisfy
($\mathcal{COF}$)K. Since $\Psi_{\epsilon}$ and $\Psi_{B}$ are particular
cases of $\Psi_{e}$, then (of course) they also satisfy ($\mathcal{COF}$)K
with the same uniform constant $K$. In addition, the property
($\mathcal{COF}$)K is stable under the product, in the sense that if $\Psi$
satisfies ($\mathcal{COF}$)${}_{K_{1}}$ and $\Phi$ satisfies
($\mathcal{COF}$)${}_{K_{2}}$, then $\Psi\Phi$ satisfies
($\mathcal{COF}$)${}_{K_{1}+K_{2}}$.

We state the precise statement of the “good-lambda” distributional inequality
that we will need in the following.

######  Lemma 4.5.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,M,κ. There exists
$\eta\in(0,1)$ that depends only on $d$ and $n$ and $C>0$ that depends on
$\lambda$, $d$ and $n$ such that the following holds.

For any a weak solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, any cut-off
function in the form $\Psi:=\Psi_{B,l,\epsilon}$ for some $\epsilon>0$, some
$l>100\epsilon$, and some boundary ball $B$ of radius $l$, and for any triplet
$\beta>0$, $\gamma>0$, $\delta\in(0,1)$, we have

(4.2) $\displaystyle|\\{x\in\mathbb{R}^{d},\widetilde{N}(\nabla
u|\Psi^{3})(x)>\beta\\}\cap E_{\beta,\gamma,\delta}|\leq
C\gamma^{2}|\\{x\in\mathbb{R}^{d},\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x)>\eta\beta\\}|,$

where

$E_{\beta,\gamma,\delta}:=\bigg{\\{}x\in\mathbb{R}^{d}:\mathcal{M}\Big{[}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{1/2}\Big{]}(x)+\delta^{1/2}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x)\\\ +\delta^{-1/2}\kappa^{1/2}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x)+\delta^{-1/2}\mathcal{M}[S(\overline{\nabla}u|\Psi)](x)\leq\gamma\beta\bigg{\\}}.$

###### Proof.

Step 1: The Whitney decomposition. We fix $\beta,\delta>0$ and we take a ball
$B\subset\mathbb{R}^{d}$ with radius $l>0$. Define

$\mathcal{E}:=\\{x\in\mathbb{R}^{d},\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x)>\eta\beta\\}.$

We notice that $(\nabla u|\Psi^{3})_{W,a}$ is continuous and $\Psi$ is
compactly supported. Hence $\mathcal{E}$ is open and bounded. We pick a ball
$B_{r}(x)$ of radius $r:=\text{dist}(x,\mathcal{E}^{c})/10$ centered at
$x\in\mathcal{E}$. Under this construction, $\mathcal{E}=\bigcup_{i\in
J}B_{r_{i}}(x_{i})$ and $\sup_{i\in J}r_{i}<\infty$. By Vitali covering lemma,
there exists a countable subcollection of balls $\\{B_{r_{i}}(x_{i})\\}_{i\in
I}$, which are disjoint and satisfy that $\mathcal{E}\subseteq\bigcup_{i\in
I}B_{5r_{i}}(x_{i})$. For each $i\in I$, we set $B_{i}:=B_{10r_{i}}(x_{i})$
and thus there exists a

(4.3) $y_{i}\in\overline{B_{i}}\cap\mathcal{E}^{c}$, in particular
$\mathcal{M}[\widetilde{N}_{a}(\nabla u|\Psi^{3})](y_{i})\leq\eta\beta$.

We define the set $F_{\beta}^{i}$ such that

$\displaystyle
F^{i}=F^{i}_{\beta,\gamma,\delta}:=\\{x\in\overline{B_{i}},\,\widetilde{N}_{a}(\nabla
u|\Psi^{3})(x)>\beta\\}\cap E_{\beta,\gamma,\delta}.$

It suffices to prove that for each $i\in I$,

(4.4) $\displaystyle|F^{i}|\lesssim C\gamma^{2}|B_{i}|$

because $\sum_{i\in I}|B_{i}|\leq 10^{d}\sum_{i\in I}|B_{r_{i}}(x_{i})|\leq
10^{d}|\mathcal{E}|.$ The inequality (4.4) is trivial when
$F_{\beta}^{i}=\emptyset$. Hence we assume that
$F_{\beta}^{i}\supset\\{x_{i}\\}$ is non-empty in the sequel of the proof.

Step 2: Localization of $\widetilde{N}(\nabla u|\Psi^{3})$ in $B_{i}$. In this
step, we show that if $x\in F^{i}$, then $\widetilde{N}(\nabla u|\Psi^{3})(x)$
has to reach its maximum value at a point $(z,r)\in\Gamma(x)$ verifying $r\leq
r_{i}$. Indeed, take $x\in F^{i}$ and then $(z,r)\in\Gamma(x)$ such that
$r>r_{i}$. Notice that $(z,r)\in\bigcup_{y\in B_{r}(z)}\Gamma(y)$, so

(4.5) $\displaystyle(\nabla u|\Psi^{3})_{W}(z,r)\leq\widetilde{N}(\nabla
u|\Psi^{3})(y)\ \ \text{for all $y\in B_{r}(z)\subset B_{20r}(y_{i})$}.$

Therefore, for a constant $C$ that depends only on $d$,

$(\nabla u|\Psi^{3})_{W}(z,r)\leq\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](y_{i})\leq C\eta\beta<\beta$

by (4.3), if $\eta$ is small enough (depending only on $d$). So it means that
for any $x\in F^{i}$, we have

(4.6) $\beta<\widetilde{N}(\nabla
u|\Psi^{3})(x)=\sup_{(z,r)\in\Gamma(x)}{\mathds{1}}_{r\leq r_{i}}(\nabla
u|\Psi^{3})_{W}(z,r)\qquad\text{ for }x\in F^{i}.$

We construct the cut-off function
$\Phi_{i}(y,s):=(1-\Psi_{K_{i}r_{i}})\Psi_{F^{i}}$ where
$\Psi_{K_{i}r_{i}}:=\Psi_{\epsilon_{i}}$ for $\epsilon_{i}:=K_{i}r_{i}$ and
$\Psi_{F^{i}}:=\Psi_{e^{i}}$ for the $1$-Lipschitz function
$e^{i}(x):=\operatorname{dist}(y,F^{i})/M_{i}$. The constants $10$ and $K_{i}$
in the construction of $\Phi_{i}$ are large enough so that $\Phi_{i}(y,t)=1$
whenever $(y,t)\in W(z,r)$ for $(z,r)\in\Gamma(x)\cap\\{r\leq r_{i}\\}$ and
$x\in F_{i}$. With such a choice and by (4.6), we have that

(4.7) $\widetilde{N}(\nabla u|\Psi^{3}\Phi_{i}^{3})(x)=\widetilde{N}(\nabla
u|\Psi^{3})(x)>\beta\qquad\text{ for }x\in F^{i}.$

We let a little bit of freedom on the choice of $K_{i}$ to avoid some future
complication. Notice that

(4.8)
$\operatorname{supp}(\nabla\Psi_{K_{i}r_{i}})\cap\operatorname{supp}(\Psi_{F_{i}})\subset
S_{i}:=(K_{i}+1)B_{i}\times\\{s\in\mathbb{R}^{n-d},\,K_{i}r_{i}/2<|s|<K_{i}r_{i}\\}.$

We first try $K_{i}=4$, which is large enough for (4.7) to be satisfied. If
$S_{i}$ intersects $\\{\Psi\neq 0\\}\cap\\{\Psi\neq 1\\}$, then we test
$K_{i}=8$ instead. If $S_{i}$ still intersects $\\{\Psi\neq
0\\}\cap\\{\Psi\neq 1\\}$, we multiply $K_{i}$ by 2 and we stop at the first
time when

(4.9) $S_{i}\cap\\{\Psi\neq 0\\}\cap\\{\Psi\neq 1\\}=\emptyset,\ \text{ i.e.
either }\ S_{i}\subset\\{\Psi\equiv 1\\}\text{ or }S_{i}\subset\\{\Psi\equiv
0\\}.$

Since $\Psi=\Psi_{B,\epsilon}$ is constructed from the product of three cut-
off function $\Psi_{e}$ where $e$ is either constant or a a slowly growing
$1/100$-Lipschitz function, while $\Psi_{F_{i}}$ is constructed with a faster
growing $1/10$-Lipschitz function, $K_{i}$ can only take a uniformly finite
number of values (i.e. we think that $K_{i}\leq 2^{7}$ and we say that
$K_{i}\leq 2^{10}$ to have some error margin).

Step 3: Catching the level sets of $\widetilde{N}(\nabla u|\Psi^{3})$. Let
$h_{\beta}:=h_{\beta}((\nabla u|\Psi^{3}\Phi_{i}^{3})_{W})$. Lemma 4.2 and
(4.7) entails that

(4.10) $\displaystyle
c\beta\leq\mathcal{M}\bigg{[}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y\in
B_{h_{\beta}(.)/2}(.)}\int_{s\in\mathbb{R}^{n-d}}|\nabla
u|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}\bigg{)}^{\frac{1}{2}}\bigg{]}(x)\qquad\text{
for }x\in F^{i}.$

We know from (4.9) that either $S_{i}\subset\\{\Psi\equiv 1\\}$ or
$S_{i}\subset\\{\Psi\equiv 0\\}$. We set

(4.13)
$\displaystyle\vec{\alpha}_{i}:=\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{S_{i}}(\nabla
u)\Psi^{3/2}\,dy\,ds=\left\\{\begin{array}[]{l}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{S_{i}}(\nabla
u)\,dy\,ds\quad\text{ if }S_{i}\subset\\{\Psi\equiv 1\\}\\\ 0\quad\text{
otherwise}\end{array}\right.$

and we want to show that $|\vec{\alpha}_{i}|$ is smaller than $c\beta/2$,
where $c$ is the constant in (4.10). We select $N$ points
$\\{z_{j}\\}_{j=1}^{N}\in 2K_{i}B_{i}$ such that
$S_{i}\subset\bigcup_{j=1}^{N}W(z_{j},K_{i}r_{i})$. We can always to so with a
uniformly bounded number $N$ of points, because $K_{i}$ is itself uniformly
bounded (between 4 and $2^{10}$). So we easily have by simply using the
definition of $\vec{\alpha}_{i}$, $(\nabla u|\Psi^{3})_{W}$,
$\widetilde{N}(\nabla u|\Psi^{3})$ and then (4.3) that

(4.14) $|\vec{\alpha}_{i}|\leq C\sum_{j=1}^{N}(\nabla
u|\Psi^{3})_{W}(z_{j},K_{i}r_{i})\leq
C\sum_{j=1}^{N}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{K_{i}r_{i}}(z_{j})}\widetilde{N}(\nabla
u|\Psi^{3})dx\\\ \leq C^{\prime}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{30K_{i}B_{i}}\widetilde{N}_{a}(\nabla
u|\Psi^{3})dx\leq C^{\prime}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](y_{i})\leq C^{\prime}\eta\beta\leq c\beta/2,$

if $\eta$ is small enough (depending only on $d$). The combination of (4.10)
and (4.14) infers that

(4.15) $\displaystyle
c\beta/2\leq\mathcal{M}\bigg{[}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y\in
B_{h_{\beta}(.)/2}(.)}\int_{s\in\mathbb{R}^{n-d}}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}\bigg{)}^{\frac{1}{2}}\bigg{]}(x)\
\text{ for }x\in F^{i}.$

Step 4: From a pointwise estimate to integral estimates. The result (4.15)
from the previous step implies that

$|F^{i}|\lesssim\frac{1}{\beta^{2}}\bigg{\|}\mathcal{M}\bigg{[}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y\in
B_{h_{\beta}(.)/2}(.)}\int_{s\in\mathbb{R}^{n-d}}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}\bigg{)}^{\frac{1}{2}}\bigg{]}\bigg{\|}^{2}_{2}\\\
\lesssim\frac{1}{\beta^{2}}\int_{\mathbb{R}^{d}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{y\in
B_{h_{\beta}(x)/2}(x)}\int_{s\in\mathbb{R}^{n-d}}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}dx$

thanks to $L^{2}$ boundedness of the Hardy-Littlewood maximal operator
$\mathcal{M}$. According to Lemma 4.1, the function $h_{\beta}$ is 1-Lipschitz
, that is, $|h_{\beta}(x)-h_{\beta}(y)|\leq|x-y|$. If $y\in
B_{h_{\beta}(x)/2}(x)$, then the Lipschitz condition implies that
$|h_{\beta}(x)-h_{\beta}(y)|\leq|x-y|\leq h_{\beta}(x)/2$ and thus
$h_{\beta}(x)/2\leq h_{\beta}(y)\leq 3h_{\beta}(x)/2$. Consequently, by
Fubini’s theorem,

$|F^{i}|\lesssim\frac{1}{\beta^{2}}\iint_{\Omega}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}\bigg{(}\int_{x\in
B_{h_{\beta}(y)}(y)}h_{\beta}(x)^{-d}dx\bigg{)}\\\
\lesssim\frac{1}{\beta^{2}}\iint_{\Omega}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}.$

Recall that

$\Psi\Phi_{i}=\Psi_{B}\Psi_{F^{i}}\Psi_{\epsilon}(1-\Psi_{K_{i}r_{i}})(1-\Psi_{2l}).$

By (4.1), $\partial_{r}[\Psi_{e}\Psi_{F^{i}}\Psi_{\epsilon}]\geq 0$ and thus
the product rule implies

$\displaystyle\Psi^{3}\Phi^{3}_{i}\partial_{r}[\chi_{\beta}^{3}]\leq\partial_{r}[\Psi^{3}\Phi^{3}_{i}\chi_{\beta}^{3}]+\partial_{r}[\Psi^{3}_{K_{i}r_{i}}]\Psi^{3}_{F^{i}}\Psi^{3}\chi^{3}_{\beta}+\partial_{r}[\Psi^{3}_{2l}]\Psi_{B}^{3}\Psi^{3}_{\epsilon}\Phi_{i}^{3}\chi^{3}_{\beta}.$

It follows that:

$|F^{i}|\lesssim\frac{1}{\beta^{2}}\iint_{\Omega}|\nabla
u-\vec{\alpha}_{i}|^{2}\partial_{r}[\Psi^{3}_{K_{i}r_{i}}]\Psi^{3}_{F^{i}}\Psi^{3}\chi^{3}_{\beta}\frac{dsdy}{|s|^{n-d-1}}\\\
+\frac{1}{\beta^{2}}\iint_{\Omega}|\nabla
u-\vec{\alpha}_{i}|^{2}\partial_{r}[\Psi^{3}_{2l}]\Psi_{B}^{3}\Psi^{3}_{\epsilon}\Phi_{i}^{3}\chi^{3}_{\beta}\frac{dsdy}{|s|^{n-d-1}}\\\
+\frac{1}{\beta^{2}}\iint_{\Omega}|\nabla
u-\vec{\alpha}_{i}|^{3}\partial_{r}[\Psi^{3}\Phi^{3}_{i}\chi_{\beta}^{3}]\frac{dsdy}{|s|^{n-d-1}}:=\textup{I}+\textup{II}+\textup{III}.$

In order to prove the claim (4.4), and hence the lemma, it suffices to show
$\textup{I}+\textup{II}+\textup{III}\leq C\gamma^{2}|B_{i}|$ with a constant
$C$ that depends only on $\lambda$, $d$, and $n$.

Step 5: We treat I. We recall that
$S_{i}\supset\operatorname{supp}(\partial_{r}\Psi_{K_{i}r_{i}})\cap\operatorname{supp}(\Psi_{F_{i}})$,
see (4.8), and $|\nabla\Psi_{K_{i}r_{i}}|\lesssim|t|$ since $\Psi$ satisfies
($\mathcal{COF}$). Therefore,

$\textup{I}\lesssim\frac{|B_{i}|}{\beta^{2}}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{S_{i}}|\nabla
u-\vec{\alpha}_{i}|^{2}\Psi^{3}\,ds\,dy=\frac{|B_{i}|}{\beta^{2}}{\mathds{1}}_{S_{i}\cap\operatorname{supp}\Psi}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{S_{i}}|\overline{\nabla}u-\vec{\alpha}_{i}|^{2}ds\,dy$

since we chose $K_{i}$ so that $\Psi$ is either constant equal to 0 or
constant equal to 1 in $S_{i}$, see (4.9), and since changing $\nabla$ to
$\overline{\nabla}$ is just rewriting a vector with a different system of
coordinates (and of course we rewrite $\vec{\alpha}_{i}$ in this system of
coordinates too). If $\Psi\equiv 0$ on $S_{i}$, the bound $I=0\leq
C\gamma^{2}|B_{i}|$ is trivial. So we assume for the rest of the step that
$\Psi\equiv 1$ on $S_{i}$. In this case, since $\vec{\alpha}_{i}$ is the
average of $\nabla u$ on $S_{i}$, the Poincaré inequality yields that:

(4.16)
$\textup{I}\lesssim\frac{r_{i}^{2}|B_{i}|}{\beta^{2}}\mathchoice{{\vbox{\hbox{$\textstyle{-}\mkern-3.5mu{-}$}}\kern-9.63316pt}}{{\vbox{\hbox{$\scriptstyle{-}\mkern-3.5mu{-}$}}\kern-5.21329pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-3.3858pt}}{{\vbox{\hbox{$\scriptscriptstyle{-}\mkern-3.5mu{-}$}}\kern-2.6208pt}}\\!\iint_{S_{i}}|\nabla\overline{\nabla}u|^{2}dsdy\lesssim\frac{|B_{i}|}{\beta^{2}}\iint_{S_{i}}|\nabla\overline{\nabla}u|^{2}\Psi^{3}\frac{ds\,dy}{|s|^{n-2}}$

because $\Psi\equiv 1$ on $S_{i}$. We adapt the argument that we used to
establish (4.14). We pick a collection of points $\\{z_{j}\\}_{j=1}^{N}\in
2K_{i}B_{i}$ such that

$S_{i}\subset\bigcup_{j=1}^{N}B_{K_{i}r_{i}/4}(z_{j})\times\\{K_{i}r_{i}/2<|s|<K_{i}r_{i}\\}.$

We can choose the collection so that $N$ is uniformly bounded. Since

$B_{K_{i}r_{i}/4}(z_{j})\times\\{K_{i}r_{i}/2<|s|<K_{i}r_{i}\\}\subset\widehat{\Gamma}(x)\
\text{ for }x\in B_{K_{i}r_{i}/4}(z_{j}),$

we have

(4.17)
$\textup{I}\lesssim\frac{|B_{i}|}{\beta^{2}}\sum_{j=1}^{N}\iint_{B_{K_{i}r_{i}/4}(z_{j})\times\\{K_{i}r_{i}/2<|s|<K_{i}r_{i}\\}}|\nabla\overline{\nabla}u|^{2}\Psi^{3}\frac{ds\,dy}{|s|^{n-d-2}}\\\
\lesssim\frac{|B_{i}|}{\beta^{2}}\sum_{j=1}^{N}\left(\fint_{B_{K_{i}r_{i}/4}(z_{j})}S(\overline{\nabla}u|\Psi^{3})(x)\,dx\right)^{2}\lesssim\frac{|B_{i}|}{\beta^{2}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{30K_{i}B_{i}}S(\overline{\nabla}u|\Psi^{3})(x)\,dx\right)^{2}\\\
\leq\frac{|B_{i}|}{\beta^{2}}\left(\mathcal{M}[S(\overline{\nabla}u|\Psi)](x_{i})\right)^{2}\leq\gamma^{2}|B_{i}|,$

where $x_{i}$ is any point of the non-emptyset $F^{i}\subset
E_{\beta,\gamma,\delta}$ and the last inequality comes from the fact that
$x_{i}\in E_{\beta,\gamma,\delta}$ (we could even have
$\textup{I}\lesssim\gamma^{2}\delta^{2}|B_{i}|$).

Step 6: We deal with II. Observe that

$\operatorname{supp}\\{\Psi_{B}\partial_{r}\Psi_{2l}\\}\subset
500B\times\\{\,l\leq|s|\leq 2l\\}\ \text{and }\
\operatorname{supp}\\{\Phi_{i}\\}\subset\\{\operatorname{dist}(y,B_{i})/20\leq|s|\leq
K_{i}r_{i}\\}$

since we know that $K_{i}\leq 2^{10}$. The integral II is non-zero only if the
(interior of the) two above supports intersect, and in this case, we
necessarily have

(4.18) $l<K_{i}r_{i}\leq 2^{10}r_{i}\ \text{ and }\
\operatorname{dist}(500B,F_{i})<40l$

which we know assume. So $500B\subset 2^{20}B_{i}$ and we can find a boundary
point $x_{i}\in\mathbb{R}^{d}$ such that

(4.19) $x_{i}\in F_{i}\cap 550B.$

By the triangle inequality and the fact that $|\nabla\Psi_{2l}|\lesssim l$ on
$\operatorname{supp}(\nabla\Psi_{2l})$, we have

$\displaystyle\textup{II}\lesssim\frac{|2^{20}B_{i}|}{\beta^{2}}\fint_{500B}\fint_{l\leq|s|\leq
2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy+\frac{|2^{20}B_{i}||\vec{\alpha}_{i}|^{2}}{\beta^{2}}\Psi_{B}^{3}=\textup{II}_{1}+\textup{II}_{2}.$

We want to bound $\textup{II}_{1}$ with the help of the Hardy Littlewood
maximal function of $x\to\Big{(}\fint_{B_{l}(x)}\fint_{l<|s|<2l}|\nabla
y|^{2}dsdy\Big{)}^{1/2}$. So we proceed like we already several times, see
around (4.14) and (4.17). We take a uniformly finite collection of points
$\\{z_{j}\\}_{j=1}^{N}\in 501B$ such that $500B\subset\bigcup B_{l/2}(z_{j})$,
and since $\fint_{B_{l/2}(z_{j})}|\nabla u|^{2}\lesssim\fint_{B_{l}(x)}|\nabla
u|^{2}$ for any $x\in B_{l/2}(z_{j})|\nabla u|^{2}$, we have

(4.20)
$\textup{II}_{1}\lesssim\frac{|B_{i}|}{\beta^{2}}\sum_{j=1}^{N}\fint_{B_{l/2}(z_{j})}\fint_{l<|s|<2l}|\nabla
u|^{2}\Psi_{B}^{3}ds\,dy\\\
\lesssim\frac{|B_{l}|}{\beta^{2}}\sum_{j=1}^{N}\left(\fint_{B_{l/2}(z_{j})}\Big{(}\fint_{B_{l}(x)}\fint_{l<|s|<2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{\frac{1}{2}}dx\right)^{2}\\\
\lesssim\frac{|B_{i}|}{\beta^{2}}\left(\fint_{B_{2000l}(x_{i})}\Big{(}\fint_{B_{l}(x)}\fint_{l<|s|<2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{\frac{1}{2}}dx\right)^{2}\\\
\leq\frac{|B_{i}|}{\beta^{2}}\left(\mathcal{M}\Big{[}\fint_{B_{l}(.)}\fint_{l<|s|<2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{\frac{1}{2}}\Big{]}(x_{i})\right)^{2}\leq\gamma^{2}|B_{i}|,$

because $x_{i}\in E_{\beta,\gamma,\delta}$. It remains to bound
$\textup{II}_{2}$, but that one will be easy. Without loss of generality, we
can assume the support of the function $\phi$ used to construct $\Psi_{2l}$ to
be exactly $[1,2]$ and hence the support of $\partial_{r}\Psi_{2l}$ to be
exactly $\\{l\leq|s|\leq 2l\\}$. But the set $S_{i}$ defined in (4.8) and used
to build $\vec{\alpha}_{i}$ has to be included by construction in either
$\\{\Psi\equiv 1\\}$ or $\\{\Psi\equiv 0\\}$. Combined with (4.18), it forces
$S_{i}\subset\\{\Psi\equiv 0\\}$, and thus
$\textup{II}_{2}=|\vec{\alpha}_{i}|=0$.

Step 7: We bound III to conclude. As discussed at the end of Step 4, we needed
to bound I, II, and III by $C\gamma^{2}|B_{i}|$ to finish the proof of the
lemma. We already proved the desired estimates of I and II in Steps 5 and 6,
so it remains to show $\textup{III}\lesssim\gamma^{2}|B_{i}|$.

We did not use Section 3 at this point, so as one could expect, it will appear
in this last Step. We easily have that

(4.21) $\displaystyle\|\widetilde{N}(\nabla
u-\vec{\alpha}_{i}|\Psi^{3}\Phi_{i}^{3})\|^{2}_{2}\leq\|\widetilde{N}(\nabla
u|\Psi^{3}\Phi^{3}_{i})\|^{2}_{2}+|\vec{\alpha}_{i}|^{2}\|\widetilde{N}(1|\Phi_{i})\|^{2}_{2}.$

Lemma 4.4 shows that $\Psi^{3}\Phi_{i}^{3}\chi_{\beta}^{3}$ satisfies
($\mathcal{COF}$)with a constant that depends only on $d$ and $n$. Thus we
apply Lemma 3.3 to the term III. Together with (4.21), we deduce that

$\textup{III}\leq\frac{1}{\beta^{2}}\Big{\\{}\delta\|\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})\|^{2}_{2}+\delta|\vec{\alpha}_{i}|^{2}\|\widetilde{N}(1|\Phi_{i})\|^{2}_{2}\\\
+C(1+\delta^{-1}K^{2})\kappa\|\widetilde{N}(\nabla
u|\Psi\Phi_{i})\|^{2}_{2}+C(1+\delta^{-1}K^{2})\|S(\overline{\nabla}u|\Psi\Phi_{i})\|^{2}_{2}\Big{\\}}$

Let $v$ be any function for which $\widetilde{N}(v|\Phi_{i})$ or
$S(v|\Phi_{i})$ makes sense, and in this situation, the non-tangential maximal
function $\widetilde{N}(v|\Phi_{i})$ and the square function $S(v|\Phi_{i})$
are supported in a ball $CB_{i}$, where $C$ is universal. Why? Because
$\Phi_{i}$ is supported in a saw-tooth region on top of $F^{i}\subset B_{i}$,
which is truncated above by $K_{i}r_{i}$. Hence the Whitney box $W(z,r)$ for
which $(v|\Phi_{i})_{W(z,r)}\neq 0$ are such that $r\leq 2K_{i}r_{i}$ and then
$z\in 10K_{i}B_{i}$, which means $\operatorname{supp}N(v|\Phi_{i})\subset
10K_{i}B_{i}$. Similarly, a point $(y,t)$ for which $\nabla v(y,t)\neq 0$ are
such that $|t|\leq K_{i}r_{i}$ and then $y\in 3K_{i}B_{i}$, which implies that
$\operatorname{supp}S(v|\Phi_{i})\subset 3K_{i}B_{i}$. Altogether

(4.22)
$\operatorname{supp}S(v|\Phi_{i})\cup\operatorname{supp}N(v|\Phi_{i})\subset
10K_{i}B_{i}\subset B^{*}_{i}:=2^{14}B_{i}.$

With this observation, we have

$\textup{III}\lesssim\frac{|B_{i}|}{\beta^{2}}\Big{\\{}\delta|\vec{\alpha}_{i}|^{2}+\delta\big{[}\sup_{B^{*}_{i}}\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})\big{]}^{2}+\delta^{-1}\kappa\big{[}\sup_{B^{*}_{i}}\widetilde{N}(\nabla
u|\Psi\Phi_{i})\big{]}^{2}+\delta^{-1}\big{[}\sup_{B^{*}_{i}}S(\overline{\nabla}u|\Psi^{3}\Phi_{i}^{3})\big{]}^{2}\Big{\\}}\\\
:=\textup{III}_{1}+\textup{III}_{2}+\textup{III}_{3}+\textup{III}_{4}.$

The three terms above are handled in a similar manner. Recall that $\Phi_{i}$
is supported in a saw-tooth region over $F_{i}$ truncated at $K_{i}r_{i}$. If
$(\nabla u|\Psi^{3}\Phi_{i}^{3})_{W}(z^{\prime},r^{\prime})\neq 0$, then
$W(z^{\prime},r^{\prime})\cap{\rm supp}\\{\Phi_{i}\\}\neq\emptyset$ and there
exists a $x_{i}\in F^{i}\subset B_{i}$ such that $|x_{i}-z^{\prime}|\leq
1000r^{\prime}\leq 2^{20}r_{i}$. It follows that for all
$(z^{\prime},r^{\prime})\in\mathbb{R}^{d+1}_{+}$,

(4.23) $(\nabla
u|\Psi^{3}\Phi_{i}^{3})_{W}(z^{\prime},r^{\prime})\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r^{\prime}}(z^{\prime})}\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})(z)dz\\\ \lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{1000r^{\prime}}(x_{i})}\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})(z)dz\leq\mathcal{M}[\widetilde{N}_{a}(\nabla
u|\Psi^{3})](x_{i}).$

Consequently, for each $z\in\mathbb{R}^{d}$, there exists a $x_{i}\in
F_{\beta}^{i}$ such that

(4.24) $\displaystyle\delta\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})(z)\lesssim\delta\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x_{i})\leq\gamma\beta$

where the last inequality follows from the fact that $x_{i}\in
E_{\beta,\gamma,\delta}$. We easily deduce

$\textup{III}_{2}:=\frac{\delta|B_{i}|}{\beta^{2}}\big{[}\sup_{B^{*}_{i}}\widetilde{N}(\nabla
u|\Psi^{3}\Phi_{i}^{3})\big{]}^{2}\lesssim\gamma^{2}|B_{i}|.$

Similarly, we have

$\textup{III}_{3}:=\frac{\delta^{-1}\kappa|B_{i}|}{\beta^{2}}\big{[}\sup_{B^{*}_{i}}\widetilde{N}(\nabla
u|\Psi\Phi_{i})\big{]}^{2}\lesssim\gamma^{2}|B_{i}|.$

The term $\textup{III}_{4}$ follows the same lines. If $y\in F^{i}$, then
$S(\overline{\nabla}u|\Psi\Phi_{i})(y)\leq
S(\overline{\nabla}u|\Psi\Phi_{i})(y)\leq\gamma\beta$. If $y\notin F_{i}$, we
take $x_{i}\in F^{i}$ such that
$r_{y}:=\operatorname{dist}(y,F_{i})=|y-x_{i}|$. We know from the construction
of $\Psi_{F^{i}}$ that

(4.25) $\Phi_{i}(z,s)\equiv 0\ \text{ for $|z-y|<r_{i}/10$ and
$|s|<r_{i}/400$.}$

We cover $B_{r_{i}/20}(y)$ by a uniformly finite collection of balls
$\\{B_{r_{i}/800}(z_{j})\\}_{j=1}^{N}$, and we notice that for any collection
$\\{w_{j}\\}_{j=1}^{N}$ of points satisfying $w_{j}\in B_{r_{i}/800}(z_{j})$,
we have

$\widehat{\Gamma}(y)\cap\operatorname{supp}\Phi_{i}\subset\bigcup_{j=1}^{N}\widehat{\Gamma}(w_{j}).$

We conclude that

$S(\overline{\nabla}u|\Psi\Phi_{i})(y)\leq\sum_{j=1}^{N}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r_{i}/800}(z_{j})}S(\overline{\nabla}u|\Psi\Phi_{i})(w)\,dw\\\
\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2r_{i}}(x_{i})}S(\overline{\nabla}u|\Psi)(w)\,dw\leq\mathcal{M}\Big{[}S(\overline{\nabla}u|\Psi)\Big{]}(x_{i})\leq\frac{\delta^{1/2}}{\gamma}\beta.$

and then $\textup{III}_{4}\lesssim\gamma^{2}|B_{i}|$ as desired.

It remains to bound $\textup{III}_{1}$, We apply the same argument as of
$(\ref{eqGL06})$ using $x_{i}\in F^{i}$ instead of $y_{i}$. So we have

(4.26)
$\displaystyle\delta|\vec{\alpha}_{i}|^{2}\lesssim\delta\big{|}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})](x_{i})\big{|}^{2}\lesssim\gamma^{2}\beta^{2},$

because $x_{i}\in E_{\beta,\gamma,\delta}$, from which we easily deduce
$\textup{III}_{1}\lesssim\gamma^{2}|B_{i}|$. The lemma follows. ∎

The “good-lambda” distributional inequality (4.2) can be used to derive the
$L^{p}-L^{p}$ boundedness result.

######  Lemma 4.6.

Let $p>1$ and $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,M,κ. For
any a weak solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, any cut-off
function in the form $\Psi:=\Psi_{B,l,\epsilon}$ for some $\epsilon>0$, some
$l>100\epsilon$, and some boundary ball $B$ of radius $l$, we have

$\|\widetilde{N}(\nabla u|\Psi^{3})\|^{p}_{p}\leq
C_{p}\bigg{\|}\bigg{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}ds\,dy\bigg{)}^{1/2}\bigg{\|}^{p}_{p}+C_{p}\kappa^{p/2}\|\widetilde{N}(\nabla
u|\Psi)\|^{p}_{p}\\\ +C_{p}\|S(\overline{\nabla}u|\Psi)\|^{p}_{p}$

where $C_{p}>0$ depends only on $\lambda$, $d$, $n$, and $p$.

######  Remark 4.7.

The limitation $p>1$ comes from the fact that we used the maximal function
$\mathcal{M}$ in Lemma 4.5. However, with the same arguments, we could prove
an analogue of (4.2) where we replace $\mathcal{M}$ by $\mathcal{M}_{q}$
defined as $\mathcal{M}_{q}[f]:=\big{(}\mathcal{M}[f^{q}]\big{)}^{1/q}$ for
any $q>0$ (with a constant $C$ depending now also on $q$). Then we could
establish Lemma 4.6 for any $p>0$ by invoking (4.2) that used
$\mathcal{M}_{q}$ with $0<q<p$.

###### Proof.

We apply the distribution inequality (4.2) to obtain that there exists a
$\eta>0$ such that for any $\gamma,\delta\in(0,1)$, we have

$\begin{split}\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{p}_{p}&=c_{p}\int_{0}^{\infty}\beta^{p-1}|\\{\widetilde{N}(\nabla
u|\Psi^{3})>\beta\\}|d\beta\\\ &\leq
c_{p}\int_{0}^{\infty}\beta^{p-1}|\\{\widetilde{N}(\nabla
u|\Psi^{3})>\beta\\}\cap
E_{\beta,\gamma,\delta}|d\beta+c_{p}\int_{0}^{\infty}\beta^{p-1}|E^{c}_{\beta,\gamma,\delta}|d\beta\\\
&\lesssim
c_{p}\gamma^{2}\int_{0}^{\infty}\beta^{p-1}|\\{\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})]>\eta\beta\\}|d\beta+c_{p}\int_{0}^{\infty}\beta^{p-1}|E^{c}_{\beta,\gamma,\delta}|d\beta\\\
:=\textup{I}+\textup{II}\end{split}$

where the implicit constant depends only on $p$. But in one had, we have

$\textup{I}=\frac{\gamma^{2}}{\eta^{p}}\|\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})]\|^{p}_{p}\lesssim\frac{\gamma^{2}}{\eta^{p}}\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{p}_{p}$

by the $L^{p}$-boundedness of the Hardy-Littlewood maximal operator. On the
other hand,

$\begin{split}\textup{II}&=\gamma^{-p}\Big{\|}\mathcal{M}\Big{[}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{1/2}\Big{]}+\delta^{1/2}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi^{3})]\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\delta^{-1/2}\kappa^{1/2}\mathcal{M}[\widetilde{N}(\nabla
u|\Psi)]+\delta^{-1/2}\mathcal{M}[S(\overline{\nabla}u|\Psi)]\Big{\|}_{p}^{p}\\\
&\lesssim\gamma^{-p}\Big{\|}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{1/2}+\delta^{1/2}\widetilde{N}(\nabla
u|\Psi^{3})+\delta^{-1/2}\kappa^{1/2}\widetilde{N}(\nabla u|\Psi)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\delta^{-1/2}S(\overline{\nabla}u|\Psi)\Big{\|}_{p}^{p}\end{split}$

again using the $L^{p}$-boundedness of the Hardy-Littlewood maximal operator.
Altogether, we have

$\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{p}_{p}\lesssim\Big{(}\frac{\gamma^{2}}{\eta^{p}}+\frac{\delta^{p/2}}{\gamma^{p}}\Big{)}\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{p}_{p}+\gamma^{-p}\Big{\|}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{l}(.)}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{l\leq|s|\leq 2l}|\nabla
u|^{2}\Psi_{B}^{3}\,ds\,dy\Big{)}^{1/2}\Big{\|}_{p}^{p}\\\
+\delta^{-p/2}\kappa^{p/2}\gamma^{-p}\|\widetilde{N}(\nabla
u|\Psi)\|^{p}_{p}+\delta^{-p/2}\gamma^{-p}\Big{\|}S(\overline{\nabla}u|\Psi)\Big{\|}_{p}^{p}.$

The lemma follows by taking $\gamma$ and then $\delta$ (depending only on
$\lambda$, $d$, $n$, and $p$) such that
$(\gamma^{2}\eta^{-p}+\delta^{p/2}\gamma^{-p})$ is small enough, so that the
first term on the right-hand side above can be hidden in the left-hand side
(which is allowed because all the terms are finite, due to the use of the
compactly supported cut-off function $\Psi$). ∎

##  5\. $S\leq N$ Local Estimates

In this section, we aim to establish that the square function is locally
bounded by the non-tangential maximal function, result that is eventually
given in Lemma 5.5 below.

Remember that we have three different directional derivatives to deal with,
which are the tangential derivatives $\nabla_{x}$, angular derivatives
$\nabla_{\varphi}$, and radial derivative $\partial_{r}$. To prove these
estimates, we first bound the square function of the radial derivative by the
square functions of the tangential and angular derivatives, and we shall rely
on Proposition 2.3, i.e. the expression of the equation in cylindrical
coordinates. Then, we treat the tangential and angular directional
derivatives, and a key point is the fact that those derivatives verify
$\partial_{r}|t|=\partial_{\varphi}|t|=0$.

######  Lemma 5.1.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$ and any radial cut-off
function $\Psi\in C^{\infty}_{0}(\Omega,[0,1])$, we have

(5.1) $\|S(\partial_{r}u|\Psi^{3})\|^{2}_{2}\leq
C\Big{(}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}+\|S(\nabla_{\varphi}u|\Psi^{3})\|^{2}_{2}\Big{)},$

where the constant $C>0$ depends only on $\lambda$ and the dimensions $d$ and
$n$.

###### Proof.

This is basically an outcome of the equation: some derivatives can be
represented in terms of others. Observe that

$|\nabla\partial_{r}u|\leq|\nabla_{x}\partial_{r}u|+|\nabla_{\varphi}\partial_{r}u|+|\partial_{r}^{2}u|\leq|\nabla\nabla_{x}u|+|\nabla\nabla_{\varphi}u|+\frac{1}{|t|}|\nabla_{\varphi}u|+|\partial_{r}^{2}u|$

because $\nabla_{x}$ and $\partial_{r}$ commute and the commutator
$[\partial_{r},\partial_{\varphi}]$ is $-\frac{1}{|t|}\partial_{\varphi}$ (see
Proposition 2.4). But since $u$ is a weak solution to $Lu=0$, 2.1 implies that

$\begin{split}|\partial_{r}^{2}u|&=\Big{|}-\operatorname{div}_{x}(\mathcal{A}_{1}\nabla_{x}u)-\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{r}u)-\frac{1}{2}\sum_{i,j=d+1}^{n}\partial_{\varphi_{ij}}^{2}u\Big{|}\\\
&\lesssim|\nabla_{x}\mathcal{A}||\nabla
u|+\lambda^{-1}|\nabla_{x}\partial_{r}u|+\lambda^{-1}|\nabla_{x}\nabla_{x}u|+|\nabla_{\varphi}^{2}u|\leq|\nabla_{x}\mathcal{A}||\nabla
u|+\lambda^{-1}|\nabla\nabla_{x}u|+|\nabla\nabla_{\varphi}u|\end{split}$

by using again the fact that $\nabla_{x}$ and $\partial_{r}$ commute. By
combining the two inequalities above, we obtain

$|\nabla\partial_{r}u|\lesssim|\nabla\nabla_{x}u|+|\nabla\nabla_{\varphi}u|+\frac{1}{|t|}|\nabla_{\varphi}u|+|\nabla_{x}\mathcal{A}||\nabla
u|$

Now, (3.2) entails that

$\|S(\partial_{r}u|\Psi^{3})\|_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\|S(\nabla_{x}u|\Psi^{3})\|_{L^{2}(\mathbb{R}^{d})}^{2}+\|S(\nabla_{\varphi}u|\Psi^{3})\|_{L^{2}(\mathbb{R}^{d})}^{2}\\\
+\iint_{\Omega}|\nabla_{\varphi}u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}+\iint_{\Omega}|t|^{2}|\nabla_{x}\mathcal{A}|^{2}|\nabla
u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}.$

However, since $|t||\nabla_{x}\mathcal{A}|\in CM(\kappa)$, the Carleson
inequality (3.3) implies that

$\iint_{\Omega}|t|^{2}|\nabla_{x}\mathcal{A}|^{2}|\nabla
u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}\lesssim\kappa\|\widetilde{N}(u|\Psi^{3})\|^{2}_{L^{2}}.$

In addition, Proposition 2.7 applied with $\Phi=|t|^{d-n}\Psi^{3}$ infers that

$\iint_{\Omega}|\nabla_{\varphi}u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}\lesssim\iint_{\Omega}|\nabla\nabla_{\varphi}u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\lesssim\|S(\nabla_{\varphi}u|\Psi^{3})\|^{2}_{L^{2}(\mathbb{R}^{d})}$

by (3.2). The lemma follows. ∎

In order to deal with the tangential and angular directional derivatives, we
will first prove a generalized result that works for both of them. Let us
write $\partial_{v}$ for either a tangential derivative $\partial_{x_{i}}$, an
angular derivative $\partial_{\varphi_{ij}}$, or the radial derivative
$\partial_{r}$. The key step is to use the equation $Lu=0$ properly. Since we
want to estimate the gradient of solutions, we should study the commutators
$[L,\partial_{v}]$ and try to bound them in a clever way. In the next lemma,
we will estimate the square function of $\partial_{v}u$ and we are able to see
how the commutator $[L,\partial_{v}]$ plays an important role in the
estimates.

It will be convenient to introduce the bilinear form
$\mathcal{B}(\cdot,\cdot)$ defined for $f\in L^{1}_{loc}(\Omega)$ and $\Psi\in
C_{0}^{\infty}(\Omega)$,

(5.2)
$\displaystyle\mathcal{B}(f,\Psi):=-\frac{1}{2}\iint_{\Omega}\partial_{r}[|t|f]\,\partial_{r}\Psi\frac{dtdx}{|t|^{n-d-1}}.$

Beware that $\mathcal{B}(f,\Psi)$ may be negative even when the function $f$
is positive. We are now ready for our next lemma.

######  Lemma 5.2.

Let $L:=-\operatorname{div}(|t|^{d+1-n}\mathcal{A}\nabla)$ be an elliptic
operator satisfying (1.23) and (1.26). For any weak solution $u\in
W^{1,2}_{loc}(\Omega)$ to $Lu=0$ and any radial cut-off function $\Psi\in
C^{\infty}_{0}(\Omega,[0,1])$, we have

(5.3)
$\frac{7}{8}\lambda\|S(\partial_{v}u|\Psi^{3})\|_{2}^{2}\leq\iint_{\Omega}|\partial_{v}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+C\iint_{\Omega}|\partial_{v}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\\\
+\mathcal{B}(|\partial_{v}u|^{2},\Psi^{3})+\iint_{\Omega}\Big{(}[L,\partial_{v}]u\Big{)}\Big{(}\Psi^{3}|t|\partial_{v}u\Big{)}dtdx,$

where $C>0$ depends only on the ellipticity constant $\lambda$ and the
dimensions $d$ and $n$.

The bound (5.3) may look a bit cryptic. The last term of (5.3) is the one that
contains the commutator $[L,\partial_{v}]$, and will be removed in the next
lemmas. The first term in the right-hand side is the “trace” term, that is the
term that will become $\operatorname{Tr}(\partial_{v}u)$ when we take
$\Psi\uparrow 1$. The two other quantities are “error” terms that contain
derivatives of the cut-off function $\Psi$, and that will eventually disappear
when we take $\Psi\uparrow 1$.

###### Proof.

To lighten the notation, we write $V$ for $\partial_{v}u$. First of all, since
$\mathcal{A}$ satisfies the uniform ellipticity condition (1.23), we have

$\displaystyle\lambda\|S(V|\Psi^{3})\|^{2}_{2}=\lambda\iint_{\Omega}|\nabla
V|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}\leq\iint_{\Omega}\mathcal{A}\nabla
V\cdot\nabla V\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}$

By product rule,

$\iint_{\Omega}\mathcal{A}\nabla V\cdot\nabla
V\,\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}=\iint_{\Omega}\mathcal{A}\nabla
V\cdot\nabla\Big{(}V\Psi^{3}|t|\Big{)}\frac{dtdx}{|t|^{n-d-1}}\\\
-\iint_{\Omega}\mathcal{A}\nabla
V\cdot\nabla\Psi^{3}\,V\frac{dtdx}{|t|^{n-d-2}}-\iint_{\Omega}\mathcal{A}\nabla
V\cdot\nabla(|t|)\,V\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}=\textup{I}+\textup{II}+\textup{III}.$

We start from the term I. Recall that $u$ is a weak solution to the equation
$Lu=0$. It follows that:

$\displaystyle
LV=L(\partial_{v}u)=\partial_{v}(Lu)+[L,\partial_{v}]=[L,\partial_{v}]\
\text{a.e. in $\Omega$.}$

Consequently,

$\displaystyle\textup{I}=\iint_{\Omega}\Big{(}[L,\partial_{v}]u\Big{)}\Big{(}V\Psi^{3}|t|\Big{)}\,dtdx,$

which is one of the term from the right-hand side of (5.3). For the term II,
since matrix is in the form of (1.26),

$\textup{II}=-\iint_{\Omega}\mathcal{A}_{1}\nabla_{x}V\cdot\nabla_{x}\Psi^{3}\,V\frac{dtdx}{|t|^{n-d-2}}-\iint_{\Omega}\mathcal{A}_{2}\frac{t}{|t|}\nabla_{t}V\cdot\nabla_{x}\Psi^{3}\,V\frac{dtdx}{|t|^{n-d-2}}\\\
-\iint_{\Omega}\nabla_{t}V\cdot\nabla_{t}\Psi^{3}\,V\frac{dtdx}{|t|^{n-d-2}}=:\textup{II}_{1}+\textup{II}_{2}+\textup{II}_{3}.$

The terms $\textup{II}_{1}$ and $\textup{II}_{2}$ are estimated together by

$\displaystyle\textup{II}_{1}+\textup{II}_{2}\leq\frac{1}{8}\lambda\|S(V|\Psi^{3})\|^{2}_{2}+C_{\lambda}\iint_{\Omega}|V|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}.$

We hide the term $\frac{1}{8}\lambda\|S(V|\Psi^{3})\|^{2}_{2}$ in the left-
hand side of (5.3), and the second term in the right-hand side above stays on
the right-hand side of (5.3).

As for $\textup{II}_{3}$, since $\Psi$ is radial, we have that

$\nabla_{t}V\cdot\nabla_{t}\Psi^{3}=(\nabla_{t}V\cdot\nabla_{t}|t|)\,\partial_{r}\Psi^{3}=(\partial_{r}V)(\partial_{r}\Psi^{3})$

and thus,

$\textup{II}_{3}=-\frac{1}{2}\iint_{\Omega}(\partial_{r}V^{2})(\partial_{r}\Psi^{3})\,\frac{dtdx}{|t|^{n-d-2}}=\mathcal{B}(V^{2},\Psi^{k})+\frac{1}{2}\iint_{\Omega}V^{2}\partial_{r}\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}$

by definition of $\mathcal{B}(.,.)$, see (5.2). The last term III is similar,
because we have

$\textup{III}=-\iint_{\Omega}\partial_{r}V\,V\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}=-\frac{1}{2}\iint_{\Omega}(\partial_{r}V^{2})\,\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}=\frac{1}{2}\iint_{\Omega}V^{2}\,\partial_{r}\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}$

thanks to the integration by parts given in Proposition 2.2. The lemma
follows. ∎

Now we bound the square function of the tangential derivatives by applying
Lemma 5.2 with $\partial_{v}=\partial_{x}$. Recall that we write
$\partial_{x}$ for any tangential directional derivative
$\partial_{i}:=\vec{e}_{i}\cdot\nabla$ where $i\leq d$. As we have discussed
in the previous paragraphs, the commutator $[L,\partial_{x}]$ plays an
important role in computing the square function of $\partial_{x}$. In our
particular case, an easy computation shows that

(5.4)
$[L,\partial_{x}]=\operatorname{div}_{x}(|t|^{d+1-n}\partial_{x}\mathcal{A})\nabla$

because $\partial_{x}|t|=0$.

######  Corollary 5.3.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$ and any radial cut-off
function $\Psi\in C^{\infty}_{0}(\Omega,[0,1])$, we have

(5.5)
$\frac{3}{4}\lambda\|S(\nabla_{x}u|\Psi^{3})\|_{2}^{2}\leq\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}\\\
+C\iint_{\Omega}|\nabla_{x}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}+\mathcal{B}(|\nabla_{x}u|^{2},\Psi^{3}),$

where $C>0$ depends only on $\lambda$, $d$, and $n$.

###### Proof.

The bound (5.5) is a consequence of the same bound on each of the tangential
derivative $\partial_{x}$, and then summing up. For a given tangential
derivative, (5.5) is an immediate consequence of Lemma 5.2 and the bound

(5.6)
$\left|\iint_{\Omega}\Big{(}[L,\partial_{x}]u\Big{)}\big{(}\Psi^{3}|t|\partial_{x}u\big{)}dtdx\right|\leq\frac{1}{8}\lambda\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}+C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}\\\
+C\iint_{\Omega}|\nabla_{\varphi}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}.$

for any tangential derivative $\partial_{x}$. So we fix a tangential
directional derivative $\partial_{x}$, and by (5.4) and then integration by
parts, we have

(5.7)
$\iint_{\Omega}\Big{(}[L,\partial_{x}]u\Big{)}\Big{(}(\partial_{x}u)\Psi^{3}|t|\Big{)}dtdx=\iint_{\Omega}\Big{(}\operatorname{div}(|t|^{d+1-n}\partial_{x}\mathcal{A})\nabla
u\Big{)}\Big{(}\Psi^{3}|t|\partial_{x}u\Big{)}dtdx\\\
=-\iint_{\Omega}(\partial_{x}\mathcal{A})\nabla
u\cdot\nabla(\partial_{x}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}-\iint_{\Omega}(\partial_{x}\mathcal{A})\nabla
u\cdot\nabla\big{(}|t|\Psi^{3}\big{)}\partial_{x}u\frac{dtdx}{|t|^{n-d-1}}\\\
=\textup{I}+\textup{II}.$

Since $|t||\nabla_{x}\mathcal{A}|\in CM(\kappa)$, the term I is bounded as
follows

(5.8)
$|\textup{I}|\leq\frac{1}{8}\lambda\|S(\partial_{x}u|\Psi^{3})\|^{2}_{2}+C_{\lambda}\int_{\Omega}|t|^{2}|\nabla_{x}\mathcal{A}|^{2}|\nabla
u|^{2}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-1}}\\\
\leq\frac{1}{8}\lambda\|S(\partial_{x}u|\Psi^{3})\|^{2}_{2}+C_{\lambda}\kappa\|\widetilde{N}_{a}(\nabla
u|\Psi^{3})\|^{2}_{2}.$

For II, remark that the special structure of $\mathcal{A}$ given in (1.26)
implies that the only derivatives that hit $|t|\Psi^{3}$ are tangential
derivative, for which $\nabla_{x}|t|=0$. Therefore,

(5.9) $|\textup{II}|=\left|\iint_{\Omega}(\partial_{x}\mathcal{A})\nabla
u\cdot(\nabla_{x}\Psi^{3})(\partial_{x}u)\frac{dtdx}{|t|^{n-d-2}}\right|\\\
\leq\iint_{\Omega}|t|^{2}|\partial_{x}\mathcal{A}|^{2}|\nabla
u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}+\iint_{\Omega}|\nabla_{x}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\\\
\leq C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla_{x}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}.$

The lemma follows. ∎

It remains to estimate the square function of the angular directional
derivatives.

######  Corollary 5.4.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$ and any radial cut-off
function $\Psi\in C^{\infty}_{0}(\Omega,[0,1])$, we have

(5.10) $\frac{3}{4}\lambda\|S(\nabla_{\varphi}u|\Psi^{3})\|_{2}^{2}\leq
C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+C\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}\\\
+C\iint_{\Omega}|\nabla_{\varphi}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}+\mathcal{B}(|\nabla_{\varphi}u|^{2},\Psi^{3}),$

where $C>0$ depends only on $\lambda$, $d$, and $n$.

###### Proof.

Fix an angular directional derivative $\partial_{\varphi}$. Thanks to Lemma
5.2, it suffices to show that

(5.11)
$\left|\iint_{\Omega}\Big{(}[L,\partial_{\varphi}]u\Big{)}\big{(}\Psi^{3}|t|\partial_{\varphi}u\big{)}dtdx+\iint_{\Omega}|\partial_{\varphi}u|^{2}\partial_{r}\Psi^{3}\,\frac{dt\,dx}{|t|^{n-d-1}}\right|\\\
\leq\frac{1}{8}\lambda\|S(\nabla_{\varphi}u|\Psi^{3})\|^{2}_{2}+C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+C\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}\\\
+C\iint_{\Omega}|\nabla_{x}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}.$

It will be important to estimate the two terms in the left-hand side of (5.11)
together, because there will be some cancellation.

We invoke Proposition 2.6 to say that

(5.12)
$\iint_{\Omega}\Big{(}[L,\partial_{\varphi}]u\Big{)}\big{(}\Psi^{3}|t|\partial_{\varphi}u\big{)}dtdx=\iint_{\Omega}\Big{(}\operatorname{div}_{x}(|t|^{d+1-n}\partial_{\varphi}\mathcal{A})\nabla
u\Big{)}\big{(}\Psi^{3}|t|\partial_{x}u\big{)}dtdx\\\
+2\iint_{\Omega}(\partial_{r}\partial_{\varphi}u)\Psi^{3}(\partial_{\varphi}u)\frac{dt\,dx}{|t|^{n-d-1}}+\iint_{\Omega}\operatorname{div}_{x}(\mathcal{A}_{2}\partial_{\varphi}u)\big{(}\Psi^{3}\partial_{\varphi}u\big{)}\frac{dt\,dx}{|t|^{n-d-2}}\\\
=:\textup{I}+\textup{II}+\textup{III}.$

By the product rule,

$\textup{III}\lesssim\iint_{\Omega}|\nabla_{x}\mathcal{A}_{2}||\nabla_{\varphi}u||\partial_{\varphi}u|\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}+\iint_{\Omega}|\mathcal{A}_{2}|\nabla_{\varphi}\nabla_{x}u||\partial_{\varphi}u|\Psi^{3}\,\frac{dtdx}{|t|^{n-d-1}}:=\textup{III}_{1}+\textup{III}_{2}.$

Since $|t||\nabla_{x}\mathcal{A}_{2}|\in CM(\kappa)$, the term
$\textup{III}_{1}$ can be estimated as follows

$\displaystyle\textup{III}_{1}\leq\epsilon\iint_{\Omega}|\nabla_{\varphi}u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}+C\epsilon^{-1}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}\leq\frac{1}{24}\lambda\|S(\partial_{\varphi}u|\Psi^{3})\|_{2}^{2}+C_{\lambda}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2},$

by using Proposition 2.7, (3.2), and by taking $\epsilon$ small enough
(depending only on $\lambda$, $d$ and $n$).

Based on the same arguments, the term $\textup{III}_{2}$ is bounded by

$\displaystyle\textup{III}_{2}\leq\epsilon\iint_{\Omega}|\nabla_{\varphi}u|^{2}\Psi^{3}\frac{dtdx}{|t|^{n-d}}+C\epsilon^{-1}\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}\leq\frac{1}{24}\lambda\|S(\partial_{\varphi}u|\Psi^{3})\|_{2}^{2}+C_{\lambda}\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}.$

The term I is analogous to the one obtained from the commutator in the proof
of Corollary 5.3. We repeat quickly the argument. By integration by parts,

$\textup{I}=-\iint_{\Omega}(\partial_{\varphi}\mathcal{A})\nabla
u\cdot\nabla_{x}(\partial_{\varphi}u)\Psi^{3}\frac{dtdx}{|t|^{n-d-2}}-\iint_{\Omega}(\partial_{\varphi}\mathcal{A})\nabla
u\cdot\nabla_{x}\Psi^{3}\,(\partial_{\varphi}u)\,\frac{dtdx}{|t|^{n-d-1}}:=\textup{I}_{1}+\textup{I}_{2}.$

The first integral is bounded by using the inequlity $2ab\leq\epsilon
a^{2}+\epsilon^{-1}b^{2}$, and the fact that $|t||\nabla_{\varphi}A|\in
CM(\kappa)$ we get similarly to (5.8) that

$|\textup{I}_{1}|\leq\frac{1}{24}\lambda\|S(\partial_{\varphi}u|\Psi^{3})\|_{2}^{2}+C_{\lambda}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|_{2}^{2}.$

As for the second integral, we proceed as in (5.9) and we obtain

$|\textup{I}_{2}|\leq C\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla_{\varphi}u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}.$

The term II cancels out with the “trace” term. Indeed, we have

$\textup{II}=\iint_{\Omega}\partial_{r}|\partial_{\varphi}u|^{2}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-1}}=-\iint_{\Omega}|\partial_{\varphi}u|^{2}\partial_{r}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-1}}$

by the integration by parts (Proposition 2.2). Observe that all our
computations proved the claim (5.11), thus the lemma follows. ∎

In the following, we combine all the previous results of this section
together. We recall that $\overline{\nabla}$ stands for the gradient in
cylindrical coordinates. Remember that we write respectively
$\|S(\nabla_{x}u|\Psi^{3})\|^{2}_{2}$ and
$\|S(\nabla_{\varphi}u|\Psi^{3})\|^{2}_{2}$ for the sums of the square
functions over all tangential derivatives and angular derivatives in $L^{2}$
norm.

######  Lemma 5.5.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. There exists
three constants $C_{1},C_{2},C_{3}>0$ depending only on $\lambda$, $d$, and
$n$ such that for any weak solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$ and
any cut-off radial function $\Psi\in C^{\infty}_{0}(\Omega,[0,1])$, we have

(5.13) $\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2}\leq
C_{1}\bigg{(}\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+\mathcal{B}(|\nabla_{x}u|^{2},\Psi^{3})\bigg{)}+C_{2}\,\mathcal{B}(|\nabla_{\varphi}u|^{2},\Psi^{3})\\\
+C_{3}\bigg{(}\kappa\|\widetilde{N}_{a}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\bigg{)}$

In addition, if $\Psi$ satisfies ($\mathcal{COF}$)K, we have

(5.14) $\displaystyle\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2}\leq
C_{K}\|\widetilde{N}(\nabla u|\Psi)\|^{2}_{2},$

where $C_{K}$ depends on $\lambda$, $n$, and $K$.

######  Remark 5.6.

Remember that the first term in the right-hand side of (5.13) is the “trace”
term, and all the other terms are meant to disappear when $\Psi\uparrow 1$.

Moreover, we have different constants because the terms that are multiplied by
$C_{1}$ and $C_{2}$ may be negative. We can say that $1\leq C_{2}\leq
C_{1}\leq C_{3}$, but nothing more, in particular taking $C_{1}=C_{2}=C_{3}$
would probably render the inequality false.

######  Remark 5.7.

The result (5.14) tells us that the sum of the square functions of all
tangential directional derivatives, angular directional derivatives, and
radial direction derivatives can be estimated locally by the non-tangential
maximal function of the full gradient.

###### Proof.

The inequality (5.13) is an immediate consequence of Lemma 5.1, Corollary 5.3,
and Corollary 5.4.

We turn to the proof of (5.14). Since $\Psi$ satisfies ($\mathcal{COF}$)K, we
have

(5.15) $|t||\nabla\Psi|\leq K\quad\text{ and
}\quad{\mathds{1}}_{\operatorname{supp}\nabla\Psi}\in CM(K),$

in particular $|t||\nabla\Psi|\in CM(K^{3})$. We deduce that

(5.16) $\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}+\iint_{\Omega}|\nabla_{x}u|^{2}|\partial_{r}\Psi^{3}|\frac{dtdx}{|t|^{n-d-1}}\\\
\leq C\iint_{\Omega}|\nabla
u|^{2}\big{[}|t|^{2}|\nabla\Psi|^{2}+|t||\nabla\Psi|\big{]}\Psi\frac{dtdx}{|t|^{n-d}}\leq
CK^{3}\|\widetilde{N}(\nabla u|\Psi)\|^{2}_{2}.$

by (5.15) and the Carleson inequality (3.3).

Consequently, it suffices to show for any $V\in L^{2}_{loc}(\Omega)$ and
$\delta\in(0,\infty)$,

(5.17)
$\displaystyle|\mathcal{B}(|V|^{2},\Psi^{3})|\leq\delta\|S(V|\Psi^{3})\|^{2}_{2}+C\delta^{-1}\|\widetilde{N}(V|\Psi)\|^{2}_{2}$

because then (5.14) follows easily by choosing $\epsilon$ small enough. From
the definition of $\mathcal{B}(.,.)$, see (5.2), and the product rule, we have

(5.18)
$|\mathcal{B}(|V|^{2},\Psi^{3})|\leq\iint_{\Omega}|V||\partial_{r}V||\partial_{r}\Psi|\Psi^{2}\frac{dtdx}{|t|^{n-d-2}}+\frac{1}{2}\iint_{\Omega}|V|^{2}|\partial_{r}\Psi|\Psi^{2}\frac{dtdx}{|t|^{n-d-1}}\\\
\leq\delta\|S(V|\Psi^{3})\|^{2}_{2}+C\delta^{-1}\iint_{\Omega}|V|^{2}|\partial_{r}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}+\frac{1}{2}\iint_{\Omega}|V|^{2}|\partial_{r}\Psi|\Psi^{2}\frac{dtdx}{|t|^{n-d-1}}.$

By using again (5.15) and the Carleson inequality, the last two terms of
(5.18) are bounded by $K^{3}\|\widetilde{N}(V|\Psi)\|^{2}_{2}$. Hence (5.17)
follows. ∎

Let us get a little bit further, since it will help us when we pass from local
to global estimates.

######  Lemma 5.8.

Let $L$ be an elliptic operator satisfying ($\mathcal{H}$)λ,κ. For any weak
solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, any cut-off function $\Psi\in
C^{\infty}_{0}(\Omega,[0,1])$ satisfying ($\mathcal{COF}$)K, any
$\delta\in(0,1)$, we have

$|\mathcal{B}(|\partial_{r}u|^{2},\Psi^{3})|=\frac{1}{2}\left|\iint_{\Omega}\Big{(}|\partial_{r}u|^{2}+|t|\partial_{r}(|\partial_{r}u|^{2})\Big{)}\partial_{r}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-1}}\right|\\\
\leq(\delta+\delta^{-1}K^{2}\kappa)\|\widetilde{N}(\nabla
u|\Psi)\|_{2}^{2}+C\delta^{-1}K^{2}\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2},$

where $C$ depends on $\lambda$, $n$, $\delta$ and $K+M+\kappa$.

###### Proof.

The equality is just the product rule and the definition of
$\mathcal{B}(|\partial_{r}u|^{2},\Psi^{3})$, see (5.2). The bound

$\left|\iint_{\Omega}|\partial_{r}u|^{2}\partial_{r}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-1}}\right|\leq(\delta+\delta^{-1}K^{2}\kappa)\|\widetilde{N}(\nabla
u|\Psi)\|_{2}^{2}+C\delta^{-1}K^{2}\|S(\overline{\nabla}u|\Psi^{3})\|_{2}^{2}$

was established in (3.6). It remains to show a bound on

$\textup{I}:=\left|\iint_{\Omega}\partial_{r}(|\partial_{r}u|^{2})\partial_{r}\Psi^{3}\,\frac{dt\,dx}{|t|^{n-d-2}}\right|,$

but that is similar to (5.18). Indeed, we use $|t||\nabla\Psi|\in CM(K)$,
(3.3), and the inequality $ab\leq\delta a^{2}+b^{2}/4\delta$ to obtain

$I=2\left|\iint_{\Omega}(\partial_{r}^{2}u)(\partial_{r}u)\partial_{r}\Psi^{3}\frac{dt\,dx}{|t|^{n-d-2}}\right|\leq\delta\|\widetilde{N}(\partial_{r}u|\Psi)\|_{2}^{2}+CK\delta^{-1}\|S(\partial_{r}u|\Psi^{3})\|_{2}^{2}.$

The lemma follows. ∎

######  Lemma 5.9.

Let $\kappa\in(0,1)$ and let $L$ be an elliptic operator satisfying
($\mathcal{H}$)λ,κ. Take $\Psi\in C^{\infty}_{0}(\Omega)$ satisfying
($\mathcal{COF}$)K. There exist four constants $c_{0},C_{1},C_{2},C_{3}>0$
depending on $\lambda$ and $n$ \- the first one being small and the last three
being large - such that, after defining

$|\nabla
u|^{2}_{\kappa}:=C_{1}|\nabla_{x}u|^{2}+C_{2}|\nabla_{\varphi}u|^{2}+\kappa^{1/2}c_{0}K^{-2}|\partial_{r}u|^{2},$

we have, for any weak solution $u\in W^{1,2}_{loc}(\Omega)$ to $Lu=0$, that

(5.19) $\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2}\leq
C_{1}\iint_{\Omega}|\nabla_{x}u|^{2}\partial_{r}\Psi^{3}\frac{dtdx}{|t|^{n-d-1}}+\mathcal{B}(|\nabla
u|^{2}_{\kappa},\Psi^{3})\\\ +C_{3}\bigg{(}\kappa\|\widetilde{N}(\nabla
u|\Psi^{3})\|^{2}_{2}+\iint_{\Omega}|\nabla
u|^{2}|\nabla_{x}\Psi|^{2}\Psi\frac{dtdx}{|t|^{n-d-2}}\bigg{)}.$

###### Proof.

By Lemma 5.5, we have three constants
$C^{\prime}_{1},C^{\prime}_{2},C^{\prime}_{3}$ depending only on $\lambda$,
$d$, and $n$ such that

(5.20) $\|S(\overline{\nabla}u|\Psi^{3})\|^{2}_{2}\leq