Optimal Regularity of Solutions to No-Sign Obstacle-Type Problems for the Sub-Laplacian

TL;DR

This paper proves the optimal interior regularity of solutions to a sub-Laplacian obstacle problem in stratified groups under minimal regularity assumptions on the data, extending classical Euclidean results to a subelliptic setting.

Contribution

It establishes the sharp $C_{H}^{1,1}$ regularity of solutions to obstacle-type problems for the sub-Laplacian in stratified groups, generalizing previous Euclidean and special case results.

Findings

Proves $C_{H}^{1,1}$ regularity under minimal assumptions on $f$

Recovers known regularity results for obstacle problems in stratified groups

Extends classical Euclidean regularity results to subelliptic setting

Abstract

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ \Delta_{H}u=f\chi_{\{u\ne0\}}, \] where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*\Gamma$ is $C_{H}^{1,1}$, where $\Gamma$ is the fundamental solution of $\Delta_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is nonnegative and satisfies the above equation we recover the $C_{H}^{1,1}$ regularity of solutions to the obstacle problem in stratified groups, that was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the $C^{1,1}$ regularity result due to Andersson, Lindgren and Shahgholian.

Equations

$${\Delta }_{H}u=f{\chi }_{\{u\ne 0\}},$$

$${\Delta }_{H}u=f{\chi }_{\{u\ne 0\}}$$

$$\{\begin{array}{ll}\Delta u=f{\chi }_{\{u>0\}}& \text{in }{B}_1,\\ u\ge 0& \text{in B1,}\\ u=g& \text{on }\partial B_1,\end{array}$$

$$\{\begin{array}{ll}\Delta u=f{\chi }_{\{u\ne 0\}}& \text{in }{B}_1,\\ u=g& \text{on }\partial B_1.\end{array}$$

$$\Vert D_h^{2}u{\Vert }_{L^{\infty }(B_{1/4})}\le C\left(\Vert D_h^2(f\ast \Gamma ){\Vert }_{L^{\infty }(B_1)}+\Vert u{\Vert }_{L^{\infty }(B_1)}\right).$$

$$\mathcal{G}=V_1\oplus V_2\oplus \cdots \oplus V_{\iota }\text{and}[V_1,V_j]=V_{j+1}$$

$$X_{m_{j-1}+1},\dots ,X_{m_j}$$

$$d_j:=k.$$

$$xy=x+y+BCH(x,y),$$

$${\delta }_{r}x=(r{x}_1,\dots ,r{x}_m,r^2{x}_{m+1},\dots ,r^2{x}_{m_2},\dots ,r^{\iota }{x}_{m_{\iota -1}+1},\dots ,r^{\iota }{x}_n)=\sum _{j=1}^n{r}^{d_j}{x}_j{e}_j$$

$${\delta }_r(xy)=({\delta }_{r}x)({\delta }_{r}y)$$

$$|{\delta }_r(A)|=r^Q|A|$$

$$\dot{\gamma }(t)=\sum _{i=1}^m{b}_i(t)X_i(\gamma (t))$$

$$d(x,y)=\inf\left\{T>0\Big{|}\gamma:[0,T]\to\mathbb{R}^{n},\ \gamma\in\mathcal{H}(x,y)\right\}$$

$$B_r(x)=x{\delta }_r(B_1).$$

$$p({\delta }_{r}x)=r^{k}p(x)\text{for all x∈Rn and r>0.}$$

$$X_j={\partial }_{x_j}+\sum _{i=m_{d_j}+1}^n{a}_{ji}{\partial }_{x_i}$$

$$\nabla_{h}u=(X_{1}u,\ldots,X_{m}u)\quad\text{and}\quad D_{h}^{2}u=\left(\begin{array}[]{cccc}X_{1}X_{1}u&X_{1}X_{2}u&\cdots&X_{1}X_{m}u\\ X_{2}X_{1}u&X_{2}X_{2}u&\cdots&X_{2}X_{m}u\\ \vdots&\vdots&\ddots&\vdots\\ X_{m}X_{1}u&\cdots&\cdots&X_{m}X_{m}u\end{array}\right),$$

$$X^{I}u:=X_1^{i_n}\cdots X_n^{i_1}u.$$

$$|X_{j}f(x)-X_{j}f(y)|\le Cd(x,y{)}^{\alpha }$$

$$D_h^{2,s}u=\frac{1}{2}\left(D_h^{2}u+D_h^2{u}^T\right).$$

$${\Delta }_{H}u=\sum _{j=1}^m{X}_j^{2}u.$$

$$c_0^{-1}{d}_G(x)\le d(x,0)\le c_0{d}_G(x)$$

$$B_r^G(x)=\left\{y\in {\mathbb{R}}^n:d_G(x,y)<r\right\}.$$

$$B_r^G(x)\subset B_{c_{0}r}(x)$$

$${\mathrm{dist}}_G({\Omega }^{\prime },{\Omega }^c):=\inf \left\{d_G(x,y):x\in {\Omega }^{\prime },y\in {\Omega }^c\right\}>h.$$

$$|X^I\vartheta (x)|\le C_{I,h}\Vert \vartheta {\Vert }_{L^1(\Omega )}$$

$$\vartheta (x)={\int }_{B_h^G(x)}{\varphi }_h(x^{-1}y)\vartheta (y)dy={\int }_{\Omega }{\hat{\varphi }}_h(y^{-1}x)\vartheta (y)dy.$$

$$|X^I\vartheta (x)|=\left|{\int }_{\Omega }{X}^I{\hat{\varphi }}_h(y^{-1}x)\vartheta (y)dy\right|\le \Vert X^I{\hat{\varphi }}_h{\Vert }_{L^{\infty }({\mathbb{R}}^n)}\Vert \vartheta {\Vert }_{L^1(\Omega )}$$

$${\int }_{\Omega }u(y)(X_{j_1}\cdots X_{j_k}\varphi )(y)dy=(-1{)}^k{\int }_{\Omega }{v}_{j_1,\dots ,j_k}(y)\varphi (y)dy$$

Full text

[TABLE]

where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely the group convolution $f*\Gamma$ is $C_{H}^{1,1}$, where $\Gamma$ is the fundamental solution of $\Delta_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is nonnegative and satisfies the above equation we recover the $C_{H}^{1,1}$ regularity of solutions to the obstacle problem in stratified groups, that was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the $C^{1,1}$ regularity result due to Andersson, Lindgren and Shahgholian.

V.M. acknowledges the support of the University of Pisa, Project PRA 2018 49. A.M. acknowledges the support of the Knut and Alice Wallenberg Foundation, Project KAW 2015.0380.

1. Introduction

The main question we consider in this paper is the optimal interior regularity of distributional solutions to the no-sign obstacle-type problem

[TABLE]

on some domain of a stratified group $\mathbb{G}$, see Section 2 for notation and terminology. In the Euclidean setting, the obstacle problem is among the most studied topics in the field of Free Boundary Problems, see for instance the monographs by Rodrigues [Rod87], Friedman [Fri88], and Petrosyan et al. [PSU12]. It asks which properties can be deduced about a function with given boundary values and that minimizes the Dirichlet energy, under the constraint of lying above a given function. This is the classical obstacle problem, that can be studied through the theory of variational inequalities, using the Dirichlet energy, see for instance [KS00, Fre72]. The variational approach, after subtracting the obstacle from the solution, leads to the following PDE formulation of the problem

[TABLE]

where $B_{1}$ denotes the metric unit ball with respect to the Carnot–Carathéodory distance (Definition 2.1). Our problem is a non-variational counterpart of (1.2), that is

[TABLE]

We point out that (1.3) — which is called a no-sign obstacle-type problem — naturally appears also when considering the so-called quadrature domains [Sak91, GS05].

Two important questions on this problem concern the regularity of solutions to (1.3), and the regularity of the free boundary. In Euclidean space, the analysis of both questions is essentially complete [Sak91, CKS00, PS07, ALS13]. In particular, in relation to the regularity of solutions, Andersson et al. [ALS13] show that $u$ has the optimal $C^{1,1}$ regularity if the linear problem $\Delta v=f$ has a $C^{1,1}$ solution. This is the minimal regularity assumption on $f$ in order to establish the $C^{1,1}$ regularity of solutions.

The main result of this paper is the sharp regularity of solutions to (1.1) also in the subelliptic setting of stratified groups.

Theorem 1.1** ($C_{H}^{1,1}$ regularity).**

Let $u\in L^{\infty}(B_{1})$ be a distributional solution to (1.1) in the unit ball $B_{1}$. Let $f:B_{1}\to\mathbb{R}$ be locally summable such that $f*\Gamma\in C_{H}^{1,1}(B_{1})$. Then there exists a universal constant $C>0$ such that, after a modification on a negligible set, we have $u\in C_{H}^{1,1}(B_{1/4})$ and

[TABLE]

In our setting, the natural counterpart of the Euclidean $C^{1,1}$ regularity is the $C^{1,1}_{H}$ regularity, where the horizontal derivatives are required to be Lipschitz continuous (Definition 2.2). The function $\Gamma$ denotes the fundamental solution of $\Delta_{H}$ (Definition 2.3). For further notation and terminology, we address the reader to Section 2.

We wish to emphasize that $u$ satisfies (1.1) also in the strong sense. Indeed, the distributional equality $\Delta_{H}(f*\Gamma)=-f$ joined with the assumed $C^{1,1}_{H}$ regularity of $f*\Gamma$ show that $f\in L^{\infty}(B_{1})$. Therefore $f\chi_{\{u\neq 0\}}\in L^{\infty}(B_{1})$ and by the regularity result of Folland, [Fol75, Theorem 6.1], we get $u\in W^{2,p}_{H,\text{loc}}(B_{1})$ for every $1\leq p<\infty$. The $C_{H}^{1,1}$ regularity of solutions to the obstacle problem in stratified groups was obtained by Danielli et al. [DGS03], using the variational formulation of the problem. The regularity of the free boundary was subsequently established in step two groups [DGP07]. Further results in this area have been obtained for Kolmogorov operators and parabolic non-divergence form operators of Hörmander type [DFPP08, FNPP10, FGN12, Fre13]. The no-sign obstacle-type problem in terms of the equation (1.1) does not seem to have been considered before in the subelliptic setting.

Our arguments are remarkably different from the ones used for the obstacle problem. For instance, in this problem without a forcing term the solution is automatically superharmonic with respect to $\Delta_{H}$, while in our setting we have no such sign condition that would yield a superharmonic solution. We initiate our analysis observing that second order horizontal derivatives of solutions to (1.1) satisfy certain $BMO$ estimates, that have been established by Bramanti et al. [BB05, BF13]. The subsequent step is to construct suitable approximating polynomials, starting from the second order horizontal derivatives of the solution. Indeed these polynomials yield a subquadratic growth estimate (4.12) at small scales. We point out that this estimate is valid for any bounded and $W^{2,p}$ regular function, with bounded sub-Laplacian, so it might be of independent interest. As a consequence, we perform a suitable rescaling of the equation and then infer the crucial decay estimate of the measure of the coincidence set (Proposition 4.6), when the horizontal Hessian of the approximating polynomial is sufficiently large. More details on this procedure can be found at the beginning of Section 4.

Although our ideas mainly follow the path set up by Andersson et al. [ALS13], and Figalli and Shahgholian [FS12], there are several difficulties related to the subelliptic setting. The basic one is concerned with the fact that the sub-Laplacian $\Delta_{H}$ is degenerate elliptic. In addition, since the operator $\Delta_{H}$ is written in terms of Hömander vector fields, we can only consider the horizontal Hessian (2.5) of the solution, that is a nonsymmetric matrix. Then the construction of the approximating polynomials starting from the average of the second order noncommuting derivatives $X_{i}X_{j}u$ becomes more delicate and requires some preliminary algebraic work, see Section 3. Notwithstanding the technical complications, the proof has become more streamlined: we can stay clear of the projection operator used in [ALS13], and this simplifies several technical points. A suitable quantitative decay estimate of the zero-level set (4.24) can be obtained also in our setting. Finally, we adapt Caffarelli’s polynomial iteration technique of [Caf89] to find explicit estimates of the second order horizontal derivatives, see (1.4).

We finish the introduction by giving an overview of the paper. In Section 2 we introduce some basic notions on stratified groups and the related function spaces. In Section 3 we construct suitable second order homogeneous polynomials (Definition 3.2), that have an assigned horizontal Hessian (Corollary 3.3). Then some important $W^{2,p}$ and $BMO$ estimates are presented. Finally, we provide the crucial scaling estimates of Lemma 3.8. In Section 4 we prove a subquadratic growth estimate of the difference between a solution and its approximating polynomial. Then we apply the subquadratic growth estimate to get a suitable decay of the measure of the zero-level set. Finally, we establish the $C_{H}^{1,1}$ regularity in quantitative terms, according to (1.4).

Acknowledgements. A substantial part of this work was accomplished during the time A.M. spent at Scuola Normale Superiore in Pisa, with the support of the Knut and Alice Wallenberg Foundation. V.M. gratefully thanks Henrik Shahgholian, along with KTH Royal Institute of Technology, for support and hospitality during December 2018, where important parts of the present work were carried out. He also acknowledges the Institutional Research Grant from the University of Pisa. The authors wish to thank Marco Bramanti for fruitful discussions concerning some a priori $BMO$ estimates. They are also indebted to the referee for several useful remarks.

2. Basic facts and notation

A stratified group is a simply connected, real nilpotent Lie group $\mathbb{G}$, whose Lie algebra $\mathcal{G}$ has a special stratification. We denote by $V_{i}$ the subspaces of $\mathcal{G}$, having the properties:

[TABLE]

for $j=1,\ldots,\iota$ and $V_{\iota+1}={0}.$ Let us denote by $n$ the topological dimension of $\mathbb{G}$ and by $m$ the dimension of $V_{1}$. We choose a *graded basis $X_{1},X_{2},\ldots,X_{n}$ *of $\mathcal{G}$, that is characterized by the property that

[TABLE]

is a basis of $V_{j}$ for all $j=1,\ldots,\iota$, where we have set $m_{0}=0$, $m_{1}=m$ and $m_{j}=\sum_{i=1}^{j}\dim V_{i}$. We notice that $m_{\iota}=n$ and with these definitions, if $m_{k-1}<j\leq m_{k}$, then $k\in\mathbb{N}$ is uniquely determined and we define the positive integer

[TABLE]

Through the exponential mapping of $\mathbb{G}$, one can construct a diffeomorphism from $\mathbb{R}^{n}$ to $\mathbb{G}$. Hence we have defined a *graded basis *$e_{1},e_{2},\ldots,e_{n}$ of $\mathbb{R}^{n}$ and graded coordinates $x_{1},x_{2},\ldots,x_{n}$ that define the point $x=(x_{1},x_{2},\ldots,x_{n})$ of $\mathbb{G}$. This allows us to identify $\mathbb{G}$ with $\mathbb{R}^{n}$, as it will be understood in the sequel.

In addition, one may also verify that the Lebesgue measure of $\mathbb{R}^{n}$ through the graded coordinates yields the Haar measure of the group $\mathbb{G}$. The notation $|A|$ denotes the Lebesgue measure of a measurable set $A\subset\mathbb{R}^{n}$.

The diffeomorphism associated to graded coordinates has also the property that the group operation on $\mathbb{G}$, when read in $\mathbb{R}^{n}$, is given by a special polynomial group operation

[TABLE]

where the precise form of the vector polynomial $BCH:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is given by the important Baker–Campbell–Hausdorff formula, in short BCH formula, see for instance [Var84]. The $degree$ of $x_{j}$ is the integer $d_{j}$ defined in (2.1) and we define intrinsic dilations as follows

[TABLE]

for any $r>0$. The notion of degree fits the algebraic properties of dilations, since

[TABLE]

for all $x,y\in\mathbb{R}^{n}$. By the form of dilations, for every measurable set $A\subset\mathbb{R}^{n}$ we have

[TABLE]

for all $r>0$, where $Q=\sum_{j=1}^{n}d_{j}$ can be proved to be the Hausdorff dimension of $\mathbb{G}$.

The metric structure of $\mathbb{R}^{n}$ is given by a control distance. We say that $\gamma:[0,T]\to\mathbb{R}^{n}$, an absolutely continuous curve, is *admissible *if for a.e. $t\in[0,1]$ there holds

[TABLE]

and $\sum_{i=1}^{m}b_{i}(t)^{2}\leq 1.$ We denote by $\mathcal{H}(x,y)$ the family of all admissible curves whose image contains ${x,y}$. By Chow’s theorem, $\mathcal{H}(x,y)$ is nonempty for every $x,y\in\mathbb{R}^{n}$, hence the following “control distance”

[TABLE]

is well defined. It is also possible to check that $d$ is actually a distance, corresponding to the well known Carnot–Carathéodory distance.

Since left translations preserve the “horizontal velocity”, $d$ is also left invariant, namely $d(x,y)=d(zx,zy)$ for all $x,y,z\in\mathbb{R}^{n}$. Furthermore, dilations are Lie group homomorphisms, hence the Carnot–Carathéodory distance is homogeneous in the sense that $d(\delta_{r}x,\delta_{r}y)=rd(x,y)$ for every $x,y\in\mathbb{R}^{n}$ and $r>0.$

Definition 2.1** (Metric balls).**

For $x\in\mathbb{R}^{n}$ and $r>0$, we denote by $B_{r}(x)$ the open ball of center $x$ and radius $r>0$ with respect to $d$. Precisely, this is the set $\{y\in\mathbb{R}^{n}:d(x,y)<r\}$. When $x=0$, we use the notation $B_{r}\vcentcolon=B_{r}(0)$.

From the properties of $d$ and $\delta_{r}$, it is easy to observe that

[TABLE]

Dilations also allow us to introduce a natural notion of homogeneity, so we may say that a polynomial $p:\mathbb{R}^{n}\to\mathbb{R}$ is $k$-homogeneous if

[TABLE]

The number $k\in\mathbb{N}$ is the degree of $p$. Moreover, any vector field $X_{j}$ of the fixed graded basis can be identified with a first order differential operator of the form

[TABLE]

for every $j=1,\ldots,n$. The functions $a_{ji}:\mathbb{R}^{n}\to\mathbb{R}$ are homogeneous polynomials of degree $d_{i}-d_{j}\geq 1$ and in particular $X_{j}(0)=e_{j}$ for all $j=1,\ldots,n$. In the sequel $\Omega$ will be understood as an open bounded subset of $\mathbb{G}$, that can be also identified with an open subset of $\mathbb{R}^{n}$, if not otherwise stated.

Given a function $u:\Omega\to\mathbb{R}$ and considering the vector fields $X_{j}$ as differential operators, we may introduce the horizontal gradient and the horizontal Hessian

[TABLE]

respectively, whenever they are pointwise defined. More generally, we can define higher order differential operators considering for $I=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n}$ the function

[TABLE]

Definition 2.2** (Folland–Stein spaces).**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. We denote by $C^{1}_{H}(\Omega)$ the space of all functions $u:\Omega\to\mathbb{R}$ such that the horizontal derivatives $X_{j}u$ exist on $\Omega$ for all $j=1,\ldots,m$ and are continuous. If $0<\alpha\leq 1$, then $C^{1,\alpha}_{H}(\Omega)$ is the space of functions $u$ in $C^{1}_{H}(\Omega)$ such that there exists $C>0$ with the property that

[TABLE]

for every $x,y\in\Omega$ and $j=1,\ldots,m$.

Notice that $D_{h}^{2}u$ is not symmetric, since the vector fields $X_{j}$ do not commute in general. We say that $X_{j}u$ are the *horizontal derivatives *and $X_{i}X_{j}u$ are the second order horizontal derivatives. The symmetrized horizontal Hessian is defined as

[TABLE]

The sub-Laplacian is defined as

[TABLE]

Functions satisfying $\Delta_{H}u=0$ are called as usual harmonic functions.

Definition 2.3** (Fundamental solution).**

We say that $\Gamma\in C^{\infty}(\mathbb{G}\operatorname{\setminus}\{0\})$ is a fundamental solution for $\Delta_{H}$ if it is locally summable, it vanishes at infinity and satisfies $\Delta_{H}\Gamma=-\delta_{0}$, where $\delta_{0}$ denotes the Dirac distribution centered at the origin.

The fundamental solution $\Gamma$ defines a gauge $d_{G}=\Gamma^{1/(2-Q)}$, that is 1-homogeneous with respect to dilations and continuous on $\mathbb{R}^{n}$. We can readily check that there exists a constant $c_{0}>1$ such that

[TABLE]

for all $x\in\mathbb{R}^{n}$. Defining $d_{G}(x,y)\vcentcolon=d_{G}(x^{-1}y)$ we also introduce the gauge ball

[TABLE]

The previous estimates clearly imply that

[TABLE]

for every $r>0$ and $x\in\mathbb{R}^{n}$.

Proposition 2.4**.**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $\vartheta$ be harmonic in $\Omega$. We consider an open set $\Omega^{\prime}\subset\Omega$ and $h>0$ such that

[TABLE]

Then $\vartheta\in C^{\infty}(\Omega)$ and for every multiindex $I$ there exists a constant $C_{I,h}>0$ such that

[TABLE]

Proof.

We consider the function $\phi$ defined in [BLU07, (5.50e)], where we choose $\varphi$ appearing in the definition of $\phi$, such that $\varphi\in C_{c}^{\infty}(]3^{-1},1[)$, $\varphi\geq 0$ and $\int_{\mathbb{R}}\varphi(t)dt=1$. It follows that $\phi$ is smooth and bounded on $\mathbb{R}^{n}$, along with all of its derivatives, and it is compactly supported in $B^{G}_{1}$, see the definition (2.7). We also consider $\phi_{r}(z)\vcentcolon=r^{-Q}\phi(\delta_{1/r}z)$, that is compactly supported on $B^{G}_{r}$. We finally set $\hat{\phi}_{r}(z)\vcentcolon=\phi_{r}(z^{-1})$ for all $z\in\mathbb{R}^{n}$ and $r>0$. Thus, using [BLU07, (5.50a),(5.50d)], for every $x\in B_{\lambda}$, we get

[TABLE]

We can differentiate the last integral an arbitrary number of times, due to the smoothness of $\hat{\phi}_{h}$, getting the smoothness of $\vartheta$ and the following estimate

[TABLE]

for all $x\in\Omega^{\prime}$. This concludes the proof. ∎

In our setting, we need the notion of Sobolev function adapted to the horizontal vector fields $X_{1},\ldots,X_{m}$, see [Fol75]. The horizontal Sobolev space $W_{H}^{k,p}(\Omega)$ consists of those functions $u\in L^{p}(\Omega)$ for which, for all $j_{s}\in\{1,\ldots,m\}$ and $s\in\{1,\ldots,k\}$, there exists a function $v_{j_{1},\ldots,j_{k}}\in L^{p}(\Omega)$ such that

[TABLE]

for any function $\phi\in C_{c}^{\infty}(\Omega)$. Also in the more general setting of Hörmander vector fields some Sobolev embedding theorems hold, see [GN96, Theorem 1.11 and (3.19)], or [Lu96, Theorem 1.1]. The next theorem specializes these embedding results for stratified groups.

Theorem 2.5**.**

Let $p>Q$, where $Q$ is the Hausdorff dimension of $\mathbb{G}$ and let $\Omega^{\prime}\Subset\Omega$ be any open and relatively compact subset. Then there exists $C>0$, depending on $\Omega^{\prime}$, such that for every $u\in W_{H}^{1,p}(\Omega)$, up to a modification of $u$ on a negligible set, we have

[TABLE]

for every $x,y\in\Omega^{\prime}$.

The following (1,1)-Poincaré inequality holds,

[TABLE]

for every $u\in C^{1}(\overline{B_{r}(x)}).$ This inequality follows from [Jer86], see also [LM00].

For any measurable function $u$ that is summable on a measurable set $A\subset\Omega$, we use the notation

[TABLE]

Definition 2.6**.**

For $u\in L^{1}(\Omega)$, we define the $BMO$ seminorms

[TABLE]

and for $1\leq p<\infty$ the corresponding $BMO^{p}$ norms

[TABLE]

The spaces $BMO^{p}(\Omega)$ and $BMO_{\text{loc}}^{p}(\Omega)$ consist of all $L^{p}$ functions on $\Omega$ with finite $BMO^{p}$ and $BMO_{\text{loc}}^{p}$ norm, respectively. See [BF13] for more information on $BMO$ functions in the subelliptic setting.

3. Preparatory results

We first study the relationship between the coefficients of a 2-homogeneous polynomial and its second order horizontal derivatives. Then, by some $W_{H}^{2,p}$ and $BMO$ estimates, we show how to control the horizontal Hessian of a Sobolev function by the horizontal Hessian of a suitable 2-homogeneous harmonic polynomial (Corollary 3.7). Finally, in Lemma 3.8 we establish a quantitative control on the growth of these polynomials at small scales.

We need first to find 2-homogeneous polynomials with assigned second order horizontal derivatives. To do this, we first observe that (2.2), combined with (2.3) and (2.1), setting

[TABLE]

imply that any $q_{j}$ is a homogeneous polynomial of degree $d_{j}$. Due to the BCH formula, one can also prove that $q_{l}$ is a 2-homogeneous polynomial with respect to the variables $x_{1},\ldots,x_{m},y_{1},\ldots,y_{m}$ for all $l=m+1,\ldots,m_{2}$ and

[TABLE]

From the definition of left invariant vector field, we get

[TABLE]

for all $j=1,\ldots,m$ and $l=m+1,\ldots,m_{2}$. As a consequence, we get

[TABLE]

for all $i,j=1,\ldots,m$ and $l=m+1,\ldots,m_{2}$. Notice that the partial derivatives in the previous equalities are all constant functions. Equalities (3.1) will be important in the proof of Proposition 3.1.

Every polynomial on $\mathbb{R}^{n}$, thought of as equipped with dilations $\delta_{r}$, is the sum of homogeneous polynomials and the maximum among these degrees is the degree of the polynomial. Polynomials of degree one are just affine functions $\ell$ of the form

[TABLE]

with $\beta=(\beta_{1},\beta_{2},\ldots,\beta_{m})\in\mathbb{R}^{n}$, $\alpha\in\mathbb{R}$ and we have used the projection

[TABLE]

A homogeneous polynomial of degree two must have the form

[TABLE]

where $c_{ij}$ and $c_{l}$ are real numbers, with $c_{ij}=c_{ji}$ for all $i,j=1,\ldots,m$.

Proposition 3.1**.**

Let $p:\mathbb{R}^{n}\to\mathbb{R}$ be a 2-homogeneous polynomial of the form (3.3) and let us consider the basis $X_{m+1},\ldots,X_{m_{2}}$ of $V_{2}$. Then we have

[TABLE]

where $\gamma_{ij}^{l}$ are proportional to the structure constants of the Lie algebra, namely

[TABLE]

and $i,j=1,\ldots,m$.

Proof.

We first define the symmetrized second order derivative

[TABLE]

so that we can write

[TABLE]

for every $i,j=1,\ldots,m$. Since $X_{i}X_{j}$ and $X_{l}$ are homogeneous differential operators of order $-2$ and $p$ has degree 2, the horizontal derivatives $X_{i}X_{j}p$ and $X_{l}p$ are constants.

By (3.3) and (2.4), we get

[TABLE]

for $j=1,\ldots,m$. As a consequence, taking into account the form of the vector fields (2.4) and of the polynomial (3.3), for any $i,j=1,\ldots,m$ and $l=m+1,\ldots,m_{2}$, we get

[TABLE]

To establish the previous equality, we have also observed that the polynomials $a_{ji}$ are homogeneous of degree $d_{i}-d_{j}=1$, therefore they are only depending on their first $m$ variables. In particular, all the partial derivatives $\partial_{x_{l}}a_{ji}$ are vanishing whenever the integers $l$ and $i$ take values from $m+1$ to $m_{2}$ and $j=1,\ldots,m$. Combining (3.6) and (3.1), we also obtain the first of the following equalities

[TABLE]

with $1\leq i,j\leq m$ and $m+1\leq l\leq m_{2}$. The latter directly follows from the form of (3.3). In conclusion, by virtue of (3.4), (3.5) and (3.6), we have obtained that

[TABLE]

hence concluding the proof. ∎

Definition 3.2**.**

For $B_{r}(x_{0})\Subset\Omega$ and $u\in W_{H,\text{loc}}^{2,1}(\Omega)$, we define the matrix

[TABLE]

where $I_{m}$ stands for the identity matrix and the $(i,j)$ entry of $(D_{h}^{2}u)_{B_{r}(x_{0})}$ is the average $(X_{i}X_{j}u)_{B_{r}(x_{0})}$. Associated to the ball $B_{r}(x_{0})$, we also define the coefficients

[TABLE]

These numbers define the 2-homogeneous polynomial

[TABLE]

that we will show to be related to $P_{r}^{x_{0}}$.

Corollary 3.3**.**

In the assumptions of Definition 3.2, the 2-homogeneous polynomial $p_{r}^{x_{0}}$ is harmonic and

[TABLE]

Proof.

By Proposition 3.1, we have

[TABLE]

where $\gamma_{ii}^{l}=0$ and by definition of $c_{ii}^{r,x_{0}}$ we get

[TABLE]

Finally, we observe that

[TABLE]

having taken into account that $[X_{i},X_{j}]=\sum_{l=m+1}^{m_{2}}2\gamma_{ij}^{l}X_{l}$ from Proposition 3.1. ∎

The following $W_{H}^{2,p}$ estimates go back to the work of Folland [Fol75], see also the work by Bramanti and Brandolini [BB00] for more general hypoelliptic operators.

Theorem 3.4** ([BB00]).**

Let $1<p<\infty$ and consider two bounded open sets $\Omega$ and $\Omega^{\prime}$ with $\Omega^{\prime}\Subset\Omega$. Then there exists a constant $C>0$ such that for every $u\in W_{H}^{2,p}(\Omega)$ it holds

[TABLE]

It is well known that even for the classical Laplacian operator $\Delta$, it is not true that $L^{\infty}$ bounds on $\Delta u$ imply the boundedness of second order horizontal derivatives. Indeed our starting point is that bounds on the $BMO$ norm of the sub-Laplacian $\Delta_{H}u$ show that the $BMO$ norm of the horizontal Hessian of $u$ is bounded, according to the results of Bramanti et al. [BB05], [BF13].

Theorem 3.5** ([BF13, Theorem 2.10]).**

Let $1<p<\infty,$ $0<\sigma<1$, $u\in BMO^{p}_{\text{loc}}(B_{1})$ and let $\Delta_{H}u\in BMO^{p}_{\text{loc}}(B_{1})$. Then $X_{i}X_{j}u\in BMO^{p}(B_{\sigma})$ for $i,j=1,\ldots,m$ and there exists a universal constant $C(\sigma,p)>0$ such that

[TABLE]

Remark 3.6*.*

Note that the nonvariational form of the operator in [BF13, Theorem 2.10] needs a priori that the solution $u$ and its horizontal derivatives are $BMO$. For our purposes, it is very important that the $BMO$ regularity of $u$ is established with no a priori assumptions. This can be obtained for the sub-Laplacian operator, since its distributional form allows us to apply a mollification argument.

In the sequel, we will also use the Frobenius norm $\lvert M\rvert$ for a matrix $M$ of coefficients $m_{ij}$, setting

[TABLE]

With this definition we easily notice that $\lvert\Delta_{H}u\rvert\leq\lvert D_{h}^{2}u\rvert$.

Corollary 3.7**.**

Let $1<p<\infty$ and $0<\sigma<1$ be fixed. There exists $C(\sigma,p)>0$ such that for all $u\in BMO^{p}_{\text{loc}}(B_{1})$ that satisfy the condition $\Delta_{H}u\in L^{\infty}(B_{1})$, we have $X_{i}X_{j}u\in BMO^{p}(B_{\sigma})$ for $i,j=1,\ldots,m$ and whenever $x_{0}\in B_{\sigma}$, $0<r<1-\sigma$, it holds

[TABLE]

where the matrix $P_{r}^{x_{0}}$ is introduced in Definition 3.2.

Proof.

Theorem 3.5 immediately implies that $X_{i}X_{j}u\in BMO^{p}(B_{\sigma})$ and (3.8) holds. Thus, we obtain the following estimates

[TABLE]

This completes the proof. ∎

Heuristically, if $D_{h}^{2}u$ is not bounded around $x_{0}$, since the difference of $D_{h}^{2}u$ and $P_{r}^{x_{0}}$ is controlled, the $BMO$ estimate tells us that also $P_{r}^{x_{0}}$ becomes unbounded as $r\to 0^{+}$. Hence we will turn our attention to $P_{r}^{x_{0}}$. In the following lemma, we will derive a general “scaling estimate” for the difference $\lvert P_{r_{1}}^{x_{0}}-P_{r_{2}}^{x_{0}}\rvert$. In particular, when $r_{2}=2r_{1}$ we get a uniform bound on the growth of $|P_{r}^{x_{0}}|$ on dyadic scales.

Lemma 3.8** (Scaling estimates).**

Let $1<p<\infty$ and $0<\lambda_{1}<1$ be fixed. Then there exists a universal constant $C(\lambda_{1},p)>0$ such that for all $u\in BMO^{p}_{\text{loc}}(B_{1})$ that satisfy the condition $\Delta_{H}u\in L^{\infty}(B_{1})$ the following holds. We have $X_{i}X_{j}u\in L^{1}_{\text{loc}}(B_{1})$, where $i,j=1,\ldots,m$ and for $x_{0}\in B_{\lambda_{1}/3}$ the matrices of the form

[TABLE]

with $0<r_{1}<\min\{2\lambda_{1}/3,1-\lambda_{1}\}$ and $r_{1}<r_{2}<1-\lambda_{1}$, satisfy the following inequality

[TABLE]

Proof.

Due to the $BMO$ estimate (3.9) with $\sigma=\lambda_{1}$, we can estimate $\lvert P_{r_{1}}^{x_{0}}-P_{r_{2}}^{x_{0}}\rvert$ as follows

[TABLE]

The last inequality follows by taking into account our conditions on the radii $r_{1}$ and $r_{2}$. Indeed, we have

[TABLE]

Finally, with a slight abuse of notation, we denote the constant $2C(\lambda_{1},p)$ again by $C(\lambda_{1},p)$ in the inequality of the lemma, concluding the proof. ∎

4. Proof of $C^{1,1}_{H}$ regularity

This section represents the core of the paper. We establish the sub-quadratic growth of the difference

[TABLE]

on the ball $B_{r}(x_{0})$, where $p_{r}^{x_{0}}$ is the harmonic polynomial introduced in Definition 3.2. We show that when the norm of $D_{h}^{2}p_{r}^{x_{0}}$ is sufficiently large, then the measure of the coincidence set $\{u=0\}$ decays in a quantitative way. This is one of the central facts, that leads us to the dichotomy argument of [ALS13] to reach the $C_{H}^{1,1}$ regularity. There are indeed two cases: (i) when $|D_{h}^{2}p_{r}^{x_{0}}|$ is uniformly bounded as $r\to 0^{+}$, we immediately infer the regularity from the subquadratic growth, (ii) if otherwise $|D_{h}^{2}p_{r}^{x_{0}}|$ grows without bound as $r\to 0^{+}$, then the coincidence set is “small” and we show that a suitable adaptation of Caffarelli’s polynomial iteration technique can lead us to the $C_{H}^{1,1}$ regularity.

In the sequel, whenever we consider a function $u$ with essentially bounded sub-Laplacian $\Delta_{H}u$, then it is understood that $u$ is chosen to be of class $C^{1}_{H}$. The following remark rigorously justifies this convention.

Remark 4.1*.*

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and let $u:\Omega\to\mathbb{R}$ be a locally summable function such that $\Delta_{H}u\in L^{\infty}(\Omega)$. From [Fol75, Theorem 6.1], we have $u\in W^{2,p^{\prime}}_{H,\text{loc}}(\Omega)$ for every $p^{\prime}>1$. In view of Theorem 2.5, by standard arguments, we can modify $u$ on a negligible set such that $u\in C^{1,\alpha}_{H}(\Omega^{\prime})$ for any relatively compact open set $\Omega^{\prime}\Subset\Omega$, where we have fixed some $p^{\prime}>Q$ and $\alpha=1-\frac{Q}{p^{\prime}}$. In particular, we have shown that, after the modification, $u\in C^{1}_{H}(\Omega)$.

Lemma 4.2** (Sub-quadratic growth).**

Assume $u\in BMO_{\text{loc}}^{p}(B_{1})$ such that $\Delta_{H}u\in L^{\infty}(B_{1})$. Let $\lambda,\sigma\in(0,1)$ and fix $p>1$. Then there exist $r_{0}>0$ and a universal constant $C(\lambda,\sigma,p)>0$, such that for any $x_{0}\in B_{\lambda}$ and $0<r\leq r_{0}$, assuming that

[TABLE]

and considering $p_{r}^{x_{0}}$, as given in Definition 3.2, the following estimate holds

[TABLE]

Proof.

We fix $x_{0}\in B_{\lambda}$ and $\lambda^{\prime}=(1+\lambda)/2,$ so that for $0<r\leq\lambda^{\prime}-\lambda,$ we have the inclusion

[TABLE]

Let us introduce the translated and rescaled function

[TABLE]

observing that it is well defined in $B_{1}$. Taking into account that $u\in W^{2,p}_{H,\text{loc}}(B_{1})$ and $\overline{B_{r}(x_{0})}\subset\overline{B_{\lambda^{\prime}}}\subset B_{1}$, then $u_{r,x_{0}}\in W^{2,p}_{H}(B_{1})$. We are in the position to apply Corollary 3.7 to $u$ with $\sigma=\lambda^{\prime}$. As a consequence of both Corollary 3.3 and (3.9), taking into account (4.2), it follows that

[TABLE]

Now we wish to apply the Poincaré inequality (2.11) to $u_{r,x_{0}}-\ell_{r,x_{0}},$ where $\ell_{r,x_{0}}$ is an affine function to be properly defined. If we let

[TABLE]

where $\pi(x)=(x_{1},\ldots,x_{m})$, it follows that

[TABLE]

since the average over $B_{1}$ of the linear part of $\ell_{r,x_{0}}$ is zero. Again, from the Poincaré inequality, using (4), we get

[TABLE]

For the sequel, we set $\hat{u}_{r,x_{0}}\vcentcolon=u_{r,x_{0}}-\ell_{r,x_{0}}$. Since both $p_{r}^{x_{0}}$ and $\ell_{r,x_{0}}$ are harmonic, we observe that

[TABLE]

for a.e. $x\in B_{1}$, where we have set $f\vcentcolon=\Delta_{H}u\in L^{\infty}(B_{1})$. We set $g_{r,x_{0}}(x)=f(x_{0}\delta_{r}x)\chi_{B_{1}}$ and we consider the decomposition $\hat{u}_{r,x_{0}}=\hat{v}_{r,x_{0}}+\hat{w}_{r,x_{0}}$, where

[TABLE]

and $\Gamma$ is the fundamental solution for $\Delta_{H}$, introduced in Definition 2.3. The explicit form of $\hat{v}_{r,x_{0}}$ allows us to get the estimate

[TABLE]

for every $x\in B_{1}$, where $Q_{0}=Q+1$ and $C>0$ can be seen as a universal constant. The previous estimate follows from the Hölder inequality, setting $q=Q_{0}/Q$ and taking into account the $(2-Q)$-homogeneity of $\Gamma$. Indeed, it holds

[TABLE]

for every $x\in B_{1}$. As a consequence, we have proved that

[TABLE]

Since $\hat{w}_{r,x_{0}}$ is harmonic, from [BLU07, (5.52)] we have the mean value type formula

[TABLE]

for any $x\in B_{\sigma}$, whenever $0<\sigma<1$ and with $c_{0}>1$ defined in (2.6). We point out that the function $\Psi$ is 0-homogeneous with respect to dilations and smooth on $\mathbb{R}^{n}\operatorname{\setminus}\left\{0\right\}$, see [BLU07, Definition 5.5.1] for more information. Notice that with our assumptions we have the inclusion $B^{G}_{(1-\sigma)/c_{0}}(x)\subset B_{1}$. For every $x\in B_{\sigma}$, it holds

[TABLE]

The constant $C(\sigma)$ only depends on $\sigma$ and it blows up as $\sigma\to 1^{-}.$ By the triangle inequality and (4.6) we obtain that

[TABLE]

We conclude from both (4.6) and (4.7) that

[TABLE]

Differentiating $\hat{v}_{r,x_{0}}$, seen as an integral, it turns out that $\hat{v}_{r,x_{0}}\in C_{H}^{1}(B_{1})$. Again arguing as in the proof of (4.5), from the Hölder inequality and the $(1-Q)$-homogeneity of $X_{j}\Gamma$, we get the estimate

[TABLE]

for every $j=1,\ldots,m$, $x\in B_{1}$ and a fixed geometric constant $\overline{C}>0.$ By Proposition 2.4, we get a constant $C_{3}(\sigma)>0$, such that

[TABLE]

Combining (4.8), (4.9) and (4.10), along with the third inequality of (4.7), we establish the first of the following inequalities:

[TABLE]

The second inequality is a consequence of (4.4). Since $u_{r,x_{0}}(0)=X_{j}u_{r,x_{0}}(0)=0$, by our assumptions on $u$, and taking into account that $p_{r}^{x_{0}}(0)=X_{j}p_{r}^{x_{0}}(0)=0$, we immediately infer from the $C_{H}^{1}$ estimate above that

[TABLE]

It follows that

[TABLE]

As a consequence, it follows that

[TABLE]

This finishes the proof. ∎

Corollary 4.3**.**

Assume $u\in L^{\infty}(B_{1})$ such that $\Delta_{H}u\in L^{\infty}(B_{1})$ and fix $0<\lambda,\sigma<1$. If we consider $\pi$ as in (3.2), then there exists $r_{0}>0$ such that the affine function

[TABLE]

satisfies the following properties. There exists a universal constant $C(\lambda,\sigma)>0$ such that for every $r\in(0,r_{0}]$ the following estimate holds

[TABLE]

where $p_{r}^{x_{0}}$ is as in Definition 3.2.

Proof.

Our assumptions allow us to apply Lemma 4.2 to $y\to u(y)-\ell^{x_{0}}(x_{0}^{-1}y)$ with $p=2$. Then there exist $r_{0},C(\lambda,\sigma)>0$ such that

[TABLE]

for every $r\in(0,r_{0}].$ In addition, we have

[TABLE]

We set $f=\Delta_{H}u\in L^{\infty}(B_{1})$ and write $v=f*\Gamma$, getting

[TABLE]

Arguing as in [GT01, Lemma 4.1], we establish

[TABLE]

therefore we have

[TABLE]

Since $u+v$ is harmonic in $B_{1}$, by (2.9), it follows that

[TABLE]

This immediately leads us to our claim. ∎

Remark 4.4*.*

Notice that under the same assumptions of Corollary 4.3, we can assume that for every $\lambda,\sigma\in(0,1)$ and any $x_{0}\in B_{\lambda}$, there exist $\tilde{r}_{0}>0$ and $C>0$, only depending on $\lambda$ and $\sigma$, such that for all $r\in(0,\tilde{r}_{0}]$ the following estimate holds

[TABLE]

for $\sigma_{0}=\sigma/c_{0}\in(0,1/c_{0})$ and additionally we have the inclusion $\overline{B^{G}_{r_{0}}(x_{0})}\subset B_{1}$. This is a consequence of the definition of $c_{0}$ in (2.8). If we set $r_{0}\vcentcolon=\tilde{r}_{0}/c_{0}$, replacing $r$ by $c_{0}r$, we may rephrase the previous estimate as follows

[TABLE]

for every $0<r\leq r_{0}$.

We introduce now the important definition of coincidence set:

[TABLE]

We will perform a blow-up of $\Lambda$ around a fixed point $x_{0}\in B_{1/2}$, considering the rescaled and translated coincidence sets

[TABLE]

for $0<r\leq r_{0}$ and some $r_{0}>0$ such that $B_{r}^{G}(x_{0})\subset B_{1}$. Notice that in the previous definition the gauge distance is used for technical reasons, related to the existence of solutions to the Dirichlet problem with respect to the sub-Laplacian.

The next result is a technical lemma, that will be used both to get the decay estimates in Proposition 4.6 and to establish the regularity in Theorem 4.8.

Lemma 4.5**.**

Let $f$ be such that $f*\Gamma\in C_{H}^{1,1}(B_{1})$ and let $u$ solve (1.1) in $B_{1}$. Then for every $0<\lambda,\sigma<1$, there exists $r_{0}>0$ such that for every $x_{0}\in B_{\lambda}$ we have $\overline{B_{r_{0}}^{G}(x_{0})}\subset B_{1}$ and the following holds. Let us consider the translated and rescaled function

[TABLE]

where $p_{c_{0}r}^{x_{0}}$ is introduced in Definition 3.2 and $\ell^{x_{0}}(z)=u(x_{0})+\langle\nabla_{h}u(x_{0}),\pi(z)\rangle$. For each $r\in(0,r_{0}]$ we also define $v_{r,x_{0}}$ as the solution to

[TABLE]

where $f_{r,x_{0}}(x)=f(x_{0}\delta_{r}x)\chi_{B^{G}_{\sigma}}$. Then there exists a universal constant $C(\lambda,\sigma)>0$, depending on $\lambda$ and $\sigma$, such that

[TABLE]

Proof.

Due to Remark 4.4, there exists $r_{0}>0$ such that for every $x_{0}\in B_{\lambda}$ we have $\overline{B_{r_{0}}^{G}(x_{0})}\subset B_{1}$ and (4.15) holds for every $r\in(0,r_{0}]$. We write the solution to the Dirichlet problem (4.18) in the form

[TABLE]

where $\zeta_{r,x_{0}}$ solves

[TABLE]

and we have defined

[TABLE]

Indeed, the open set $B_{\sigma}^{G}$ is regular with respect to $\Delta_{H},$ see [BLU07, Proposition 7.2.8]. From the identity

[TABLE]

taking into account the equality $D^{2}_{h}(f*\Gamma)(x_{0}\delta_{r}x)=D^{2}_{h}(f_{r,x_{0}}*\Gamma)(x)$ for a.e. $x\in B^{G}_{\sigma^{2}}$ and the estimate (2.9) we obtain that

[TABLE]

Now we combine the maximum principle and the Dirichlet problem (4.18) to get

[TABLE]

Due to the version of the sub-quadratic growth in (4.15), taking into account the definition (4.17) and the immediate estimate

[TABLE]

where $C>0$ only depends on $\Gamma$, it follows that

[TABLE]

In conclusion, we have established the following estimate

[TABLE]

concluding the proof. ∎

Proposition 4.6** (Decay of the coincidence set).**

Let $f$ be such that $f*\Gamma\in C_{H}^{1,1}(B_{1})$ and let $u$ solve (1.1). Then for every $\beta>0,$ there exist $C_{\beta}>0$ and $r_{0}>0$ so that if $0<r\leq r_{0}$, $x_{0}\in B_{1/2}$, and the estimate

[TABLE]

holds, we have

[TABLE]

Proof.

Our assumptions allow us to apply Lemma 4.5 with $\lambda=1/2$, where we choose $\sigma\in[1/\sqrt{2},1)$. Let $r_{0}>0$, $v_{r,x_{0}}$ and $u_{r,x_{0}}$ be as in the same lemma and define

[TABLE]

Lemma 4.5 yields a constant $C(\sigma)>0$ such that

[TABLE]

for every $r\in(0,r_{0}]$. In addition, from the definition of $w_{r,x_{0}}$ we observe that

[TABLE]

By uniqueness, it follows that

[TABLE]

where $G_{B_{\sigma}^{G}}$ is the Green function of $B_{\sigma}^{G}$, according to [BLU07, Definition 9.2.1]. From the definition of Green function, we have $G_{B_{\sigma}^{G}}\geq 0$. In addition, taking into account [BLU07, Proposition 9.2.12(iv)], we also notice that the maximum principle gives

[TABLE]

for every $x,y\in B^{G}_{\sigma}$ with $x\neq y$. Then a standard convolution estimate yields

[TABLE]

for some geometric constant $C>0.$ The $W_{H}^{2,p}$ estimates (3.7) give a universal constant $C_{1}$, depending on $\sigma$, such that

[TABLE]

The second inequality is again a consequence of a convolution estimate, joined with (4.26). Since $|\Lambda_{r}(x_{0})|\leq|B^{G}_{1}|$, the third inequality is also established. Furthermore, taking the second order horizontal derivatives in the definition (4.17), we get the equality

[TABLE]

and also $\Lambda_{r\sigma^{2}}(x_{0})=\delta_{\sigma^{-2}}(\Lambda_{r}(x_{0})\cap B_{\sigma^{2}}^{G}).$ In addition, arguing as in [GT01, Lemma 7.7], we can establish that $(D^{2}_{h}u)(x_{0}\delta_{r}x)=0$ a.e. on the coincidence set $\Lambda_{r}(x_{0})$. Taking into account all previous facts, we get

[TABLE]

Consequently,

[TABLE]

We see that the coefficient in front of $\lvert\Lambda_{r\sigma^{2}}(x_{0})\rvert$ is bigger than $2^{\beta Q}$ if

[TABLE]

By the simple inequality $\lVert D_{h}^{2}(f*\Gamma)\rVert_{L^{\infty}(B_{1})}\geq\lVert f\rVert_{L^{\infty}(B_{1})}$, a few more computations lead us to the following sufficient condition

[TABLE]

to get (4.27) to hold. Finally, we choose $\sigma=1/\sqrt{2}$, then the proof follows by choosing the constant $C_{\beta}$ in our statement equal to $\sqrt[2Q]{C_{2}(1/\sqrt{2})2^{Q}(2^{\beta Q}+2^{2Q-1})}$ and replacing $c_{0}r$ by $r$. ∎

To carry out the proof of the $C_{H}^{1,1}$ regularity, we need a Calderón type second order differentiability, according to the next definition.

Definition 4.7**.**

We say that $u\in L_{\text{loc}}^{1}(\Omega)$ is twice $L^{1}$ differentiable at $x_{0}$ if there exists a polynomial $t$ of degree less than or equal to two, such that

[TABLE]

The polynomial $t$ has the following form

[TABLE]

$x=(x_{1},\ldots,x_{n})$, $x_{0}=(x_{01},\ldots,x_{0n})$ and $c_{0},c_{ij},c_{l}\in\mathbb{R}.$

It is possible to show that any $u\in W_{H,\text{loc}}^{2,1}(\Omega)$ is twice $L^{1}$ differentiable a.e. in $\Omega$. Furthermore, if the function is twice $L^{1}$ differentiable at a Lebesgue point $x_{0}\in\Omega$ of all functions $X_{i}X_{j}u,$ $X_{j}u$ and $u$, then the corresponding polynomial is unique and it has the following form

[TABLE]

see [Mag05] for more information. We are now in the position to prove the optimal interior regularity of solutions to the no-sign obstacle-type problem (1.1).

Theorem 4.8** ($C^{1,1}$ regularity).**

Let $u\in L^{\infty}(B_{1})$ be a distributional solution to (1.1) in the unit ball $B_{1}$. Let $f:B_{1}\to\mathbb{R}$ be locally summable such that $f*\Gamma\in C_{H}^{1,1}(B_{1})$. Then there exists a universal constant $C>0$ such that, after a modification on a negligible set, we have $u\in C_{H}^{1,1}(B_{1/4})$ and

[TABLE]

Proof.

We consider $C_{\beta}$ as in Proposition 4.6 and fix $\beta=4$. We consider a priori the following constant

[TABLE]

Combining the Hölder inequality and Theorem 3.4, taking into account that the constant $C_{\beta}$ in Proposition 4.6 is bounded from below by a universal positive constant independent of $\beta$, we can find a universal constant $C_{1}^{\prime}>0$ such that

[TABLE]

Let $r_{0}>0$ be the minimum among the $r_{0}$’s of Remark 4.4, Lemma 4.5 with $\lambda=1/4$ and Proposition 4.6 with $\beta=4$. We fix an integer $i_{0}$ such that

[TABLE]

such that $2^{-i_{0}}\leq r_{0}$, where $c_{0}$ is the geometric constant appearing in (2.6). Then (4.28) provides us with a universal constant $\overline{C}_{1}\geq 1$ such that

[TABLE]

for all $y\in B_{1/4}$. Notice that $\overline{C}_{1}$ actually depends on $i_{0}$. However, this integer is fixed throughout the proof. We have chosen $i_{0}$ to satisfy also (4.29) in view of the subsequent application of Lemma 3.8 with $\lambda_{1}=3/4$. We can fix $x_{0}\in B_{1/4}$ such that $u$ is twice $L^{1}$ differentiable at $x_{0}.$ Using [Mag05, Theorem 3.8] for $p=1$ and $k=2$, the set of these differentiability points has full measure in $B_{1}$. We can further write $u=v-w$, such that

[TABLE]

on $B_{1},$ where $v=-f*\Gamma.$ By assumption $v\in C_{H}^{1,1}(B_{1})$, hence it is also a.e. twice $L^{1}$ differentiable, therefore we can further assume $v$ is twice $L^{1}$ differentiable at $x_{0}$, having the set of these points full measure in $B_{1}$. Now, only two cases may occur.

Case 1: $\displaystyle\boldsymbol{\liminf_{k\to\infty}\lvert P_{2^{-k}}^{x_{0}}\rvert\leq\overline{C}_{1}K}$. At our point $x_{0}$, we have

[TABLE]

Therefore

[TABLE]

Case 2: $\displaystyle\boldsymbol{\liminf_{k\to\infty}\lvert P_{2^{-k}}^{x_{0}}\rvert>\overline{C}_{1}K}$. Then the following integer is well defined

[TABLE]

The positive integer $k_{0}$ possibly depends on $x_{0}$. We notice that from the definition of $k_{0}$, we have $\lvert P_{2^{-k_{0}+1}}^{x_{0}}\rvert\leq\overline{C}_{1}K$. The strict inequality $k_{0}>i_{0}$ follows by (4.30). In view of our choice of $i_{0}$, that satisfies (4.29) and then $i_{0}>3$, we can apply Lemma 3.8 with $\lambda_{1}=3/4$. Indeed, we have $B_{1/4}=B_{\lambda_{1}/3}$ and

[TABLE]

so Lemma 3.8 with $r_{1}=2^{-k_{0}}$ and $r_{2}=2^{-k_{0}+1}$ yields

[TABLE]

We consider the “rescaled function” defined in Lemma 4.5:

[TABLE]

This function coincides with $u_{2^{-k_{0}},x_{0}}$ of the same lemma. Now we set $f_{0}(x)\vcentcolon=f(x_{0}\delta_{2^{-k_{0}}}x),$ that is also defined on $B_{1}^{G}.$ We can find a harmonic function $h_{0}$ such that

[TABLE]

and $v_{0}$ satisfies the Dirichlet problem

[TABLE]

with $0<\sigma<1$. Notice that $v_{0}$ is also twice $L^{1}$ differentiable at $0,$ being a consequence of the twice $L^{1}$ differentiability of $v$ at $x_{0}.$ For the same reason, the twice $L^{1}$ differentiability of $u$ at $x_{0}$ gives the twice $L^{1}$ differentiability of $u_{0}$ at [math]. From Lemma 4.5 with $\lambda=1/4$, there exists $C_{\sigma}>0$ such that

[TABLE]

For the sequel, it is now important to remark that the difference

[TABLE]

is twice $L^{1}$ differentiable at the origin. Then we know the existence of a polynomial

[TABLE]

such that we get

[TABLE]

as $r\to 0^{+}$ and for an arbitrary $\kappa>0$. The definition of $w_{0}$ immediately gives

[TABLE]

Claim: for a fixed $0<\alpha<1$, there exist $l_{0}\geq 1$ and $C>0$, depending on $\alpha$ and on universal constants, such that for $\tau=2^{-l_{0}}$ and for every $k\in\mathbb{N}\operatorname{\setminus}\left\{0\right\}$, there exist harmonic polynomials $q_{k}$ with the property that

[TABLE]

where the constants are independent of $x_{0}$.

To prove (4.37) by induction, we need first to establish the case $k=1$. Here we choose the null harmonic polynomial $q_{1}=0.$ We consider the decomposition (4.35) and observe that standard $L^{\infty}$ estimates for $v_{0}$ are available, since it solves (4.33). Indeed, we may further decompose $v_{0}$ into the sum of $z_{0}=-f_{0}*\Gamma$ and of a harmonic function $h_{0}$ such that $h_{0}|_{\partial B^{G}_{\sigma}}=u_{0}-z_{0}$. Then we apply the sub-quadratic growth estimate (4.15) of Remark 4.4 where we fix $\lambda=1/4$. This leads us to the following estimate

[TABLE]

Then using again estimate (4.15), we obtain

[TABLE]

Taking $\sigma=\tau=2^{-l_{0}}$, the estimate (4.37) is established for $k=1$. We may take $l_{0}\in\mathbb{N}$ possibly larger, such that

[TABLE]

In view of Proposition 2.4, there exists a universal constant $c>0$ such that

[TABLE]

for any harmonic function $H$ on $B_{1}^{G}$. Now we assume the statement (4.37) is true for any fixed $k\geq 1$ and define

[TABLE]

on $\overline{B_{1}^{G}}$. We choose the harmonic function $h_{k}$ such that

[TABLE]

From the definition of $w_{k}$, we get

[TABLE]

on $B_{1}^{G}$, and the induction assumption yields

[TABLE]

Clearly $w_{k}-h_{k}$ vanishes on $\partial B_{1}^{G}$. Taking into account our choice of $i_{0}$ such that $2^{-i_{0}}\leq r_{0}$, the decay estimate of the coincidence set (4.24) applies in particular for $\beta=4$ and for every $r\in(0,2^{-k_{0}}]$, that is

[TABLE]

Arguing as before for $v_{0}$, we can decompose $w_{k}-h_{k}$ into the sum of a harmonic function and a convolution with the fundamental solution, therefore standard convolution estimates yield the first of the following inequalities

[TABLE]

where the third inequality is a consequence of (4.41) and the last inequality follows from (4.38). Combining the estimate (4.39) for harmonic functions, the maximum principle and our induction assumption as stated in (4.40), we get

[TABLE]

Define $\overline{q}_{k}(x)$ as the second order Taylor polynomial of $h_{k}$ at the origin. In particular, $\overline{q}_{k}$ is harmonic. The previous estimates joined with the application of the stratified Taylor inequality stated in [FS82, Corollary 1.44 with $k=2$ and $x=0$] give

[TABLE]

where we have chosen $l_{0}$ possibly larger, such that the following conditions

[TABLE]

also hold. The constants $b$ and $C^{\prime}_{2}$ are from [FS82, Corollary 1.44 with $k=2$ and $x=0$] and this corollary is applied with the gauge distance $d_{G}$. We stress that $l_{0}$ does not depend on either $k_{0}$ or $x_{0}$. This is very important for the final estimate of $D_{h}^{2}u(x_{0})$. As a consequence, we obtain

[TABLE]

Taking into account the definition of $w_{k},$ we have proved that

[TABLE]

from which we infer that

[TABLE]

If we define the new polynomial

[TABLE]

then the induction step is proved and this concludes the proof of our claim. By the same previous argument, we have another universal constant $c^{\prime}>0$ such that

[TABLE]

We introduce the following notation:

[TABLE]

From the definition of $\bar{q}_{k}$ and taking into account the estimates (4.45), we get

[TABLE]

Consequently, differentiating the equality

[TABLE]

with the differential operators $X_{i}$, $X_{i}X_{j}$ for $i,j=1,\ldots,m$ and $X_{l}$ for $l=m+1,\ldots,m_{2}$, and evaluating all the equalities at the origin, we get

[TABLE]

Representing any of these coefficients as $\gamma^{k}$, we notice that they are Cauchy sequences converging to some $\gamma$. In addition, for any of them we have

[TABLE]

In these estimates, we have set $l=0$ when $\gamma^{k}=$ $c_{l}^{k},c_{ij}^{k},$ $l=1$ for $\gamma^{k}=b_{i}^{k}$** **and $l=2$ for $\gamma^{k}=a_{i}^{k}.$ As a consequence, the polynomials $q_{k}$ uniformly converge on compact sets to a polynomial $\tilde{q}$, that has the form

[TABLE]

Any coefficient $\gamma$ of $\tilde{q}$, can be written for instance as $\gamma^{2}+\gamma-\gamma^{2}.$ By (4.47), we can find a universal estimate for $\gamma-\gamma^{2}$, that depends on $\alpha$. We also observe that the coefficients of $q_{2}$ are given by the formula $q_{2}=\overline{q}_{1}\circ\delta_{\tau^{-1}}$. The estimate (4.46) for $k=2$ and the fact that $\tau=2^{-l_{0}}$ can be universally fixed, independently of $x_{0}$, finally lead us to the estimate

[TABLE]

for a suitable geometric constant $C_{\tau,\alpha}>0$ depending on the constants of (4.47) and (4.46). From (4.47), defining

[TABLE]

we can establish the following quantitative estimate on small balls:

[TABLE]

where we have used the proper intrinsic homogeneity of all the monomials $x_{i}$, $x_{i}x_{j}$ and $x_{l}$, with $i,j=1,\ldots,m$ and $l=m+1,\ldots,m_{2}$. We would like to prove that

[TABLE]

Let us consider $r=\tau^{k}$ and then

[TABLE]

goes to zero as $k\to\infty.$ If we choose $\kappa_{0}>0$ sufficiently small, then we have proved that

[TABLE]

By the uniqueness of the second order polynomial $R$ satisfying (4.36) with $\kappa=\kappa_{0}$, we get $\tilde{q}=R$. Taking into account (4.34) with $\sigma=\tau$ and (4.48) with $\alpha=1/2$, we obtain a new constant $C>0,$ such that

[TABLE]

As a consequence, since $k_{0}>i_{0}$ and $i_{0}$ satisfies (4.29) we can apply Lemma 3.8 with $r_{1}=2^{-k_{0}}$ and $r_{2}=c_{0}2^{-k_{0}}$, so that taking into account the estimate (4.31) and the definition (4.32) of $u_{0}$, we finally obtain a possibly larger constant, that we still denote by $C>0$, such that

[TABLE]

concluding the proof. ∎