arXiv subjects

Joseph Feneuil

Publications and source records attributed to Joseph Feneuil.

At least 19 recordsLinked to original sources

arXiv2026

Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion

In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \|\nabla f\|_p \leq C \|\Delta^\gamma f\|_p $ holds for every $p\in(1,\infty)$ and every $ 0<\gamma<\gamma^*(p):=\frac{1}{D+1}+\frac{D-1}{D+1}\,\frac{1}{p}, $ while it fails whenever $p\in(1,\infty)$ and $\gamma^*(p)<\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\gamma=\gamma^*(p)$. This provides the first example of an $L^p$-bounded ``super-Riesz transform'', namely an operator of the form $\nabla \Delta^{-\gamma}$ with $\gamma$ strictly larger than the Euclidean threshold $\frac12$. To achieve this, we establish a more general result linking the diffusion escape rate and a Poincar\'e inequality on balls to the validity of the reverse Riesz-like inequality $\|\Delta^\gamma f\|_p \leq C \|\nabla f\|_p.$

math.FA↗

arXiv2026

Stability of $L^p$ Dirichlet problem under small bi-Lipschitz transformations of domains

We show that small bi-Lipschitz deformations of a Lipschitz domain (with possibly large Lipschitz constant) preserve the solvability of the Dirichlet problem for the Laplacian with boundary data in $L^p$, for the same value of $p>1$. As a consequence, for all $p\in(1,\infty)$, we obtain the solvability of the $L^p$ Dirichlet problem for small Lipschitz perturbations of convex domains, thereby unifying two fundamentally different settings in which such results were previously known: convex and $C^1$ domains. The key ingredient and novelty of our approach is a construction of a change of variables based on a non-constant basis derived from the Green function, which encodes the geometry of the base domain.

math.AP↗

arXiv2025

In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$, $p\in (2,\infty)$

In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters $ p \in (1, \infty) $ and $ \gamma \in [0, 1] $ for which the quantities $ \|\Delta^\gamma f\|_p $ and $ \|\nabla f\|_p $ can be compared. In particular, we prove that in such metric spaces, the Riesz transform $ \nabla \Delta^{-1/2} $ is unbounded on $ L^p $ for all $ p \in (2, \infty) $, thereby demonstrating a clear departure from the behavior observed in the Euclidean setting.

math.FA↗

arXiv2024

Carleson perturbations of locally Lipschitz elliptic operators

In one-sided Chord-Arc Domains $\Omega$, we demonstrate that the $A_\infty$-absolute continuity of the elliptic measure with respect to the surface measure remains stable under $L^2$ Carleson perturbations. This stability holds provided that either the elliptic operator $L_0=-\operatorname{div} A_0\nabla$, which is being perturbed, or the perturbed operator $L_1=-\operatorname{div} A_1\nabla$ satisfies the condition $\sup_{X\in \Omega }\operatorname{dist}(X,\partial \Omega)|\nabla A_i(X)| <\infty$ on its coefficients. $L^2$ Carleson perturbations are slightly more general than those previously discussed in the literature. The proof hinges on the availability of a comprehensive elliptic theory and a domain $\Omega$ that allows uniform non-tangential access to any point on its boundary. Consequently, while the current theory of $L^2$ Carleson perturbations can be extended to more general contexts, we have chosen not to do so in order to simplify the presentation.

math.AP↗

arXiv2024

The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem

We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $Ω$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $Ω$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse Hölder inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem.

math.AP↗

arXiv2023

An alternative proof of the $L^p$-regularity problem for Dahlberg-Kenig-Pipher operators on $\mathbb R^n_+$

In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a (possibly large) Carleson measure condition. In addition, we slightly expand the class of operators for which the regularity problem is solvable, and establish an analogous result for weighted uniformly elliptic operators on $\mathbb{R}^n \setminus \mathbb{R}^d$, where $d < n - 1$.

math.AP↗

arXiv2023

Elliptic theory for sets with higher co-dimensional boundaries

Many geometric and analytic properties of sets hinge on the properties of harmonic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let $Γ\subset \mathbb R^n$ be an Ahlfors regular set of dimension $d<n-1$ (not necessarily integer) and $Ω= \mathbb R^n \setminus Γ$. Let $L = - {\rm div} A\nabla$ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix $A$ are bounded from above and below by a multiple of ${\rm dist}(\cdot, Γ)^{d+1-n}$. We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or $L^p$ estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to $L$, establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when $Γ$ is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator $L$ for which the harmonic measure given here is absolutely continuous with respect to the $d$-Hausdorff measure on $Γ$ and vice versa. It thus extends Dahlberg's theorem to some sets of codimension higher than 1.

math.AP↗

arXiv2023

Elliptic theory in domains with boundaries of mixed dimension

Take an open domain $Ω\subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $Ω$ can be a ball on $\mathbb R^3$, minus one of its diameters $D$, or $Ω\subset \mathbb R^3$ could be a so-called saw-tooth domain, with a boundary consisting of pieces of 1-dimensional curves intercepted by 2-dimensional spheres. Under appropriate geometric assumptions, such as the existence of doubling measures on $Ω$ and $\partial Ω$ with appropriate size conditions, we construct a class of degenerate elliptic operators $L$ adapted to the geometry, and establish key estimates of elliptic theory associated to those operators. This includes boundary Poincaré and Harnack inequalities, maximum principle, and Hölder continuity of solutions at the boundary. We introduce Hilbert spaces naturally associated to the geometry, construct appropriate trace and extension operators, and use them to define weak solutions to $Lu=0$. Then we prove De Giorgi-Nash-Moser estimates inside $Ω$ and on the boundary, solve the Dirichlet problem and thus construct an elliptic measure $ω_L$ associated to $L$. At last, we introduce Green functions, and use them to prove a comparison principle. Since our theory emphasizes measures, rather than the geometry per se, the results are new even in the classical setting of a half-plane $\mathbb R^2_+$ when the boundary $\partial \mathbb R^2_+= \mathbb R$ is equipped with a doubling measure $μ$ singular with respect to the Lebesgue measure on $\mathbb R$. Finally, the present paper provides a generalization of the celebrated Caffarelli-Sylvestre extension operator from its classical setting of $\mathbb R^{n+1}_+$ to general open sets, and hence, an extension of the concept of fractional Laplacian to Ahlfors regular boundaries and beyond.

math.AP↗

arXiv2023

A Green function characterization of uniformly rectifiable sets of any codimension

In this paper, we obtain a unified characterization of uniformly rectifiable sets of {\it any codimension} in terms of a Carleson estimate on the second derivatives of the Green function. When restricted to domains with boundaries of codimension 1, our result generalizes a previous result of Azzam for the Laplacian to more general elliptic operators. For domains with boundaries of codimension greater than 1, our result is completely new.

math.AP↗

arXiv2022

Green functions and smooth distances

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function $G$ behaves like a distance function to the boundary, in the sense that $\Big|\frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\Big|^2D(X) dX$ is the density of a Carleson measure, where $D$ is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's $α$-number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.

math.AP↗

arXiv2022

Carleson Perturbations for the Regularity Problem

We prove that the solvability of the regularity problem in $L^q(\partial Ω)$ is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same $L^q$, and if the perturbation is large, the regularity problem is solvable in $L^{r}$ for some other $r\in (1,\infty)$. We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular boundary, together with a notion of gradient on the boundary.

math.AP↗

arXiv2022

The Regularity problem in domains with lower dimensional boundaries

In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries 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".

math.AP↗

arXiv2022

Generalized Carleson perturbations of elliptic operators and applications

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{\infty}$ among elliptic measures.

math.AP↗

arXiv2022

Absolute continuity of the harmonic measure on low dimensional rectifiable sets

We consider a uniformly rectifiable set $Γ\subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $Ω= \mathbb R^n \setminus Γ$, Guy David, Svitlana Mayboroda, and the author introduced a notion of harmonic measure on $Γ$. We prove in the present article that this harmonic measure on $Γ$ satisfies the $A^\infty$-property, that is the harmonic measure and the $d$-dimension Hausdorff measure on $Γ$ are mutually absolutely continuous in a quantitative and scale invariant way. Thus, we give an alternate proof of a recent theorem of David and Mayboroda, which itself extends a result of Hofmann and Martell to the case where the uniformly rectifiable set $Γ$ is not of codimension 1. The proof is surprisingly simple - in particular does not follow the route used by David and Mayboroda, or by Hofmann and Martell - but is specific to the case when $d<n-1$.

math.AP↗

arXiv2022

A change of variable for Dahlberg-Kenig-Pipher operators

In the present article, we purpose a method to deal with Dahlberg-Kenig-Pipher (DPK) operators in boundary value problems on the upper half plane. We give a nice subclass of the weak DKP operators that generates the full class of weak DKP operators under bi-Lipschitz changes of variable on $\mathbb R^n_+$ that fixe the boundary $\mathbb R^{n-1}$. Therefore, if one wants to prove a property on DKP operators which is stable by bi-Lipschitz transformations, one can directly assume that the operator belongs to the subclass. Our method gives an alternative proof to some past results and self-improves others beyond the existing literature.

math.AP↗

arXiv2021

The Green function with pole at infinity applied to the study of the elliptic measure

In $\mathbb R^{d+1}_+$ or in $\mathbb R^n\setminus \mathbb R^d$ ($d<n-1$), we study the Green function with pole at infinity introduced by David, Engelstein, and Mayboroda. In two cases, we deduce the equivalence between the elliptic measure and the Lebesgue measure on $\mathbb R^d$; and we further prove the $A_\infty$-absolute continuity of the elliptic measure for operators that can be related to the two previous cases via Carleson measures, extending the range of operators for which the $A_\infty$-absolute continuity of the elliptic measure is known.

math.AP↗

arXiv2021

Green function estimates on complements of low-dimensional uniformly rectifiable sets

It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the "flagship" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators $L_{β,γ} =- {\rm div} D^{d+1+γ-n} \nabla$ associated to a domain $Ω\subset \mathbb R^n$ with a uniformly rectifiable boundary $Γ$ of dimension $d < n-1$, the now usual distance to the boundary $D = D_β$ given by $D_β(X)^{-β} = \int_Γ |X-y|^{-d-β} dσ(y)$ for $X \in Ω$, where $β>0$ and $γ\in (-1,1)$. In this paper we show that the Green function $G$ for $L_{β,γ}$, with pole at infinity, is well approximated by multiples of $D^{1-γ}$, in the sense that the function $\big| D\nabla\big(\ln\big( \frac{G}{D^{1-γ}} \big)\big)\big|^2$ satisfies a Carleson measure estimate on $Ω$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the "magical" distance function from a previous work from the first author, the third author, and Max Engelstein.

math.AP↗

arXiv2018

Dirichlet problem in domains with lower dimensional boundaries

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $Ω:= \mathbb R^n \setminus \mathbb R^d$ with $d 1$ provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d=n-1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n-1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

math.AP↗