arXiv subjects

Joseph Feneuil

Publications and source records attributed to Joseph Feneuil.

27 records · Page 2Linked to original sources

arXiv2018

A new elliptic measure on lower dimensional sets

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of $n-1$ dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

math.AP↗

arXiv2017

About the $L^2$ analyticity of Markov operators on graphs

Let $Γ$ be a graph and $P$ be a reversible random walk on $Γ$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $Γ$ but only a polynomial control of the volume.

math.PR↗

arXiv2017

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\mathbb R^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond. In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph $Γ$ of dimension $d$ in $\mathbb R^n$, $d<n-1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator $L$ such that the corresponding harmonic measure $ω_L$ is absolutely continuous with respect to the Hausdorff measure on $Γ$. More generally, we provide sufficient conditions on the matrix of coefficients of $L$ which guarantee the mutual absolute continuity of $ω_L$ and the Hausdorff measure.

math.AP↗

arXiv2016

Harmonic measure on sets of codimension larger than one

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the $d$-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.

math.AP↗

arXiv2016

Riesz transform on graphs under subgaussian estimates

Let $Γ$ be a doubling graph satisfying some pointwise subgaussian estimates of the Markov kernel. We introduce a space $H^1(Γ)$ of functions and a space $H^1(T_Γ)$ of 1-forms and give various characterizations of them. We prove the $H^1$-boundedness of the Riesz transform, from which we deduce the $L^p$ boundedness of the Riesz transform for any $p\in (1,2)$. In a previous work, we showed a $H^1_w$-boundedness of the Riesz transform under weaker assumptions, but the $L^p$ boundedness was not established.

math.FA↗

arXiv2015

Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound

We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on $L^p$ for $1 \textless{} p \textless{} 2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this.In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1 \textless{} p \textless{} 2$. This yields a full picture of the ranges of $p\in (1,+\infty)$ for which respectively the Riesz transform is $L^p$ -bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

math.CA↗

arXiv2015

Littlewood-Paley functionals on graphs

Let $Γ$ be a graph equipped with a Markov operator $P$. We introduce discrete fractional Littlewood-Paley square functionals and prove their $L^p$-boundedness under various geometric assumptions on $Γ$.

math.FA↗

arXiv2015

Algebra properties for Besov spaces on unimodular Lie groups

We consider the Besov space $B^{p,q}_α(G)$ on a unimodular Lie group $G$ equipped with a sublaplacian $Δ$. Using estimates of the heat kernel associated with $Δ$, we give several characterizations of Besov spaces, and show an algebra property for $B^{p,q}_α(G) \cap L^\infty(G)$ for $α>0$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$. These results hold for polynomial as well as for exponential volume growth of balls.

math.AP↗

arXiv2014

Hardy and BMO spaces on graphs, application to Riesz transform

Let $Γ$ be a graph with the doubling property for the volume of balls and $P$ a reversible random walk on $Γ$. We introduce $H^1$ Hardy spaces of functions and $1$-forms adapted to $P$ and prove various characterizations of these spaces. We also characterize the dual space of $H^1$ as a $BMO$-type space adapted to $P$. As an application, we establish $H^1$-$H^1$ and $H^1$-$L^1$ boundedness of the Riesz transform.

math.CA↗