arXiv subjects
Publications and source records attributed to Emmanuel Russ.
We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(σ\nabla u)=0$ with $L^p$-boundary data.
We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms.
We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms.
Let $G$ be a real connected Lie group with polynomial volume growth, endowed with its Haar measure $dx$. Given a $C^2$ positive function $M$ on $G$, we give a sufficient condition for an $L^2$ Poincaré inequality with respect to the measure $M(x)dx$ to hold on $G$. We then establish a non-local Poincaré inequality on $G$ with respect to $M(x)dx$.
We prove a fractional version of Poincaré inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.
Let $Ω$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $Ω$ in weighted Lebesgue and Sobolev spaces on $Ω$, and rely this invertibility to a geometric characterization of $Ω$ and to weighted Poincaré inequalities on $Ω$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $Ω$ is Lipschitz or, more generally, when $Ω$ is a John domain, and focus on the case of $s$-John domains.
Let $Γ$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $Γ$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert (I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1 2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.
Let $M$ be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces $H^p$ of differential forms on $M$ and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the $H^p$-boundedness for Riesz transforms on $M$, generalizing previously known results. Further applications, in particular to $H^{\infty}$ functional calculus and Hodge decomposition, are given.
Let $Ω$ be a bounded $C^{2}$ domain in $\R^n$, and let $Ω^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $Ω$. Consider the operator $L=-÷(A\nabla)+v\cdot \nabla +V$ on $Ω$ with Dirichlet boundary condition. We prove that minimizing the principal eigenvalue of $L$ when the Lebesgue measure of $Ω$ is fixed and when $A$, $v$ and $V$ vary under some constraints is the same as minimizing the principal eigenvalue of some operators $L^*$ in the ball $Ω^*$ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in $Ω$ and the new ones in $Ω^*$ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when $Ω$ is not a ball.
Let $Ω$ be a bounded $C^{2,α}$ domain in $\R^n$ ($n\geq 1$, $0<α<1$), $Ω^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $Ω$, $τ\geq 0$ and $v\in L^{\infty}(Ω,\R^n)$ with $\left\Vert v\right\Vert\_{\infty}\leq τ$. If $λ\_{1}(Ω,τ)$ denotes the principal eigenvalue of the operator $-Δ+v\cdot\nabla$ in $Ω$ with Dirichlet boundary condition, we establish that $λ\_{1}(Ω,v)\geq λ\_{1}(Ω^{\ast},τe\_{r})$ where $e\_{r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $Ω=Ω^{\ast}$ and $v=τe\_{r}$. This result can be viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian.