Riesz meets Sobolev

Thierry Coulhon; Adam Sikora

Displaying similar documents to “Riesz meets Sobolev”

Riesz potentials and amalgams

Michael Cowling, Stefano Meda, Roberta Pasquale (1999)

Annales de l'institut Fourier

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Let $(M, d)$ be a metric space, equipped with a Borel measure $μ$ satisfying suitable compatibility conditions. An amalgam $A_{p}^{q} (M)$ is a space which looks locally like $L^{p} (M)$ but globally like $L^{q} (M)$ . We consider the case where the measure $μ (B (x, ρ)$ of the ball $B (x, ρ)$ with centre $x$ and radius $ρ$ behaves like a polynomial in $ρ$ , and consider the mapping properties between amalgams of kernel operators where the kernel $ker K (x, y)$ behaves like $d (x, y)^{- a}$ when $d (x, y) \leq 1$ and like $d (x, y)^{- b}$ when $d (x, y) \geq 1$ . As an application, we describe Hardy–Littlewood–Sobolev type regularity...

Some remarks on gradient estimates for heat kernels.

Dungey, Nick (2006)

Abstract and Applied Analysis

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Large time behaviour of heat kernels on non-compact manifolds: fast and slow decays

Thierry Coulhon (1998)

Journées équations aux dérivées partielles

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In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^{2}$ isoperimetric profile. The main point is to connect the decay of the $L^{1} - L^{\infty}$ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods....

Heat kernels on non-compact riemannian manifolds : a partial survey

Thierry Coulhon (1996-1997)

Séminaire de théorie spectrale et géométrie

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Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

Thierry Coulhon, Xuan Thinh Duong, Xiang Dong Li (2003)

Studia Mathematica

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We study the weak type (1,1) and the $L^{p}$ -boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions and ℋ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that and ℋ are bounded in $L^{p}$ , 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum....

$L^{p}$ boundedness of Riesz transforms for orthogonal polynomials in a general context

Liliana Forzani, Emanuela Sasso, Roberto Scotto (2015)

Studia Mathematica

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Nowak and Stempak (2006) proposed a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions, and proved their boundedness on L². Following them, we give easy to check sufficient conditions for their boundedness on $L^{p}$ , 1 < p < ∞. We also discuss the symmetrized version of these transforms.

Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator

Adam Nowak, Luz Roncal, Krzysztof Stempak (2010)

Colloquium Mathematicae

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We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding $L^{p}$ spaces, 1 < p < ∞.

Remark on the inequality of F. Riesz

W. Łenski (2005)

Banach Center Publications

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We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${| φ ₙ |}^{p}$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

An inversion formula and a note on the Riesz kernel

Andrejs Dunkels (1976)

Annales de l'institut Fourier

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For potentials $U_{K}^{T} = K * T$ , where $K$ and $T$ are certain Schwartz distributions, an inversion formula for $T$ is derived. Convolutions and Fourier transforms of distributions in $(D_{L}^{'} p)$ -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order $α$ , $0 < α < m$ , of a compact subset $E$ of $R^{m}$ has the following property: its restriction to the interior of $E$ is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M, g)$ for the following general heat equation $u_{t} = Δ_{V} u + a u log u + b u$ where $a$ is a constant and $b$ is a differentiable function defined on $M \times [0, \infty)$ . We suppose that the Bakry-Émery curvature and the $N$ -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. ...

Riesz transforms on connected sums

Gilles Carron (2007)

Annales de l’institut Fourier

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Assume that $M_{0}$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_{0}$ satisfies a Sobolev inequality of dimension $ν > 3$ . Let $M$ be a complete Riemannian manifold isometric at infinity to $M_{0}$ and let $p \in (ν / (ν - 1), ν)$ . The boundedness of the Riesz transform of $L^{p} (M_{0})$ then implies the boundedness of the Riesz transform of $L^{p} (M)$

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...