Riesz meets Sobolev

Sobolev meets Poincark

We prove that a very weak form of the PoincarC inequality implies a Sobolev-Poincare ineq~iality in the abstract setting of metric spaces. Sobolev rencontre Poincare Version jkzngaise abrigie L e but de cette Note est la demonstration du theoreme 1 qui affirme que, dans le cadre tres general des espaces metriques, une inegalite fiaible de Poincare entraine une inegalite de Sobole\--Poincare. Ce...

Riesz Bases in Sobolev Spaces 1792 Hb

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L2(R) of [50] to a pair of dual Sobolev spaces H(R) and H−s(Rd). In terms of masks for φ, ψ, . . . , ψ ∈ H(R) and φ̃, ψ̃, . . . , ψ̃ ∈ H−s(Rd), simple sufficient conditions are given to ensure that (X(φ;ψ, . . . , ψ), X−s(φ̃; ψ̃, . . . , ψ̃)) forms a pair of dual wavelet frames in (Hs(Rd),H−s(Rd)), where X(φ;ψ, . . . , ψ) := {φ(· − k) : k ∈ Zd} ...

Hölder Continuity of Sobolev Functions and Riesz Potentials

Let v be a distribution on RN with gradient in Lp for some 1 ≤ p <∞ and let γ ∈ (0, 1) if p ≤ N, γ ∈ [1−N/p, 1) if p > N. The main result of this paper states that if |x|(1−γ)p−N ∗ |∇v|p ∈ L∞, then v ∈ C0,γ(RN ). The trivial case p > N and γ = 1−N/p is Morrey’s theorem. An investigation of the condition |x|(1−γ)p−N ∗|∇v|p ∈ L∞ produces other special cases that do not restrict p. Some rely on “u...

Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs

Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖p and ∥(I − P )f ∥∥ p uniformly in f for 1 < p < +∞. These conditions are different for p < 2 and p > 2. The proofs rely on recent techniques developed to hand...