Pointwise regularity criteria
type: Item
instance of: scholarly article
title: Pointwise regularity criteria
zbMATH Open document ID: 1074.42006
author: Stéphane Jaffard
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-22T00:00:00Z
review text: For \(p\in (1,\infty)\), \(u>-d/p\) and \(x_0\in \mathbb{R}^d\) fixed, the Calderón-Zygmund class \(T^p_u(x_0)\) consists of those \(f\in L^p_{\text{ loc}}(\mathbb{R}^d)\) for which there is a polynomial \(P\) of degree less than \(u\) such that \[ \biggl(\frac{1}{r^d}\int_{B(x_0,r)} | f(x)-P(x-x_0)| ^p\, dx\bigg)^{1/p}\leq Cr^u \qquad(r>0). \] Suppose that one has a finite family \(\psi^{\epsilon}\), \(\epsilon \in E\), of compactly supported mother wavelets of suitable regularity that generate an orthonormal wavelet basis for \(L^2(\mathbb{R}^d)\). Let \(\psi_{I}^\epsilon\) be the \(L^\infty\) normalized dilate \(\psi^\epsilon_I(x) = \psi^\epsilon(2^jx-k)\) in which \(I=\prod_{i=1}^d 2^{-j}[k_i,\,k_i+1)\). One defines the local square function \[ S_f (j, x_0)=\left(\sum_{I\subset 3I_j(x_0); \,\varepsilon} | \langle f,\, \psi_I^\varepsilon\rangle| ^2 \chi_{I}(x)\right)^{1/2}. \] Here \(I_j(x_0)\) is the dyadic interval of sidelength \(2^{-j}\) containing \(x_0\), and the sum extends over all mother wavelet indices and over all dyadic subcubes of \(3I_j(x_0)\), the cube with the same center and three times the sidelength of \(I_j(x_0)\). It is proved that if \(f\in T^p_u(x_0)\), \(p\in (0,\infty)\), then there is a \(C\geq 0\) such that, for all \(j\geq 0\), \(\| S_f(j,x_0)\| _p\leq C 2^{-j(u+d/p)}\). When \(p=\infty\) the condition \(f\in T^\infty_u(x_0)\) becomes \[ \sum_{J\subset I;\, \epsilon} | J|\, | \langle f,\,\psi_J^\epsilon\rangle| ^2\leq C^2 2^{-uj}| I| \] whenever \(I\subset 3I(j,x_0)\) is dyadic. This condition is sharper than the usual two-microlocal condition.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2136414
zbMATH Keywords: Hölder regularity
zbMATH Keywords: wavelets
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.crma.2004.10.011
OpenAlex ID: W2035196628
cites work: Q3280291
cites work: Pointwise smoothness, two-microlocalization and wavelet coefficients
cites work: Q4664795
cites work: Wavelet analysis of fractal boundaries. II: Multifractal analysis
cites work: Q3995454
DOI: 10.1016/J.CRMA.2004.10.011
description: scientific article; zbMATH DE number 2136414