A parabolic Littlewood-Paley inequality with applications to parabolic equations
type: Item
instance of: scholarly article
title: A parabolic Littlewood-Paley inequality with applications to parabolic equations
zbMATH Open document ID: 0839.35017
DOI: 10.12775/TMNA.1994.033
published in: Topological Methods in Nonlinear Analysis
publication date: 1996-06-25T00:00:00Z
review text: The main result of this article is the following. Let \(K\in (0, \infty)\) and \(\psi(x)\) a \(C^\infty(\mathbb{R}^d)\) integrable function such that \[ \int_{\mathbb{R}^d} \psi dx= 0,\;|\psi|_{L_1(\mathbb{R}^d)}+ ||x|\psi|_{L_1(\mathbb{R}^d)}+ |\nabla \psi|_{L_1(\mathbb{R}^d)}+ |x\cdot \psi|_{L_1(\mathbb{R}^d)}\leq K. \] Define \(\Psi_t g(x)= t^{- d/2} \psi(x/\sqrt t)* g(x)\). Let \(H\) be a separable Hilbert space, \(p\in [2, \infty)\), \(- \infty\leq a< b\leq \infty\) and \(g\in L_p((a, b)\times \mathbb{R}^d, H)\). Then \[ \int_{\mathbb{R}^d} \int^b_a \Biggl[ \int^t_a |\Psi_{t- s} g(s, \cdot)(x)|^2_H {ds\over t- s}\Biggr]^{p/2} dt dx\leq N \int_{\mathbb{R}^d} \int^b_a|g(t, x)|^p_H dt dx, \] where the constant \(N\) depends only on \(d\), \(p\) and \(K\). The author also gives an extension of this result to parabolic equations with coefficients depending only on \(t\).
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 828288
zbMATH Keywords: Littlewood-Paley inequality
author: N. V. Krylov
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 828288