Littlewood-Paley functions associated to second order elliptic operators
type: Item
instance of: scholarly article
title: Littlewood-Paley functions associated to second order elliptic operators
zbMATH Open document ID: 1067.42013
published in: Mathematische Zeitschrift
publication date: 2004-07-01T00:00:00Z
review text: Let \(A\) be an \(n\times n\) matrix of complex, \(L^\infty\) coefficients defined on \({\mathbb R}^n\), and satisfying the ellipticity condition. Define a divergence form operator \[ Lf\equiv-\text{div}(A\nabla f). \] The generalized vertical Littlewood-Paley \(G_L\)-function associated with the divergence form operator \(L\) is defined by \[ G_L(f)(x)=\left(\int_0^\infty t| \nabla e^{-t\sqrt{L}}f(x)| ^2\,dt\right)^{1/2}. \] Let \(p_n=\frac{2n}{n+2}\). In this paper, the author proves that \(G_L\) is bounded on \(L^q({\mathbb R}^n)\) for \(p_n<q\leq 2\), and \(G_L\) is of weak-type \((p_n,p_n)\). Note that if \(L\) is the Laplacian \(-\triangle\) on \({\mathbb R}^n\), then \(G_L\) is the classical Littlewood-Paley \(g_x\)-function defined by \[ g_x(f)(x)=\left(\int_0^\infty | \nabla_x u(x,y)| ^2y\,dy\right)^{1/2}, \] where \(u(x,y)=P_y\ast f(x)\) and \(P_y(x)\) is the Poisson kernel.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 2078068
zbMATH Keywords: Littlewood-Paley function
zbMATH Keywords: second order elliptic operator
zbMATH Keywords: divergence form operator
author: Lixin Yan
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: s00209-003-0606-z
OpenAlex ID: W2028058856
DOI: 10.1007/S00209-003-0606-Z
description: scientific article; zbMATH DE number 2078068