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Hormander Type Multipliers on anisotropic Hardy spaces

Acta Mathematica Sinica,English Series 2019年第11期35卷 1841-1853页

作者： Jiao CHEN Liang HUANG School of Mathematical Sciences Chongqing Normal UniversityChongqing 401131P.R.China School of Mathematical Sciences Beijing Normal UniversityBeijing 100875P.R.China

The main purpose of this paper is to establish, using the Littlewood–Paley–Stein theory(in particular, the Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp(Rn;A) associated with expensive dilation A:■Our main Theorem is the following: Assume that m(ξ) is a function on Rn satisfying ■with s > ζ--1(1/p-1/2). Then Tm is bounded from Hp(Rn;A) to Hp(Rn;A) for all 0 anisotropic sobolev space associated with expansive dilation A* on Rn.

关键词： Hormander multiplier Littlewood-Paley’s inequality anisotropic Hardy space anisotropic sobolev spaces

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Polynomials,Higher Order sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein spaces on Stratified Groups

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Acta Mathematica Sinica,English Series 2000年第3期16卷 405-444页

作者： Guozhen Lu Department of Mathematics,Wayne State University,Detroit,MI 48202,USA Department of Mathematics Wayne State University Detroit USA

This paper consists of three main *** of them is to develop local and global sobolev interpolation inequalities of any higher order for the nonisotropic sobolev spaces on stratified nilpotent Lie *** the extensive research after Jerison’s work[3]on Poincaré-type inequalities for Hrmander’s vector fields over the years,our results given here even in the nonweighted case appear to be *** interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector *** main tools to prove such inequalities are approximating the sobolev func- tions by polynomials associated with the left invariant vector fields on *** very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own *** the existence of such polynomials is the second main part of this *** results of these two parts have been announced in the author’s paper in Mathematical Research Letters[38]. The third main part of this paper contains extension theorems on anisotropic sobolev spaces on stratified groups and their applications to proving sobolev interpolation inequalities on(εδ)domains. Some results of weighted sobolev spaces are also given *** construct a linear extension operator which is bounded on different sobolev spaces *** particular,we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given sobolev *** are stated and proved for weighted anisotropic sobolev spaces on stratified groups.

关键词： Poincaré inequalities Extension theorems Interpolation inequalities anisotropic sobolev spaces A_p weights (ε,δ)domains Vector fields Polynomials on stratified groups

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anisotropic estimates for sub-elliptic operators

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Science China Mathematics 2008年第4期51卷 509-522页

作者： John BLAND Tom DUCHAMP Department of Mathematics University of Toronto Department of Mathematics Box 354350University of Washington

In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.

关键词： sub-elliptic operators anisotropic estimates anisotropic sobolev spaces Rumin complex contact manifolds 35H20 35B45 53D10 32V20

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