some Results on Embedding Spaces of Sobolev Type into Lorentz Spaces

Lorentz spaces L^(s,l) provide a deeper insight into integrable functions than L^s spaces do. Various results of the embedding of the following anisotropic spaces B_(p,).~ _p^and Ⅳ_p~ in- to L^(s,l) together with a counterexample are obtained.

A PRIORI ESTIMATES TO THE MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH ANISOTROPIC GROWTH CONDITIONS

In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.

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

This paper consists of three main parts.One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups.Despite 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 new.Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector fields.The main tools to prove such inequalities are approximating the Sobolev func- tions by polynomials associated with the left invariant vector fields on G.Some very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights.Finding the existence of such polynomials is the second main part of this paper.Main 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 here.We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously.In 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 functions.Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

Hormander Type Multipliers on Anisotropic Hardy Spaces

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 < p ≤ 1 and ■where A* denotes the transpose of A. Here we have used the notations mj(ξ) = m(A*jξ)φ(ξ) and φ(ξ) is a suitable cut-off function on Rn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on Rn.

Anisotropic estimates for sub-elliptic operators

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.

A Note on the Illposedness for Anisotropic Nonlinear Schrdinger Equation

In this short note, we show the illposedness of anisotropic Schroedinger equation in L^2 if the growth of nonlinearity is larger than a threshold power pc which is also the critical power for blowup, as Fibich, Ilan and Schochet have pointed out recently. The illposedness in anisotropic Sobolev space Hk,d-d^2s,s where 0 〈 s 〈 sc, sc =d/2-k/4-2/p-1, and the illposedness in Sobolev space of negative order H^s, s 〈 0 are also proved.