Guozhen Lu

Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications.

In this paper we mainly prove weighted Poincare inequalities for vector fields satisfying Hormanders condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the local regularity of weak solutions for certain classes of strongly degenerate differential operators formed by vector fields.

A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type

The purpose of this note is to study the relationship between the validity of L1 versions of Poincare’s inequality and the existence of representation formulas for functions as (fractional) integral transforms of first-order vector fields. The simplest example of a representation formula of the type we have in mind is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space R:

Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations

This paper proves Harnacks inequality for solutions to a class of quasilinear subelliptic differential equations. The proof relies on various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields

A PDE Perspective of the Normalized Infinity Laplacian

X_{1},\ldots, X_{m}

Convex functions on Carnot groups

satisfying Hormanders condition. The nonlinear subelliptic equations under study include the important p-sub-Laplacian equation, e.g.,

Existence and size estimates for the green's functions of differential operators constructed from degenerate vector fields

\sum_{j=1}^{m}X_{j}^{*}\left(|Xu|^{p-2}X_{j}u\right) =A|Xu|^{p}+B|Xu|^{p-1}+C|u|^{p-1}+D,\\ 1<p<\infty

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

where

Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type

|Xu|=\sum_{j=1}^{m}\left(|X_{j}u|^{2}\right)^{\frac{1}{2}}