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

Guozhen Lu — 1992

Revista Matemática Iberoamericana

In this paper we mainly prove weighted Poincaré inequalities for vector fields satisfying Hörmander's 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...

Covering lemmas and BMO estimates for eigenfunctions on Riemannian surfaces.

Guozhen Lu — 1991

Revista Matemática Iberoamericana

The principal aim of this note is to prove a covering Lemma in R. As an application of this covering lemma, we can prove the BMO estimates for eigenfunctions on two-dimensional Riemannian manifolds (M, g). We will get the upper bound estimate for || log |u| ||, where u is the solution to Δu + λu = 0, for λ > 1 and Δ is the Laplacian on (M, g). A covering lemma on homogeneous spaces is also obtained in this note.

Unique continuation with weak type lower order terms: The variable coefficient case.

Guozhen Lu — 1995

Publicacions Matemàtiques

This paper deals with the unique continuation problems for variable coefficient elliptic differential equations of second order. We will prove that the unique continuation property holds when the variable coefficients of the leading term are Lipschitz continuous and the coefficients of the lower order terms have small weak type Lorentz norms. This will improve an earlier result of T. Wolff in this direction.

High order representation formulas and embedding theorems on stratified groups and generalizations

Guozhen LuRichard Wheeden — 2000

Studia Mathematica

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable L 1 to L 1 Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and L 1 to L 1 Poincaré...

Representation formulas and weighted Poincaré inequalities for Hörmander vector fields

Bruno FranchiGuozhen LuRichard L. Wheeden — 1995

Annales de l'institut Fourier

We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the L 1 versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.

Subelliptic Poincaré inequalities: the case p < 1.

Stephen M. BuckleyPekka KoskelaGuozhen Lu — 1995

Publicacions Matemàtiques

We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p < 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.

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