Richard L. Wheeden

Positive harmonic functions on Lipschitz domains

0. Introduction. The results of this paper are based on a study of certain kernel functions associated with Lipschitz domains D. These functions are related to harmonic measure in D and to the ideal boundary of D as defined by R. S. Martin (see [6]), and are analogous to the Poisson kernel. Let DczEn + i he a Lipschitz domain with a point P0 fixed. We say that u is a kernel function at Q0 e 8D if u(P) is positive and harmonic for P e D with u(P0) = 1 and u(P) vanishes continuously as P-+ Q for each Q e 3D, Q^QoOne fundamental result of the paper is a uniform estimate for various approximations to kernel functions and another is the uniqueness of kernel functions. We use the uniform estimate to show that functions arising from several different constructions are in fact kernel functions, and the uniqueness then leads to further results. The first application is to kernels related to harmonic measure. Recall that the generalized solution of the Dirichlet problem in D for continuous boundary values f(Q) is f f(Q)K(P, Q) dcopo(Q), PeD, JdD

Weighted norm inequalities for singular and fractional integrals

Inequalities of the form 11 lxl cTf 1,q < C 11 lxl af 11, are proved for certain well-known integral transforms, T, in En. The transforms considered include Calder6n-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.

Weighted sobolev-poincaré inequalities for grushin type operators

(1994). Weighted sobolev-poincare inequalities for grushin type operators. Communications in Partial Differential Equations: Vol. 19, No. 3-4, pp. 523-604.

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:

Hölder continuity of weak solutions to subelliptic equations with rough coefficients

Introduction Comparisons of conditions Proof of the general subellipticity theorem Reduction of the proofs of the rough diagonal extensions of Hormanders theorem Homogeneous spaces and subrepresentation inequalities Appendix Bibliography.

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.

Fractional Integrals on Weighted H p and L p Spaces

We study the two weight function problem IIafII Hq <C cfl11P , 0 <p p q < o0, for fractional integrals on Hardy spaces. If u and In satisfy the doubling condition and 0 < p < 1, we obtain a necessary and sufficient condition for the norm inequality to hold. If 1 < p < oc we obtain a necessary condition and a sufficient condition, and show these are the same under various additional condi- tions on u and v. We also consider the corresponding problem for Lq and LP, and obtain a necessary and sufficient condition in some cases.

Degenerate Sobolev spaces and regularity of subelliptic equations

We develop a notion of degenerate Sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients. These Sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local Holder continuity of solutions, extending our results in an earlier work, where we studied regularity of classical weak solutions. In cases when the quadratic forms arise from collections of rough vector fields, we study containment relations between the degenerate Sobolev spaces and the corresponding spaces defined in terms of weak derivatives relative to the vector fields.

Weighted Poincaré Inequalities for Hörmander Vector Fields and local regularity for a class of degenerate elliptic equations

In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnacks inequality) for a class of degenerate elliptic equations with measurable coefficients.

WEIGHTED NORM INEQUALITIES FOR INTEGRAL OPERATORS

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of “testing type,” like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type (Lp, Lq) estimates. We show that in such a space it is possible to characterize these estimates by testing them only over “cubes”. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.