Guy David

A boundedness criterion for generalized Calderon-Zygmund operators

In 1965, A. P. Calderon showed the L2-boundedness of the so-called first Calderon commutator. This is one of the first examples of a non-convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the Hilbert transform. These properties, together with the L2-boundedness, imply the LP-boundedness for all ps in ]1, + oo[. Many operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral operator on a curve, are associated with kernels having these properties. For these operators, one of the major questions is if they are bounded on L2. We are going to give necessary and sufficient conditions for such an operator to be bounded on L2. They are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in BMO. In the case of the aforementioned first commutator this can be checked by an integration by parts. In the first section we present some basic notions and state the theorem, which is proved in Sections 2 and 3. In Section 4 we show how to recover some classical results. In Sections 5 and 6 we construct a functional calculus for small perturbations of A, and in Section 7 we show a connection between the theory of Calderon-Zygmund operators and Katos conjecture. It is a pleasure to express our thanks to R. R. Coifman and Y. Meyer for suggesting many elegant simplifications in our proofs and most of the applications. We also wish to thank Stephen Semmes for several pertinent remarks.

Analysis of and on uniformly rectifiable sets

The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant substitute for the classical notion of rectifiability; as the answer (sometimes only conjecturally) to certain geometric questions in complex and harmonic analysis; as a condition which ensures the parametrizability of a given set, with estimates, but with some holes and self-intersections allowed; and as an achievable baseline for information about the structure of a set. This book is about understanding uniform rectifiability of a given set in terms of the approximate behaviour of the set at most locations and scales. In addition to being a general reference on uniform rectifiability, the book also poses many open problems, some of which are quite basic.

Unrectictifiable 1-sets have vanishing analytic capacity

We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature, and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)-theorem to some spaces that are non necessarily of homogeneous type.

Removable sets for Lipschitz harmonic functions in the plane

The main motivation for this work comes from the century-old Painleve problem: try to characterize geometrically removable sets for bounded analytic functions in C.

The Problem of the Calissons

A calisson is a French sweet that looks like two equilateral triangles meeting along an edge. Calissons could come in a box shaped like a regular hexagon, and their packing would suggest an interesting combinatorial problem. Suppose a box with side of length n is filled with sweets of sides of length 1. The long diagonal of each calisson in the box is parallel to one of three different lines, as in the picture.

Analytic capacity, Calderón-Zygmund operators, and rectifiability

For

Uniform rectifiability and quasiminimizing sets of arbitrary codimension

K\subset\Bbb C

Quantitative rectifiability and Lipschitz mappings

compact, we say that

s-Numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions

K

Hausdorff dimension of uniformly non flat sets with topology

has vanishing analytic capacity (or