Stephen Semmes

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.

On the nonexistence of bilipschitz parameterizations and geometric problems about

How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of Rn are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A8 weight.

A_\infty

(1994). A primer on hardy spaces, and some remarks on a theorem of evans and muller. Communications in Partial Differential Equations: Vol. 19, No. 1-2, pp. 277-319.

-weights

Abstract We introduce a notion for hypersurfaces in R d + 1 that is analogous to the chord-arc condition with small constant for curves in the plane. We give various equivalent operator-theoretic, function-theoretic, and geometrical characterizations of these surfaces. We not know any nice potential-theoretic characterizations. In a companion paper we address the issue of finding good parameterizations for these surfaces.

A primer on hardy spaces, and some remarks on a theorem of evans and müller

V. V. Peller has characterized the Hankel operators which belong to the Schatten class, 1≦p<∞. We extend this characterization to 0<p<1 and give applications involving commutators and rational functions.

Chord-arc surfaces with small constant, I☆

A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a metric space admits a quasisymmetric parametrization by providing examples of spaces with many Eucledian-like properties which are nonetheless substantially different from Euclidean geometry. These examples are geometrically self-similar versions of classical topologically self-similar examples from geometric topology, and they can be realized as codimension 1 subsets of Euclidean spaces. Unlike earlier examples going back to Rickman, these sets enjoy good bounds on their geodesic distance functions and good mass bounds (Ahlfors regularity). They are also smooth except for reasonably tame degenerations near small sets, they are uniformly rectifiable, and they have good properties in terms of analysis (like Sobolev and Poincare inequalities). The construction also produces uniform domains which have many nice properties but which are not quasiconformally equivalent to balls.

Trace ideal criteria for Hankel operators, and applications to Besov spaces

Most problems in the ensuing list are of fairly recent origin. None of them seem easy and some are likely to be very difficult. The formulation of each problem is such that it can be answered by one word only: either yes or no. (Strictly speaking, it is conceivable that within the same question, the answer sometimes depends on the dimension.) We offer no conjectures or guesses. In many cases, the particular question is just a chosen concise representative from a whole group of related open problems. Whenever known, we shall point out the original source of a question. Otherwise, the question is either a folk question, a modification of a folk question, or suggested by one or both of the authors. We apologize in advance for all omissions and misquotes. Practically all the problems require some background definitions; many concepts that are being used have only recently been introduced, and are perhaps not so widely known. Typically, the question is stated first, and the relevant definitions and references are given right afterwords. To keep this essay brief, we give little or no motivation here. For this purpose, we kindly invite the reader to consult the literature as referred to in the text. This list of questions was born at the Institut des Hautes Études Scientifiques in August 1996. The choices we made were necessarily partial but still somewhat arbitrary. There certainly are many more problems around these topics that we deem equally worthy.

Good metric spaces without good parameterizations

On donne des conditions geometriques sur une hypersurface de R n pour que certaines integrales singulieres sur cette hypersurface definissent des operateurs bornes sur L 2

Thirty-three yes or no questions about mappings, measures, and metrics

Abstract The direct way to estimate the singular values of a compact operator is to decompose it as a sum of orthogonal rank one pieces. However, such decompositions can generally not be found in practice. We give a variation of the decomposition using nearly weakly orthonormal (NWO) sequences. The NWO decomposition is easier to do in examples but is still strong enough to give good singular value estimates. We illustrate this by giving sharp trace ideal estimates for the double layer potential and for the first and higher commutators of multiplication operators and Calderon-Zygmund operators. In particular, NWO sequences seem well suited for dealing with operators which depend nonlinearly on their symbol. We also show that the class of operators which admit NWO expansions is well behaved under composition, under conjugation by weights, and under conjugation by changes of variable.

A criterion for the boundedness of singular integrals on hypersurfaces

Abstract Every function having bounded mean oscillation (BMO) on the line is written as a sum of coefficients times normalized rational functions with the coefficients satisfying a Carleson measure packing condition. This decomposition theorem and related techniques are used to obtain operator norm estimates for first and second order commutators and direct and converse singular value estimates for Hankel operators and products of Hankel operators.