Mitchell H. Taibleson

The characterization of the Triebel-Lizorkin spaces forp=∞

We establish the characterization of the weighted Triebel-Lizorkin spaces for p=∞ by means of a “generalized” Littlewood-Paley function which is based on a kernel satisfying “minimal” moment and Tauberian conditions. This characterization completes earlier work by Bui et al. The definitions of the Ḟ∞,qα spaces are extended in a natural way to Ḟ∞,∞α and it is proven that this is the same space as Ḃ∞,∞α, which justifies the standard convention in which the two spaces are defined to be equal. As a consequence, we obtain a new characterization of the Hölder-Zygmund space Ḃ∞,∞α.

Estimates for Finite Expansions of Gegenbauer and Jacobi Polynomials

Publisher Summary This chapter presents estimates for certain Gegenbauer polynomial and Jacobi polynomial expansions. The primary interest is on expansions of a form that are needed to extend a result of E.M. Stein. The chapter concludes with the construction of functions on Σ n , the n-dimensional sphere in R n+l and on other compact rank 1 symmetric spaces.

Brownian motion characterization of some Besov-Lipschitz spaces on domains

We characterize the Besov-Lipschitz spaces with zero boundary conditions on bounded smooth domains. We prove that the appropriate first and second difference norms are equivalent to the norm given in terms of the transition kernel of the Brownian motion killed upon exit from the domain.

Harmonic functions on cartesian products of trees with finite graphs

Abstract Let G be a graph which is the Cartesian product of an infinite, locally finite tree T and a finite, connected graph A . On G , consider a stochastic transition operator P giving rise to a transient random walk and such that positive transitions occur only along the edges of G . We construct a matrix-valued kernel on T , which extends naturally in the second variable to the space of ends Ω of T . This kernel is used to derive a unique integral representation over Ω of all—not necessarily positive—functions on G which are harmonic with respect to P . We explain the relation with the Martin boundary and the positive harmonic functions and, as a particular case, we show what happens when A arises from a finite abelian group and P is compatible with the structure of A .

Harmonic measure of the planar Cantor set from the viewpoint of graph theory

Abstract This paper outlines a graph-theoretical approach to the study of the harmonic measure on the two-dimensional Cantor set. The Cantor set is regarded as the space of ends of a (nonplanar) graph with a tree-like structure. The method is based upon the combinatorics of the random walk with internal states induced on this graph by Brownian motion, and it could be used for numerical approximation.