A basic theme in the wonderful books and surveys of Stein, Weiss, and Zygmund is that Hilbert transforms, Poisson kernels, heat kernels, and related objects are quite interesting and fundamental. I certainly like this point of view. There is a variety of ways in which things can be interesting or fundamental, of course. In the last several years there have been striking developments connected t...

It is well known that the disconnectedness of a non-Archimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy to that difficulty and study measure theory and integration on the Levi-Civita field. After reviewing basic elements of calculus on the field, we introduce a measure that proves to be a natural gene...

Let ? be a closed set in R n with the Lebesgue measure j?j = 0. The rst aim of the paper is to give a Fourier analytical characterization of the Hausdorr dimension of ?. Let 0 < d < n. If there exist a Borel measure with supp ? and constants c 1 > 0 and c 2 > 0 such that c 1 r d (B(x; r)) c 2 r d for all 0 < r < 1 and all x 2 ?, where B(x; r) is a ball with centre x and radius r, then ? is call...

It is well-known that for every N ≥ 1 and d ≥ 1 there exist point sets x1, . . . , xN ∈ [0, 1] whose discrepancy with respect to the Lebesgue measure is of order at most (logN)d−1N−1. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1] there exist points x1, . . . , xN whose discrepancy with respect to μ is of order at most (lo...

Gaussian measure is constructed for any given hyperplane in an infinite dimensional Hilbert space, and this is used to define a generalization of the Radon transform to the infinite dimensional setting, using Gauss measure instead of Lebesgue measure. An inversion formula is obtained and a support theorem proved.

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-intege...

Define quantified S4, QS4 [first-order S4, FOS4], by combining the axioms and rules of inference of propositional S4 with the axioms and rules of classical first order logic without identity [with identity]. In the 1950’s, Rasiowa and Sikorski extended the algebraic semantics for propositional S4 to a constant-domain algebraic semantics for QS4, and showed that QS4 is sound and complete for thi...

Given an (ordinary) super-Brownian motion (SBM) % on R d of dimension d = 2; 3; we consider a (catalytic) SBM X % on R d with \local branching rates" %s(dx). We show that X % t is absolutely continuous with a density function % t ; say. Moreover, there exists a version of the map (t; z) 7 ! % t (z) which is C 1 and solves the heat equation oo the catalyst %, more precisely, oo the (zero set of)...