Sharp Inequalities

Purdue University, West Lafayette IN

Investigators

Abstract

The thrust of this project concerns the introduction of new probabilisitic techniques to study properties of certain discrete transformations that play an important role in the theory of harmonic analysis and its applications in signal processing in the time/frequency domains. The transformations in question include several versions of the Hilbert transform on the integers in dimension one and on the lattice in several dimensions. These basic transformations were introduced by David Hilbert at the beginning of the 20th century as simpler models of their continuous counterparts. For many applications, discrete models are much simpler from the point of view of computations. One of the goals of the project is to show that the magnitudes of the discrete transformations, as measured by certain summability properties of the sequences they transform, coincide with those of their continuous counterparts. Related geometric problems concerning the long time behavior of Brownian motion and more general stochastic processes under certain natural constrains, will be studied as well. The project deals with several problems and conjectures for sharp inequalities which lie at the interface of probability, harmonic analysis and spectral, potential theoretic, and geometric properties of the Laplacian and the fractional Laplacian. A problem of significant interest in the early part of the 20th century was the question of how the size of a periodic function controls the size of its conjugate, where the size is measured by the Lebesgue Lp-norm, where p is strictly between 1 and infinity. In 1925, M. Riesz answered this question in his celebrated paper on the Lp boundedness of the Hilbert transform and showed that this implies the same for the discrete version acting on the space of doubly infinite sequences in little lp. Shortly thereafter, E.C. Titchmarsh gave a direct proof of the boundedness of the discrete Hilbert transform and showed that the norms of these two operators are equal. The following year Titchmarsh pointed out that his argument for equality was incorrect. The question of equality had been a long-standing open problem since 1927. In a recent publication, M. Kwasnicki and the PI solved this problem by identifying the sharp lp bound for the discrete Hilbert transform which turns out to be the same as the sharp Lp bound found for the continuous version found by Pichorides in the early 70's. The first part of this project discusses closely related problems for discrete operators in one and several dimensions. The second part of the project addresses questions of sharp inequalities, known as stability (or deficit) inequalities, for exit times of Brownian motion and symmetric stable processes from subsets of finite volume in Euclidean spaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.