Uniform estimates in one- and two-dimensional time-frequency analysis

Abstract

This thesis is concerned with two special cases of the singular Brascamp-Lieb inequality, namely, the trilinear forms corresponding to the one- and two-dimensional bilinear Hilbert transform. In this work we study the uniform estimates in the parameter space of these two objects. The questions of the uniform bounds in one dimension arose from investigating Calderon's commutator, implying an alternative proof of its boundedness. Another reason for studying this problem is that, as the parameters degenerate, one can recover the bounds for the classical Hilbert transform, which is a well understood operator. Analogously, it is natural to investigate the two dimensional form, whose parameter space turns out to be considerably more involved and offering many more questions concerning the uniform bounds.<br /> The thesis consists of four chapters.<br /> In Chapter 1 we investigate the parameter space of the bilinear Hilbert transform. We complete the classification of the two dimensional form that was first given by Demeter and Thiele. We also describe the parameter space, reducing its dimensionality, and discuss the related geometry, which raises many open questions concerning the uniform bounds in two dimensions.<br /> In Chapter 2 we prove the uniform bounds for the bilinear Hilbert transform in the local L^1 range, which extends the previously known range of exponents for this problem. This a joint work with Gennady Uraltsev.<br /> In Chapter 3, which is an elaboration on Chapter 2, we prove the uniform bounds for the Walsh model of the bilinear Hilbert transform in the local L^1 range in the framework of the iterated outer L^p spaces. This theorem was already proven by Oberlin and Thiele, however, in their work they did not use the outer measure structure.<br /> Finally, Chapter 4 is dedicated to proving the uniform bounds for the Walsh model of the two dimensional bilinear Hilbert transform, in a two parameter setting in the vicinity of the triple that corresponds to the two dimensional singular integral.