Singular integrals and maximal operators related to Carleson's theorem and curves in the plane

Abstract

In this thesis we study several different operators that are related to Carleson's theorem and curves in the plane. <br /> An interesting open problem in harmonic analysis is the study of analogues of Carleson's operator that feature integration along curves. In that context it is natural to ask whether the established methods of time-frequency analysis carry over to an anisotropic setting. We answer that question and also provide certain partial bounds for the Carleson operator along monomial curves using entirely different methods.<br /> Another line of results in this thesis concerns maximal operators and Hilbert transforms along variable curves in the plane. These are related to Carleson-type operators via a partial Fourier transform in the second variable. A central motivation for studying these operators stems from Zygmund's conjecture on differentiation along Lipschitz vector fields. One of our results can be understood as proving a curved variant of this conjecture.