Date of Award
Doctor of Philosophy (PhD)
Let (mu) be a positive Borel measure on the circle group T. If, for P (GREATERTHEQ) 1, there is a q > p such that (mu) defines a bounded convolution operator from L('p)(T) to L('q)(T), then (mu) is L('p)-improving or (mu) improves L('p). Here a result of Oberlin is extended. It is shown that a large class of Cantor-Lebesgue measures improve L('p)(T) for p > 1. Also, for the members of a particular sequence of Lebesgue measures associated with some homogeneous Cantor sets, which includes the middle-third Cantor-Lebesgue measure studied by Oberlin, estimates are obtained for the amount of improvement. These results are obtained by a reduction to inequalities concerning convolution against probability measures on finite cyclic groups.
Ritter, David Lawrence, "Some Singular Measures on the Circle Which Improve L('p) Spaces." (1981). LSU Historical Dissertations and Theses. 3654.