#### Date of Award

1981

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Abstract

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.

#### Recommended Citation

Ritter, David Lawrence, "Some Singular Measures on the Circle Which Improve L('p) Spaces." (1981). *LSU Historical Dissertations and Theses*. 3654.

http://digitalcommons.lsu.edu/gradschool_disstheses/3654

#### Pages

37