Lᵖ to Lᑫ Estimates for the Circular Maximal Function

Citation

Schlag, Wilhelm (1996) Lᵖ to Lᑫ Estimates for the Circular Maximal Function. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4tq4-p076. https://resolver.caltech.edu/CaltechTHESIS:07142025-185158120

Abstract

In this thesis we establish sharp L^p → L^q bounds for the circular maximal function in the plane. This is accomplished by interpolating a L^5/2 → L⁵ endpoint estimate with Bourgain's well-known L^p → L^p bounds. The endpoint estimate is proved by combining the geometric/combinatorial method of Kolasa- Wolff with a L² inequality on a small ball. The L^p → L^q estimates for the circular maximal function established in this thesis would be a consequence of C. Sogge's sharp local smoothing conjecture for the wave equation.

Thesis Files