Lᵖ to Lᑫ Estimates for the Circular Maximal Function

Author: Schlag, Wilhelm

Year: 1996

Degree: Dissertation (Ph.D.)

Advisor: Wolff, Thomas H.

Committee Members: Wolff, Thomas H.; Makarov, Nikolai G.; Wilson, Richard M.

Option: Mathematics

DOI: 10.7907/4tq4-p076

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.

Files

• Schlag_W_1996.pdf (application/pdf)