Proposal: DMS-9801410 Principal Investigator: Nets Katz Abstract: The purpose of this project is to study directional maximal operators in the plane. The major fact underlying this study is the existence of a Besicovitch set, which rules out the boundedness of maximal operators over all directions. When the set of directions is restricted, less is understood. Katz will work on unboundedness in all L^p spaces for maximal operators over Ahlfors regular sets of directions with dimension greater than zero. He will also try to work out Wolff's conjecture about sets that contain a d-dimensional set in each direction. Both of these problems involve finding elasticity in the known constructions of Besicovitch-like sets. A large part of mathematics (as well as physics and signal processing) involves the interaction between time and frequency. In signal processing, time is a one-dimensional continuum and a signal is a function of time. Much depends upon what can be said about the instantaneous frequency of the signal at a given time. Time and frequency also occur as a metaphor in quantum mechanics. Position takes the place of time and momentum the place of frequency. The Heisenberg uncertainty principle limits how instantaneously we may specify a frequency. It says that a note requires a length of time to be heard -- its wavelength. In the quantum mechanical metaphor, there is no need for time -- now called position -- to be one-dimensional. Thus, neither is frequency so constrained. The relevant ranges needed to specify frequencies become rectangles oriented in various directions. To answer questions about this type of universe, it is necessary to understand the interaction between rectangles in different directions. The purpose of this project is to improve that understanding in the setting of two-dimensional space.
