A Weak Type Inequality for Martingale Transforms and Other Subordinate Martingales
Suh, Jiyeon
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https://hdl.handle.net/2142/86815
Description
Title
A Weak Type Inequality for Martingale Transforms and Other Subordinate Martingales
Author(s)
Suh, Jiyeon
Issue Date
2003
Doctoral Committee Chair(s)
Burkholder, Donald L.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We study a problem of finding the best constant in a weak type inequality for martingale transforms extending the result of Burkholder (1966). First, we study the inequality for the discrete-time martingale case. We present examples of martingales that give good lower estimates of the best constant. We then find a biconcave function to prove that the supremum of these lower estimates is in fact the best constant. We use this biconcave function to prove a sharp weak type inequality for differentially subordinate martingales with the same best constant, and by approximation a similar inequality for stochastic integrals. We generalize these results to the continuous-time case and give an application to harmonic functions.
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