In Search of Sharp Inequalities

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0072037Banuelos Firstly, ideas from martingales theory will be used to investigate sharp inequalities for certain singular integrals which have played a fundamental role in several different areas of harmonic analysis, partial differential equations and the calculus of variations. These investigations lead to connections of probability to a conjecture of Morrey which asserts that rank--one convexity does not imply quasiconvexity in dimensions larger than 2. Secondly, extremal problems for the hyperbolic metric will be investigated. These problems arise from the conformal invariance of Brownian motion and conditioned Brownian motion in simply connected domains in the plane. These investigations in turn lead to new problems for martingales and Brownian motion. These problems will also be explored. Morrey's conjecture is important because of its implications in constructing minimizers in several problems in the calculus of variations describing various physical systems. The connection of these problems to probability was discovered by the PI and his collaborator, G. Wang and A.J. Lindeman, while investigating sharp inequalities for stochastic integrals and more general martingales. These connections lead to several questions which if false prove Morrey's's conjecture, a long standing open problem, and if true prove a conjecture concerning the norm of certain singular integrals, another long standing open problem. The solution to the proposed problems on the hyperbolic metric will yield a deeper understanding of various isoperimetric (extremal) inequalities for Brownian motion and conditioned Brownian motion. As with previous research projects of the PI, graduate students will be (are already) involved.