Harmonic analysis and partial differential equations: Sharp geometric inequalities, fully nonlinear equations and applications

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The proposed research projects involves topics that use harmonic analysis and partial differential equations. We will focus on finding the sharp constants and extremals for Moser-Trudinger-Onofri inequalities on the Heisenberg group, stratified groups, the complex sphere and more general CR manifolds Also to derive sharp Sobolev inequalities on the Heisenberg and more general stratified groups, and optimal geometric inequalities for high order differential operators. The project will use these to study applications in CR geometry, including the regularity of degenerate elliptic Monge-Ampere equations and also parabolic Monge-Ampere equations arising from the Gauss curvature flow and the regularity of fully nonlinear equations in the sub-elliptic setting. We shall also study the properties and applications of convex functions in such settings, some geometric embedding inequalities such as the Poincare and Sobolev inequalities associated with high order sub-elliptic derivatives and their applications to partial differential equations. The successful completion of the proposed projects in sub-elliptic structures requires substantially new techniques and innovative ideas which are not available in the classical cases.The proposed projects have applications to control theory, optimization, human vision and other topics in sciences and engineering.