Nonlinear Equations, Weighted Norm Inequalities, and Best Constants in Harmonic Analysis

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ABSTRACT:The proposed research is concerned with the investigation of certain classes of nonlinear partial differential and integral equations, andrelated weighted norm inequalities. A systematic use will be made of function spaces intrinsically connected with the problems involved, dyadic linear and nonlinear models, nonlinear potential theory, quasimetrics on spaces of homogeneous and nonhomogeneous type (without the doubling property), and other means of modern analysis. The main goal is to characterize completely the solvability problem (i.e., find necessary and matching sufficient conditions), and obtain sharp estimates for solutions of nonlinear elliptic and parabolicPDEs with very general coefficients at the lower order terms and data, as well as for nonlinear integral operators with general kernels.Linear problems for operators of Schrodinger type with general potentials (possibly complex valued distributions) will be considered as well. A good control of the constants involved is an essential part of this study. In particular, a special attention will be paid to best constant inequalities which appear in related problems of harmonic analysis and operator theory.Many questions considered in the proposed research are motivated bystudies in mathematical physics, control theory, and stochastic processes. As a result, sharp inequalities and criteria of solvability will be found for equations and operators which describe important phenomena with linear and nonlinear sources appearing in quantum physics, fluid flow, heat transfer, and electromagnetism problems.