Geometric harmonic analysis for diffusive heat equations

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The interaction between harmonic analysis and partial differential equations (PDE) has a long history going back to the Dirichlet problem for the Laplace equation. Nowadays, a fruitful ground for the interplay is the study of various aspects of PDE. With special emphasis on the geometric capacity-measure theory approach, this proposed research program will be centered about the boundary between the above two core areas via the Cauchy problem for the linear-nonlinear heat equations. Particular topics to be investigated include:* Spacetime estimates for the parabolic Riesz type potentials and capacities generated by solutions to the inhomogeneous heat equations.* Wellposedness of the nonlinear heat equations such as the semilinear dissipative, dissipative quasi-geostrophic, and generalized Euler and Navier-Stokes equations.* Carleson type measures versus Riesz type capacities through a priori estimations of solutions to the homogeneous heat and Schroedinger equations in terms of the initial data.* Function-theoretic and/or operator-theoretic characterizations of various function spaces associated to the heat semigroup operators.* Sharp geometric and/or analytic inequalities (e.g. the capacitary Brunn-Minkowski and logarithmic Sobolev estimates) related to the convexity of potentials and some dissipative equations.The principle ideas and techniques will involve developing some geometric features of harmonic analysis over function spaces and their operators, building on many significant advances in complex-harmonic analysis, operator-potential theory, and convex geometry made in the last decades by a number of people. It is expected that this research will provide useful tools for the field of evolution equations and convex geometric analysis.