Partial Differential Equations

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This project is mainly concerned with the study of the regularity ofsolutions of partial differential equations that arise in the theory ofseveral complex variables and on CR manifolds. In particular we plan tostudy various properties of subelliptic multipliers and alsohypoellipticity when subellipticity fails. Related questions are Hoeldercontinuity and real analyticity of solutions. We also will continue ourstudy of global regularity on CR manifolds using microlocal methods.Partial differential equations arise naturally in physics, chemistry,engineering, economics, and in several diverse areas of mathematics.Sometimes these equations can be solved explicitly but more often theycannot and, in that case, the problem is to find whether the solutionexists and if so to find various of its properties and to approximateit. This is accomplished by means of estimates. Here we study estimatesfor certain special fundamental partial differential equations but thetechniques we develop are applicable in much greater generality. Thebasic concept that we use are "subelliptic multipliers" this is a toolwhich converts the search for estimates into problems of algebra andgeometry. This concept has proved very effective not only in provingestimates but also in the study of various algebraic and geometricproblems.