Mathematical Sciences: Non-Standard Singular Integrals

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The object of this mathematical research is the analysis of certain singular integral transforms and their mapping properties relative to various spaces of functions. In particular, work will be done to establishing the boundedness of such mappings in those cases where the kernel of the singular integral lacks the smoothness necessary to be treated by classical techniques. These include multilinear singular integrals called d-commutators and operators which fail to carry the Hardy space into integrable functions or bounded functions into BMO. Some progress has already been made. For example, a result similar to the now standard David-Journe T1-condition has been developed which permits consideration of very rough kernels. The work also lends itself to the consideration of commutators of singular integrals with mixed homogeneity, in particular parabolic homogeneity. Singular integral transformations form the basis for much of the important work under active consideration in modern harmonic analysis. Among the motivating forces behind these studies is the goal for finding comprehensive techniques for solving broad classes of partial differential equations. These equations arise as models of many important, nonlinear physical phenomena.