An Operator Approach to Problems in Analysis and Probability: Matrix Muckenhoupt Weights, Hankel and Toeplitz Operators, Singular Integrals and the Angle between Past and Future

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Abstract Treil/Volberg DMS-9622963 Treil and Volberg will continue their research on bases of classical wavelets in vector weighted spaces. They will consider some classical problems in the theory of Hankel and Toeplitz operators in spaces of analytic functions, singular integral operators. One goal is the generalizations of these problems to multi-parameter (vector) setting. Other problems they will investigate are Sarason's problem about necessary and sufficient condition for a product of two (unbounded) Toeplitz operators to be bounded, description of completely regular multivariate stationary Gaussian processes, two weights inequality for Bergman Projection, and boundedness of a product of two (big) Hankel operators in the Bergman space. Singular integrals is one of the main tools analysis brought to applications (signal processing, market models, prediction theory, etc..). In practice many such problems have a vector (multi-parameter) nature. The proposed research is based on the approach targeting exactly the multi-parameter nature of such problems and overcomes the difficulty stemming from this multi-parameter nature. Among the practical applications of the research are prediction theory for stationary Gaussian processes, wavelets (signal processing). The authors strongly feel that the developed technique can be helpful.