Mathematical Sciences: Function and Operator Theory on Holomorphic Spaces

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DMS-9622890 PI: Zhijian Wu University of Alabama @ Tuscaloosa In the past several years, the principal investigator has mainly focused his research on function and operator theory on analytic (one or several variables) or monogenic function spaces. The operators include Hankel, Toeplitz, commutators, multipliers, bilinear forms, approximation and atomic decomposition. The spaces (analytic or harmonic) are Bergman, Hardy, Dirichlet and the scale of capacity related spaces which includes the Bloch space and the space of bounded mean oscillation functions. The study also involves differential operators such as d-bar, Laplacian, Div, Curl and Dirac operators. The principal investigator proposes to continue his ongoing study, to explore more connections and applications of function and operator theory to other areas. Although there are many problems related to the study which arise in nature, the principal investigator plans to focus specifically on the aforementioned operators on potential related function spaces, the function theory of these spaces, Nehari type problems in Dirichlet spaces, minimum solution operators to some differential systems and Clifford analysis in the theory of compensated compactness. He will use function and operator theoretical tools and techniques in harmonic analysis, together with new ideas and methods in his investigation. Beside their own charm in pure mathematics, problems in this project have rich connections and applications to many fields of applied sciences. For example Hankel operators (or more generally commutators and bilinear forms) are very popular in various aspects of engineering design and system science. Their behavior provides important information to automatic control, navigation systems and robust stabilization. The study of Hankel and related operators in various underlying spaces will provide effective ways to deal with control problems in different situations. Another good example is the recent discover o f the connection between Clifford analysis and the compensated compactness phenomenon in non-linear partial differential equations of multi-variable spaces. Experience and previous achievement indicate that the principal investigator has the potential to accomplish his goal.