Clemson Analysis and PDE Seminar

Abstract: Classical Clark measures are singular measures on the unit circle defined via inner functions on the disk that are closely tied to important topics in operator theory and complex analysis. In this talk, we’ll review the classical situation, present an analogous definition for two-variable Clark measures on the bidisk, and prove some related results. Specifically, in the rational case, we’ll give exact formulas for these Clark measures, characterize when associated embeddings are unitary, and obtain natural perturbations of pairs of compressed shift operators to pairs of commuting unitary operators. (Time permitting, we'll discuss which results generalize to the non-rational case). This is joint work with John Anderson, Palak Aurora, Linus Bergqvist, Joseph Cima, Conni Liaw, and Alan Sola.

Abstract: In 2017, Nazarov, Petermichl, Treil, and Volberg introduced convex body-valued sparse operators to obtain estimates for the norms of Calderón-Zygmund operators on matrix weighted L^2. In this talk, we combine four topics in harmonic analysis to give a new proof of bounds for Calderón-Zygmund operators on matrix weighted variable Lebesgue spaces. We combine tools from variable Lebesgue spaces, matrix weights, convex-set valued analysis, and the convex body-valued sparse operators mentioned above. With these tools, the proof is much simpler than the existing proof given by myself and Zoe Nieraeth in 2025. This is joint work with Marcin Bownik, David Cruz-Uribe, and Fatih Sirin. 

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