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We investigate weak-type (1,1) boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the exact weak-(1,1) norm of these sparse operators.
The Bellman Function for Level Sets of Sparse Operators Sponsored by the Meyer Fund
This talk focuses on a fundamental concept in the field of partial differential equations - unique continuation principles. Such a principle describes the propagation of the zeros of solutions to PDEs. Specifically, it answers the question: what condition is required to guarantee that if a solution to a PDE vanishes on a certain subset of the spatial domain, then it must also vanish on a larger subset of the domain. Motivated by Hardy’s uncertainty principle, Escauriaza, Kenig, Ponce, and Vega were able to show in a series of papers that if a linear Schrödinger solution decays sufficiently fast at two different times, the solution must be trivial. In this talk, we will discuss unique continuation properties of solutions to higher-order Schrödinger equations and variable-coefficient Schrödinger equations, and extend the classical Escauriaza-Kenig-Ponce-Vega type of result to these models. This is based on joint works with S. Federico-Z. Li, and Z. Lee.
Some unique continuation results for Schrödinger equations Sponsored by the Meyer Fund
The Bellman Function for Level Sets of Sparse Operators