FALL 2021 Nonlinear Analysis Seminar Series

Professor Galia Dafni,  Concordia University

Title: Boundedness and continuity of rearrangements in BMO and VMO

Abstract:

Joint work with Almut Burchard (Toronto) and Ryan Gibara (Cincinnati). Let $f$ be a function of bounded mean oscillation (BMO) on cubes in $\mathbb{R}^n$, $n > 1$. If $f$ is rearrangeable, we show that its symmetric decreasing rearrangement$Sf$ belongs to $\mathrm{BMO}(\mathbb{R}^n)$. We also improve the bounds for the decreasing rearrangement $f^*$ by Bennett, DeVore and Sharpley, $\|f^*\|_{ \mathrm{BMO}(\mathbb{R}_+)} \leq C_n\|f\| _{\mathrm{BMO}(\mathbb{R}^n)}$, by eliminating the exponential dependence of $C_n$ on the dimension $n$. The key is to switch from cubes to a comparable family of shapes. Using a family of rectangles that is preserved under bisections, one can prove a dimension-free Calderón-Zygmund decomposition, and the boundedness of the decreasing rearrangement with the same constant. Restricting to the subspace of functions of vanishing mean oscillation (VMO), we show that these rearrangements take VMO functions to VMO functions. Furthermore, while the map from $f$ to $f^*$ is not continuous in the BMO seminorm, we prove continuity when the limit is in VMO.

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