Abstract
If $T$ is the operator given by convolution with surface measure on the sphere, $(E,F)$ is a quasi-extremal pair of sets for $T$ if $\langle T\chi_E, \chi_F \rangle\gtrsim|E|^{d/(d+1)}|F|^{d/(d+1)}$. In this article, we explicitly define a family $\mathcal{F}$ of quasi-extremal pairs of sets for $T$. We prove that $\mathcal{F}$ is fundamental in the sense that every quasi-extremal pair $(E,F)$ is comparable (in a rather strong sense) to a pair from $\mathcal{F}$. This extends work carried out by Christ for convolution with surface measure on the paraboloid.
Citation
Betsy Stovall. "Quasi-extremals for convolution with surface measure on the sphere." Illinois J. Math. 53 (2) 391 - 412, Summer 2009. https://doi.org/10.1215/ijm/1266934784
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