Illinois Journal of Mathematics

Quasi-extremals for convolution with surface measure on the sphere

Betsy Stovall

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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.

Article information

Illinois J. Math., Volume 53, Number 2 (2009), 391-412.

First available in Project Euclid: 23 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 44A35: Convolution 44A12: Radon transform [See also 92C55]


Stovall, Betsy. Quasi-extremals for convolution with surface measure on the sphere. Illinois J. Math. 53 (2009), no. 2, 391--412. doi:10.1215/ijm/1266934784.

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