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arXiv · 1710.07692

Existence of extremizers for a model convolution operator

Chandan Biswas
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Abstract

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^{2},...,t^{d})$ is a bounded operator from $L^{p}$ to $L^{q}$ if $(\frac{1}{p}, \frac{1}{q})$ lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of $T$. We also establish a relation between extremizers for $T$ at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.

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Chandan Biswas. 2017-10-20. Existence of extremizers for a model convolution operator. https://arxiv.org/abs/1710.07692

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