Sharp constant for a k-plane transform inequality

Drouot, Alexis

doi:10.2140/apde.2014.7.1237

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Sharp constant for a $k$-plane transform inequality

Alexis Drouot

Vol. 7 (2014), No. 6, 1237–1252

DOI: 10.2140/apde.2014.7.1237

Abstract

The $k$ -plane transform $ℛ_{k}$ acting on test functions on $ℝ^{d}$ satisfies a dilation-invariant $L^{p} \to L^{q}$ inequality for some exponents $p, q$ . We will make explicit some extremizers and the value of the best constant for any value of $k$ and $d$ , solving the endpoint case of a conjecture of Baernstein and Loss. This extends their own result for $k = 2$ and Christ’s result for $k = d - 1$ .

Keywords

k-plane transform, best constant, extremizers

Mathematical Subject Classification 2010

Primary: 44A12

Milestones

Received: 27 January 2012

Revised: 3 June 2014

Accepted: 27 August 2014

Published: 18 October 2014

Authors

Alexis Drouot
Department of Mathematics UC Berkeley Berkeley, 94704 United States

Vol. 7, No. 6, 2014