Sharp constant for a \(k\)-plane transform inequality. (English) Zbl 1310.44003

Summary: The \(k\)-plane transform \(\mathcal R_k\) acting on test functions on \(\mathbb R^d\) 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 A. Baernstein II and M. Loss [Rend. Semin. Mat. Fis. Milano 67, 9–26 (1997; Zbl 1016.44002)]. This extends their own result for \(k=2\) and M. Christ’s result [“Quasiextremals for a Radon-like transform”, Preprint, arXiv:1106.0722; “On extremals for a Radon-like transform”, Preprint, arXiv:1106.0728; “Extremizers of a Radon transform inequality”, Preprint, arXiv:1106.0719] for \(k=d-1\).


44A12 Radon transform


Zbl 1016.44002
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