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 $$k=d-1$$.