Quantitative form of certain k-plane transform inequalities

TL;DR

This paper provides a quantitative refinement of k-plane transform inequalities, explicitly determining extremizers and best constants for certain cases, with extensions to radial functions.

Contribution

It introduces a quantitative form of k-plane transform inequalities, including explicit extremizers and constants, extending previous results to specific cases.

Findings

Explicit extremizers and best constants for k-plane transform inequalities.

Quantitative inequalities valid for k = d - 1 and for k < d - 1 with radial functions.

Extension of known inequalities to a broader class of functions.

Abstract

Let d > 1 and 0 < k < d. The k-plane transform satisies some Lp to Lq dilation-invariant inequality. In this case the best constant and the extremizers are explicitly known. We give a quantitative form of the inequality with respect to these extremizers, that works for k = d - 1 and for k < d-1 while restricted to radial functions.