Quasiextremals for a Radon-like transform

TL;DR

This paper characterizes quasiextremals for a Radon-like transform involving convolution on a paraboloid, providing insights into functions nearly attaining the inequality bounds and advancing the inverse problem analysis.

Contribution

It provides a complete characterization of quasiextremals for the paraboloid convolution operator with quantitative bounds, a novel result in harmonic analysis.

Findings

Characterization of all quasiextremals with quantitative control

First results on inverse problems related to Radon-like transforms

Discussion of related results and future research directions

Abstract

Convolution with an appropriate surface measure on a paraboloid is known to define a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. By a quasiextremal for the associated inequality, we mean a function f for which the norm of Tf is at least a constant c times the norm of f. Our main result characterizes all quasiextremals, with some quantitative control in terms of c. Several related results are also discussed. This is the first in a series of at least four articles about a circle of questions concerning the inverse problem of deducing information about f from information about the ratio of the norm of Tf to the norm of f.