On Fractional Integrals Generated by Radon Transforms over Paraboloids

TL;DR

This paper establishes sharp $L^p$-$L^q$ bounds for Radon transforms and fractional integrals with paraboloid singularities using Fourier analysis and advanced interpolation techniques.

Contribution

It introduces new sharp estimates for Radon transforms over paraboloids, extending previous results with refined Fourier and interpolation methods.

Findings

Sharp $L^p$-$L^q$ estimates derived

Extension to more general convolution-type fractional integrals

Application of Fourier transform and interpolation techniques

Abstract

We apply the Fourier transform technique and a modified version of E. Stein's interpolation theorem communicated by L. Grafakos, to obtain sharp $L^p$-$L^q$ estimates for the Radon transform and more general convolution-type fractional integrals with the kernels having singularity on the paraboloids.