Fractional integrals and Fourier transforms

TL;DR

This paper surveys key results on fractional integrals and Fourier transforms, emphasizing estimates and alternative proofs, based on nonincreasing rearrangements, and stems from a mini-course at Barcelona University.

Contribution

It provides a concise overview of fundamental estimates and introduces alternative proof techniques for fractional integrals and Fourier transforms.

Findings

Alternative proofs of classical results

Estimates of fractional integrals and Fourier transforms

Application of nonincreasing rearrangements

Abstract

This paper gives a short survey of some basic results related to estimates of fractional integrals and Fourier transforms. It is closely adjoint to our previous survey papers \cite{K1998} and \cite{K2007}. The main methods used in the paper are based on nonincreasing rearrangements. We give alternative proofs of some results. We observe also that the paper represents the mini-course given by the author at Barcelona University in October, 2014.