Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces

Abstract

We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces. We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces. We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces. Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel–Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.