Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 897-922

FOURIER INTEGRALS AND SOBOLEV IMBEDDING ON REARRANGEMENT INVARIANT QUASI-BANACH FUNCTION SPACES

Kwok-Pun Ho

The Hong Kong Institute of Education, Department of Mathematics and Information Technology
10 Lo Ping Road, Tai Po, Hong Kong, P.R. China; vkpho 'at' ied.edu.hk

Abstract. We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangement-invariant 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.

2010 Mathematics Subject Classification: Primary 42B20, 42B35, 46B70, 46E30, 46E35, 42A96.

Key words: Fourier integral operator, Sobolev embedding, oscillatory integrals, Hausdorff-Young inequalities, restriction theorem, rearrangement-invariant, quasi-Banach function spaces, interpolation of operators, Triebel-Lizorkin spaces, Hardy spaces.

Reference to this article: K.-P. Ho: Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces. Ann. Acad. Sci. Fenn. Math. 41 (2016), 897-922.

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doi:10.5186/aasfm.2016.4157