Fourier Inequalities in Lorentz and Lebesgue Spaces

Abstract

Mapping properties of the Fourier transform between weighted Lebesgue and Lorentz spaces are studied. These are generalizations to Hausdorff-Young and Pitt’s inequalities. The boundedness of the Fourier transform on $R^n$ as a map between Lorentz spaces leads to weighted Lebesgue inequalities for the Fourier transform on $R^n$ . A major part of the work is on Fourier coefficients. Several different sufficient conditions and necessary conditions for the boundedness of Fourier transform on unit circle, viewed as a map between Lorentz $\Lambda$ and $\Gamma$ spaces are established. For a large range of Lorentz indices, necessary and sufficient conditions for boundedness are given. A number of known inequalities for generalized quasi concave functions are generalized and improved as part of the preparation for the proofs of the Fourier series results. The Lorentz space results are used to obtain conditions that guarantee the continuity of the Fourier coefficient map between weighted $L^p$ spaces. Applications to $L\log L$ and Lorentz-Zygmund spaces are also given