Abstract
We prove boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, when acting on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.
Highlights
In the paper we deduce mapping properties of step multipliers and Fourier step multipliers when acting on quasi-Banach modulation spaces
In optics, the refractive index of a material is the frequency response of a causal system whose real part gives the phase shift of the penetrating light and the imaginary part gives the attenuation
In order to obtain continuity for weighted modulation spaces with general moderate weights in the momentum variables, it is required that the Fourier transform of Gabor atoms obey even stronger regularities of Gevrey types
Summary
In the paper we deduce mapping properties of step multipliers and Fourier step multipliers when acting on quasi-Banach modulation spaces. In order to obtain continuity for weighted modulation spaces with general moderate weights in the momentum variables, it is required that the Fourier transform of Gabor atoms obey even stronger regularities of Gevrey types.
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