The spaces $(L^{q},\;l^{p})^{\alpha}$ and $M^{p,\;\alpha}$ are closely related to classical problems in Harmonic Analysis: properties of multiplier and Fourier multiplier from a Lebesgue space to another, finite $(1,p)$-energy measures. We characterize the Fourier transforms of their elements and establish criteria of compactness in these spaces.
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