Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications

Abstract

In this paper, we study Hörmander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as Paley type, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse $\beta$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an applications of the analysis, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.

Some changes were made. Accepted to JFAA

References in corpus (4)

• Fourier multipliers and their applications to PDE on the quantum Euclidean space
• A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces
• Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces