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<p>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 the Paley, 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an application of the study, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.</p>

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Hörmander Type Fourier Multiplier Theorem and Nikolskii Inequality on Quantum Tori, and Applications

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Michael Ruzhansky,
  • Serikbol Shaimardan,
  • Kanat Tulenov

摘要

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 the Paley, 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 \(\beta \) β -versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an application of the study, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.