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<p>In this paper, we study Fourier multipliers on quantum Euclidean spaces and obtain results on their <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> boundedness. On the way to get these results, we prove Paley, Hausdorff–Young–Paley, and Hardy–Littlewood inequalities on the quantum Euclidean space. As applications, we establish the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> estimate for the heat semigroup and Sobolev embedding theorem on quantum Euclidean spaces. We also obtain quantum analogues of logarithmic Sobolev and Nash type inequalities.</p>

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\(L^p\)–\(L^q\) boundedness of Fourier multipliers on quantum Euclidean spaces

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Michael Ruzhansky,
  • Serikbol Shaimardan,
  • Kanat Tulenov

摘要

In this paper, we study Fourier multipliers on quantum Euclidean spaces and obtain results on their \(L^p\) L p – \(L^q\) L q boundedness. On the way to get these results, we prove Paley, Hausdorff–Young–Paley, and Hardy–Littlewood inequalities on the quantum Euclidean space. As applications, we establish the \(L^p\) L p – \(L^q\) L q estimate for the heat semigroup and Sobolev embedding theorem on quantum Euclidean spaces. We also obtain quantum analogues of logarithmic Sobolev and Nash type inequalities.