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<p>We establish an analogue of Pitt’s inequality for Fourier series on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb T}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, extending the range of parameters by employing averages of Fourier coefficients. We show that this inequality is closely connected to sharp estimates for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> norms of the Dirichlet kernels over various subsets of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

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Pitt’s Inequalities and Bounds for Dirichlet Kernels

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月13日

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  • Erlan Nursultanov,
  • Sergey Tikhonov,
  • Ferenc Weisz

摘要

We establish an analogue of Pitt’s inequality for Fourier series on \({\mathbb T}^d\) T d , extending the range of parameters by employing averages of Fourier coefficients. We show that this inequality is closely connected to sharp estimates for the \(L_p\) L p norms of the Dirichlet kernels over various subsets of \({\mathbb Z}^d\) Z d .