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Abstract <p> We establish a criterion for the sum of a multiple (or one-dimensional) trigonometric series to belong to the Lebesgue space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_p([0,2\pi]^d)\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1&lt;p&lt;\infty\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(d\ge 1\)</EquationSource> </InlineEquation>. We consider series whose coefficients satisfy a new general monotonicity condition, which does not reduce to any of the previously known forms of general monotonicity. In particular, we present examples of sequences that belong to this new class but are not contained in any of the previously studied classes of general monotone sequences. The results extend and refine the known criteria for the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_p\)</EquationSource> </InlineEquation>-integrability of the sums of multiple and one-dimensional trigonometric series. </p>

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On \(L_p\)-Integrability of Multiple Trigonometric Series with General Monotone Coefficients

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. B. Mukanov,
  • E. D. Nursultanov

摘要

Abstract

We establish a criterion for the sum of a multiple (or one-dimensional) trigonometric series to belong to the Lebesgue space \(L_p([0,2\pi]^d)\) for \(1<p<\infty\) and \(d\ge 1\) . We consider series whose coefficients satisfy a new general monotonicity condition, which does not reduce to any of the previously known forms of general monotonicity. In particular, we present examples of sequences that belong to this new class but are not contained in any of the previously studied classes of general monotone sequences. The results extend and refine the known criteria for the \(L_p\) -integrability of the sums of multiple and one-dimensional trigonometric series.