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<p>We show that the method in the recent work by Roncal, Shrivastava, and Shuin can be adapted to show that certain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-improving bounds in the interior of the boundedness region for the single-scale maximal bilinear spherical averaging operator implies sparse bounds for the corresponding full maximal operator. More generally we show that the proof applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> exponents.</p>

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Sparse Bounds for Maximal Triangle and Bilinear Spherical Averaging Operators

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月5日

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  • Eyvindur Ari Palsson,
  • Sean R. Sovine

摘要

We show that the method in the recent work by Roncal, Shrivastava, and Shuin can be adapted to show that certain \(L^p\) L p -improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that \(L^p\) L p -improving bounds in the interior of the boundedness region for the single-scale maximal bilinear spherical averaging operator implies sparse bounds for the corresponding full maximal operator. More generally we show that the proof applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate \(L^p\) L p -improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of \(L^p\) L p exponents.