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<p>In this survey, we collect recent progress in the understanding of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> bounds for bilinear spherical averages and some associated maximal functions like the bilinear spherical maximal function and its lacunary counterpart. We describe necessary conditions satisfied by triples in the <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 region of a bilinear spherical averaging operator and the localized bilinear spherical maximal function, as well as describe the best-known boundedness regions to date. We state some open questions along the way to motivate future research on this topic, and we exploit some possible generalizations.</p>

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Survey on Bilinear Spherical Averages and Associated Maximal Operators

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月30日

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  • Tainara Borges

摘要

In this survey, we collect recent progress in the understanding of \(L^{p}\) L p bounds for bilinear spherical averages and some associated maximal functions like the bilinear spherical maximal function and its lacunary counterpart. We describe necessary conditions satisfied by triples in the \(L^{p}\) L p improving region of a bilinear spherical averaging operator and the localized bilinear spherical maximal function, as well as describe the best-known boundedness regions to date. We state some open questions along the way to motivate future research on this topic, and we exploit some possible generalizations.