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<p>We generalize the centered and uncentered Hardy-Littlewood maximal function in terms of shape of integration sets. Continuity of two global and two local generalized maximal functions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{R}^{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> is studied. The global ones are continuous. One of the local ones is continuous, provided the integration set is star-shaped with respect to its center. For the second local one, we construct a counterexample.</p>

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On continuity of generalized maximal functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月28日

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  • Juha Kapulainen

摘要

We generalize the centered and uncentered Hardy-Littlewood maximal function in terms of shape of integration sets. Continuity of two global and two local generalized maximal functions in \(\textbf{R}^{n}\) R n is studied. The global ones are continuous. One of the local ones is continuous, provided the integration set is star-shaped with respect to its center. For the second local one, we construct a counterexample.