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<p>We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(W^{1,\varphi }(\mathbb {R}^n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>φ</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under some natural assumptions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> we show that the maximal function is bounded and continuous in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(W^{1,\varphi }(\mathbb {R}^n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>φ</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Maximal Operator in Musielak–Orlicz–Sobolev Spaces

  • 期刊论文

  • 发表日期：

    2026年3月26日

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  • Piotr Michał Bies,
  • Michał Gaczkowski,
  • Przemysław Górka

摘要

We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space \(W^{1,\varphi }(\mathbb {R}^n)\) W 1 , φ ( R n ) . Under some natural assumptions on \(\varphi \) φ we show that the maximal function is bounded and continuous in \(W^{1,\varphi }(\mathbb {R}^n)\) W 1 , φ ( R n ) .