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<p>In this paper we study the endpoint Sobolev regularity for composition of maximal operators. We prove some endpoint Sobolev bounds and continuity for the composition of maximal operators. In addition, we also obtain some new endpoint Sobolev bounds and continuity for fractional maximal functions. The main new idea includes some applications of the weak (1,&#xa0;1) bound of the Hardy–Littlewood maximal operator and two basic inequalities of a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_457_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{1,\infty }(\mathbb {R}^n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</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> function.</p>

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Composition of maximal operators on the endpoint Sobolev space

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • Feng Liu,
  • Xueying Zhu

摘要

In this paper we study the endpoint Sobolev regularity for composition of maximal operators. We prove some endpoint Sobolev bounds and continuity for the composition of maximal operators. In addition, we also obtain some new endpoint Sobolev bounds and continuity for fractional maximal functions. The main new idea includes some applications of the weak (1, 1) bound of the Hardy–Littlewood maximal operator and two basic inequalities of a \(L^{1,\infty }(\mathbb {R}^n)\) L 1 , ∞ ( R n ) function.