The aim of this paper is to study the maximal commutators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1038_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> and the commutators of the maximal operator [b, M] in the total Morrey spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1038_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{p,\lambda ,\mu }(\mathbb {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on any stratified Lie group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1038_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation> when b belongs to Lipschitz spaces <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1038_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">${\dot{\Lambda }}_{\beta }(\mathbb {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi mathvariant="normal">Λ</mi> <mo>˙</mo> </mover> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Some new characterizations for certain subclasses of Lipschitz spaces <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1038_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">${\dot{\Lambda }}_{\beta }(\mathbb {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi mathvariant="normal">Λ</mi> <mo>˙</mo> </mover> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are given.

Characterizations of commutators of the maximal function in total Morrey spaces on stratified Lie groups

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发表日期：

2025年3月6日

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Vagif S. Guliyev

摘要

The aim of this paper is to study the maximal commutators $M_{b}$ $M_{b}$ and the commutators of the maximal operator [b, M] in the total Morrey spaces $L^{p,\lambda ,\mu }(\mathbb {G})$ $L^{p, λ, μ} (G)$ on any stratified Lie group $\mathbb {G}$ $G$ when b belongs to Lipschitz spaces ${\dot{\Lambda }}_{\beta }(\mathbb {G})$ ${\dot{Λ}}_{β} (G)$ . Some new characterizations for certain subclasses of Lipschitz spaces ${\dot{\Lambda }}_{\beta }(\mathbb {G})$ ${\dot{Λ}}_{β} (G)$ are given.