We are concerned with the boundedness of modified Hardy-Littlewood maximal operator Mλ and Sobolev inequalities for the variable Riesz potentials Iα(·),τf on Musielak-Orlicz spaces LΦ(X) over unbounded metric measure spaces, as an improvement of our recent paper, see T. Ohno, T. Shimomura (2025a). As an application, we give the boundedness of Mλ and Sobolev inequalities for Iα(·),τf for the multi-phase functionals with variable exponents <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_2624_Article_Equ1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="378" /> </MediaObject> <EquationSource Format="TEX">$\Phi(x,t) = t^{p(x)} + a(x) t^{q(x)}+ b(x) t^{s(x)}, \quad x \in X, \ t \geqslant 0,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>t</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>t</mi> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>t</mi> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> <mo>,</mo> </math></EquationSource> </Equation> where p(·), q(·), and s(·) are log-Hölder continuous, p(x) < q(x) ⩽ s(x) for x ∈ X, and a(·), and b(·) are nonnegative, bounded, and Hölder continuous.

Maximal and Riesz potential operators on Musielak-Orlicz spaces over unbounded metric measure spaces

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发表日期：

2025年8月1日

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Takao Ohno,
Tetsu Shimomura

摘要

We are concerned with the boundedness of modified Hardy-Littlewood maximal operator M_λ and Sobolev inequalities for the variable Riesz potentials I_α(·),τf on Musielak-Orlicz spaces L^Φ(X) over unbounded metric measure spaces, as an improvement of our recent paper, see T. Ohno, T. Shimomura (2025a). As an application, we give the boundedness of M_λ and Sobolev inequalities for I_α(·),τf for the multi-phase functionals with variable exponents $\Phi(x,t) = t^{p(x)} + a(x) t^{q(x)}+ b(x) t^{s(x)}, \quad x \in X, \ t \geqslant 0,$ $Φ (x, t) = t^{p (x)} + a (x) t^{q (x)} + b (x) t^{s (x)}, x \in X, t ⩾ 0,$ where p(·), q(·), and s(·) are log-Hölder continuous, p(x) < q(x) ⩽ s(x) for x ∈ X, and a(·), and b(·) are nonnegative, bounded, and Hölder continuous.