We prove that the Hardy–Littlewood maximal operator M is bounded on the variable Lebesgue space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{p(\cdot )}(X,d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1<p_-\le p_+<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <msub> <mi>p</mi> <mo>-</mo> </msub> <mo>≤</mo> <msub> <mi>p</mi> <mo>+</mo> </msub> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, over an unbounded space of homogeneous type <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(X,d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with a Borel-semiregular measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, if and only if the averaging operators <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_\mathcal {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi mathvariant="script">Q</mi> </msub> </math></EquationSource> </InlineEquation> are bounded on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^{p(\cdot )}(X,d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> uniformly over all families <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Q</mi> </math></EquationSource> </InlineEquation> of pairwise disjoint “cubes” from a Hytönen–Kairema dyadic system on X. This extends Diening’s well-known characterization of the boundedness of M on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^{p(\cdot )}(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </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> to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result.

The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good

期刊论文
出版模式：

开放获取
发表日期：

2026年5月21日

查看全文

Alina Shalukhina

摘要

We prove that the Hardy–Littlewood maximal operator M is bounded on the variable Lebesgue space $L^{p(\cdot )}(X,d,\mu )$ $L^{p (\cdot)} (X, d, μ)$ , with $1<p_-\le p_+<\infty $ $1 < p_{-} \leq p_{+} < \infty$ , over an unbounded space of homogeneous type $(X,d,\mu )$ $(X, d, μ)$ with a Borel-semiregular measure $\mu $ $μ$ , if and only if the averaging operators $T_\mathcal {Q}$ $T_{Q}$ are bounded on $L^{p(\cdot )}(X,d,\mu )$ $L^{p (\cdot)} (X, d, μ)$ uniformly over all families $\mathcal {Q}$ $Q$ of pairwise disjoint “cubes” from a Hytönen–Kairema dyadic system on X. This extends Diening’s well-known characterization of the boundedness of M on $L^{p(\cdot )}(\mathbb {R}^n)$ $L^{p (\cdot)} (R^{n})$ to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result.