Let X be a metric space with doubling measure and L a non-negative self-adjoint operator on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{2}\left( X\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mfenced close=")" open="("> <mi>X</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> whose heat kernels satisfying the Gaussian estimates. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="177" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi : X \times [0, \infty ) \rightarrow [0, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfy that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (x, \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an Orlicz function for any given <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$x \in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (\cdot , t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a Muckenhoupt <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {A}}_{\infty }\left( X\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">A</mi> <mi>∞</mi> </msub> <mfenced close=")" open="("> <mi>X</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> weight uniformly in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$t \in (0, \infty ).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this paper, we introduce the weak Musielak–Orlicz Hardy space <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$W H_{L}^{\varphi }\left( X\right) ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <msubsup> <mi>H</mi> <mrow> <mi>L</mi> </mrow> <mi>φ</mi> </msubsup> <mfenced close=")" open="("> <mi>X</mi> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> associated to the operator L. And we obtain the atomic characterization of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$W H_{L}^{\varphi }\left( X\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <msubsup> <mi>H</mi> <mrow> <mi>L</mi> </mrow> <mi>φ</mi> </msubsup> <mfenced close=")" open="("> <mi>X</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we establish non-tangential maximal functions characterizations of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1913_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">${WH}_{L}^{\varphi }\left( X\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="italic">WH</mi> </mrow> <mrow> <mi>L</mi> </mrow> <mi>φ</mi> </msubsup> <mfenced close=")" open="("> <mi>X</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> via the atomic decomposition.

Weak Musielak–Orlicz Hardy Spaces Associated with Operators Satisfying Gaussian Estimates on Spaces of Homogeneous Type

期刊论文
发表日期：

2025年2月6日

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Yao He

摘要

Let X be a metric space with doubling measure and L a non-negative self-adjoint operator on $L^{2}\left( X\right) $ $L^{2} (X)$ whose heat kernels satisfying the Gaussian estimates. Let $\varphi : X \times [0, \infty ) \rightarrow [0, \infty )$ $φ : X \times [0, \infty) \to [0, \infty)$ satisfy that $\varphi (x, \cdot )$ $φ (x, \cdot)$ is an Orlicz function for any given $x \in X$ $x \in X$ , and $\varphi (\cdot , t)$ $φ (\cdot, t)$ is a Muckenhoupt ${\mathbb {A}}_{\infty }\left( X\right) $ $A_{\infty} (X)$ weight uniformly in $t \in (0, \infty ).$ $t \in (0, \infty) .$ In this paper, we introduce the weak Musielak–Orlicz Hardy space $W H_{L}^{\varphi }\left( X\right) ,$ $W H_{L}^{φ} (X),$ associated to the operator L. And we obtain the atomic characterization of $W H_{L}^{\varphi }\left( X\right) $ $W H_{L}^{φ} (X)$ . Moreover, we establish non-tangential maximal functions characterizations of ${WH}_{L}^{\varphi }\left( X\right) $ ${WH}_{L}^{φ} (X)$ via the atomic decomposition.