<p>In this paper, we introduce the idea of the homogeneous (non-homogeneous) weighted Herz–Morrey type Hardy spaces denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1671_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">$HM \dot{K}_{q,\beta }^{\alpha , p}\left( \omega _1,\omega _{2}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mi>M</mi> <msubsup> <mover accent="true"> <mi>K</mi> <mo>˙</mo> </mover> <mrow> <mi>q</mi> <mo>,</mo> <mi>β</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mfenced close=")" open="("> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1671_Article_IEq2.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( HM K_{q,\beta }^{\alpha , p}\left( \omega _1,\omega _{2}\right) \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>H</mi> <mi>M</mi> <msubsup> <mi>K</mi> <mrow> <mi>q</mi> <mo>,</mo> <mi>β</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mfenced close=")" open="("> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> </mfenced> </mfenced> </math></EquationSource> </InlineEquation>. We obtain the atomic decomposition and molecular characterization of these spaces. We derive the boundedness of the maximal Bochner–Riesz operator and the Bochner–Riesz operator on these spaces as an application.</p>

Boundedness of the Bochner–Riesz Operators on the Weighted Herz–Morrey Type Hardy Spaces

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发表日期：

2025年3月5日

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Babar Sultan,
Mehvish Sultan,
Amjad Hussain

摘要

In this paper, we introduce the idea of the homogeneous (non-homogeneous) weighted Herz–Morrey type Hardy spaces denoted by $HM \dot{K}_{q,\beta }^{\alpha , p}\left( \omega _1,\omega _{2}\right) $ $H M {\dot{K}}_{q, β}^{α, p} (ω_{1}, ω_{2})$ $\left( HM K_{q,\beta }^{\alpha , p}\left( \omega _1,\omega _{2}\right) \right) $ $(H, M, K_{q, β}^{α, p}, (ω_{1}, ω_{2}))$ . We obtain the atomic decomposition and molecular characterization of these spaces. We derive the boundedness of the maximal Bochner–Riesz operator and the Bochner–Riesz operator on these spaces as an application.