Our aim in this article is to study the weighted boundedness of the centered Hardy–Littlewood maximal operator on harmonic NA groups. Closely following the approach of Antezana and Ombrosi in the setting of real hyperbolic spaces, we prove the weighted <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3762_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-boundedness, for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3762_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, of the maximal operator. Furthermore, as an endpoint case, we establish a variant of the Fefferman–Stein inequality, from which a vector-valued maximal inequality has been obtained. We also provide examples of weights to substantiate various aspects of our results. In particular, we show that certain spherical functions on harmonic NA groups serve as examples of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3762_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$A_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> weights.

Weighted estimates for Hardy–Littlewood maximal functions on harmonic NA groups

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2025年5月12日

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Pritam Ganguly,
Tapendu Rana,
Jayanta Sarkar

摘要

Our aim in this article is to study the weighted boundedness of the centered Hardy–Littlewood maximal operator on harmonic NA groups. Closely following the approach of Antezana and Ombrosi in the setting of real hyperbolic spaces, we prove the weighted $L^p$ $L^{p}$ -boundedness, for $1<p<\infty $ $1 < p < \infty$ , of the maximal operator. Furthermore, as an endpoint case, we establish a variant of the Fefferman–Stein inequality, from which a vector-valued maximal inequality has been obtained. We also provide examples of weights to substantiate various aspects of our results. In particular, we show that certain spherical functions on harmonic NA groups serve as examples of $A_p$ $A_{p}$ weights.