<p>In [<CitationRef CitationID="CR55">55</CitationRef>] Weisz investigated strong convergence of partial sums <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1087_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$S_{m,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of the two-dimensional Walsh-Fourier series in the martingale Hardy spaces, but under the condition when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1087_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$2^{-\alpha }<m/n\le 2^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo><</mo> <mi>m</mi> <mo stretchy="false">/</mo> <mi>n</mi> <mo>≤</mo> <msup> <mn>2</mn> <mi>α</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper we investigate strong convergence of the two-dimensional Walsh-Fourier series in the martingale Hardy spaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1087_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">${H_p^{\square }\left( G^2\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>p</mi> <mo>□</mo> </msubsup> <mfenced close=")" open="("> <msup> <mi>G</mi> <mn>2</mn> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1087_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$0<p<1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> without any restriction on the indices. Moreover, we also prove sharpness of our result.</p>

Sharp strong convergence result of the two-dimensional Walsh-Fourier series in martingale Hardy spaces

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发表日期：

2025年5月19日

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G. Tephnadze

摘要

In [55] Weisz investigated strong convergence of partial sums $S_{m,n}$ $S_{m, n}$ of the two-dimensional Walsh-Fourier series in the martingale Hardy spaces, but under the condition when $2^{-\alpha }<m/n\le 2^{\alpha }$ $2^{- α} < m / n \leq 2^{α}$ . In this paper we investigate strong convergence of the two-dimensional Walsh-Fourier series in the martingale Hardy spaces ${H_p^{\square }\left( G^2\right) }$ $H_{p}^{□} (G^{2})$ for $0<p<1,$ $0 < p < 1,$ without any restriction on the indices. Moreover, we also prove sharpness of our result.