<p>Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> be a von Neumann algebra equipped with a normal semifinite faithful trace <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mspace width="0.166667em" /> <mi mathvariant="script">M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the operator-valued Hardy space with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, which is first studied by T. Mei [Mem. Amer. Math. Soc. 188 (2007), vi+64 pp; MR2327840]. In this paper, the authors mainly establish some new square function characterizations of operator-valued Hardy space <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mspace width="0.166667em" /> <mi mathvariant="script">M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1109_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, which can describe the predual spaces of noncommutative BMO spaces.</p>

New square function characterizations of operator-valued Hardy spaces on the Euclidean space $\mathbb {R}^d$

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发表日期：

2025年7月31日

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Wenhua Wang,
Tiantian Zhao

摘要

Let $\mathcal {M}$ $M$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau $ $τ$ . Let $\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})$ $H_{p} (R^{d}, M)$ denote the operator-valued Hardy space with $1\le p<\infty $ $1 \leq p < \infty$ , which is first studied by T. Mei [Mem. Amer. Math. Soc. 188 (2007), vi+64 pp; MR2327840]. In this paper, the authors mainly establish some new square function characterizations of operator-valued Hardy space $\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})$ $H_{p} (R^{d}, M)$ for all $1\le p<\infty $ $1 \leq p < \infty$ , which can describe the predual spaces of noncommutative BMO spaces.

New square function characterizations of operator-valued Hardy spaces on the Euclidean space \(\mathbb {R}^d\)

摘要