<p>For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, we establish the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> boundedness of the maximal Riesz transforms in terms of the Riesz transforms on quantum tori <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_{p}(\mathbb {T}^{d}_{\theta })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>θ</mi> <mi>d</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and quantum Euclidean space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_{p}(\mathbb {R}^{d}_{\theta })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>θ</mi> <mi>d</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, the norm constants in both cases are independent of the dimension <i>d</i> when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Maximal Riesz transform in terms of Riesz transform on quantum tori and Euclidean space

期刊论文
发表日期：

2026年6月24日

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Xudong Lai,
Xiao Xiong,
Yue Zhang

摘要

For $1<p<\infty $ $1 < p < \infty$ , we establish the $L_{p}$ $L_{p}$ boundedness of the maximal Riesz transforms in terms of the Riesz transforms on quantum tori $L_{p}(\mathbb {T}^{d}_{\theta })$ $L_{p} (T_{θ}^{d})$ , and quantum Euclidean space $L_{p}(\mathbb {R}^{d}_{\theta })$ $L_{p} (R_{θ}^{d})$ . In particular, the norm constants in both cases are independent of the dimension d when $2\le p<\infty $ $2 \leq p < \infty$ .