<p>This paper examines the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> boundedness of the <i>k</i>-th order commutator of parabolic singular integrals. When the kernel function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> satisfies the condition <Equation ID="Equ34"> <EquationSource Format="TEX">$\begin{aligned} \sup \limits _{\xi \in S^{n-1}}\int _{S^{n-1}}|\Omega (y) |\left( \log \frac{1}{|\langle \xi ,y\rangle |}\right) ^{1+\gamma }d\sigma (y)<\infty \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo movablelimits="false">sup</mo> <mrow> <mi>ξ</mi> <mo>∈</mo> <msup> <mi>S</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </munder> <msub> <mo>∫</mo> <msup> <mi>S</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </msub> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mfenced close=")" open="("> <mo>log</mo> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">⟨</mo> <mi>ξ</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">⟩</mo> <mo stretchy="false">|</mo> </mrow> </mfrac> </mfenced> <mrow> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </mrow> </msup> <mi>d</mi> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mi>∞</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma >k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, we demonstrate the boundedness within the range <Equation ID="Equ35"> <EquationSource Format="TEX">$\begin{aligned} \frac{2(\gamma +1)}{2(\gamma +1)-(k+1)}<p<\frac{2(\gamma +1)}{k+1}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>-</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>This work improves several previously established results.</p>

$L^p$ Boundedness for the Commutator of Parabolic Singular Integral with Rough Kernels

期刊论文
发表日期：

2026年1月5日

查看全文

Shengrui Lin,
Jiecheng Chen,
Shaoyong He

摘要

This paper examines the $L^p(\mathbb {R}^n)$ $L^{p} (R^{n})$ boundedness of the k-th order commutator of parabolic singular integrals. When the kernel function $\Omega $ $Ω$ satisfies the condition $\begin{aligned} \sup \limits _{\xi \in S^{n-1}}\int _{S^{n-1}}|\Omega (y) |\left( \log \frac{1}{|\langle \xi ,y\rangle |}\right) ^{1+\gamma }d\sigma (y)<\infty \end{aligned}$ $\begin{matrix} sup_{ξ \in S^{n - 1}} \int_{S^{n - 1}} | Ω (y) | {(log \frac{1}{| ⟨ ξ, y ⟩ |})}^{1 + γ} d σ (y) < \infty \end{matrix}$ with $\gamma >k$ $γ > k$ , we demonstrate the boundedness within the range $\begin{aligned} \frac{2(\gamma +1)}{2(\gamma +1)-(k+1)}<p<\frac{2(\gamma +1)}{k+1}. \end{aligned}$ $\begin{matrix} \frac{2 (γ + 1)}{2 (γ + 1) - (k + 1)} < p < \frac{2 (γ + 1)}{k + 1} . \end{matrix}$ This work improves several previously established results.

\(L^p\) Boundedness for the Commutator of Parabolic Singular Integral with Rough Kernels

摘要