<p>In this paper we shall investigate the weighted <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-type estimates for a class of generalized Calderón-Zygmund type singular integrals <Equation ID="Equ30"> <EquationSource Format="TEX">$\begin{aligned} T_{\epsilon , \beta } f(x) =\int _{\left\{ |x-y|>\epsilon \right\} }\frac{\Omega (x-y)}{|x-y|^{n-\beta }}f(y)dy \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>T</mi> <mrow> <mi>ϵ</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mfenced close="}" open="{"> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo>></mo> <mi>ϵ</mi> </mfenced> </msub> <mfrac> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>β</mi> </mrow> </msup> </mfrac> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta \in (0, n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We should remark that the existence of the constant <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> in these singular integrals is the key difference from traditional Calderón-Zygmund type singular integral operators, which also renders the problems more challenging. In fact, this class of singular integrals arises from the approximation of the surface quasi-geostrophic (SQG) equation, which describes geophysical flows in the atmosphere and ocean.</p>

Weighted $L^p$-type estimates for a class of generalized Calderón-Zygmund type singular integrals

期刊论文
发表日期：

2026年2月2日

查看全文

Lihe Wang,
Fengping Yao

摘要

In this paper we shall investigate the weighted $L^p$ $L^{p}$ -type estimates for a class of generalized Calderón-Zygmund type singular integrals $\begin{aligned} T_{\epsilon , \beta } f(x) =\int _{\left\{ |x-y|>\epsilon \right\} }\frac{\Omega (x-y)}{|x-y|^{n-\beta }}f(y)dy \end{aligned}$ $\begin{matrix} T_{ϵ, β} f (x) = \int_{\{| x - y | > ϵ\}} \frac{Ω (x - y)}{{| x - y |}^{n - β}} f (y) d y \end{matrix}$ for $\epsilon >0$ $ϵ > 0$ and $\beta \in (0, n)$ $β \in (0, n)$ . We should remark that the existence of the constant $\beta $ $β$ in these singular integrals is the key difference from traditional Calderón-Zygmund type singular integral operators, which also renders the problems more challenging. In fact, this class of singular integrals arises from the approximation of the surface quasi-geostrophic (SQG) equation, which describes geophysical flows in the atmosphere and ocean.

Weighted \(L^p\)-type estimates for a class of generalized Calderón-Zygmund type singular integrals

摘要