<p>In this paper we study an extension of maximal Calderón–Zygmund type singular integral <Equation ID="Equ61"> <EquationSource Format="TEX">$T_{\beta }^{}f (x) =\sup _{\varepsilon >0} \Big |\int _{|y|\ge \varepsilon } \frac{\Omega (y)}{|y|^{n-\beta }}f(x-y)dy\Big |$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>β</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true">sup</mo> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mi>ε</mi> </mrow> </msub> <mfrac> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</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>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>y</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> </mrow> </math></EquationSource> </Equation>and non-tangential maximal Calderón–Zygmund type singular integral <Equation ID="Equ62"> <EquationSource Format="TEX">$T_{\Gamma _{\alpha },\beta }^{} f(x):= \sup _{(y,\varepsilon )\in \Gamma _{\alpha }(x)} \Big |\int _{|t|\ge \varepsilon } \frac{\Omega (t)}{|t|^{n-\beta }}f(y-t)dt\Big |,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>T</mi> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>α</mi> </msub> <mo>,</mo> <mi>β</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mo movablelimits="true">sup</mo> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="normal">Γ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </munder> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mi>ε</mi> </mrow> </msub> <mfrac> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>t</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>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma _{\alpha }(x)=\{(y,\varepsilon )\in \mathbb {R}^{n}\times (0,\infty ): |x-y|<\alpha \varepsilon \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>α</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> </mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo><</mo> <mi>α</mi> <mi>ε</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is fixed. We establish the uniform <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> estimate (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1<q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>) and weak type (1, 1) estimate of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_{\beta }^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>T</mi> <mrow> <mi>β</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> and its non-tangential maximal version. Therefore, the strong type estimate and weak type estimate of the classical maximal Calderón–Zygmund type singular integrals can be recovered by letting <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On extension of non-tangential maximal Calderón–Zygmund type singular integrals

期刊论文
发表日期：

2026年4月20日

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Ting Chen,
Wanhao Li,
Wenchang Sun

摘要

In this paper we study an extension of maximal Calderón–Zygmund type singular integral $T_{\beta }^{*}f (x) =\sup _{\varepsilon >0} \Big |\int _{|y|\ge \varepsilon } \frac{\Omega (y)}{|y|^{n-\beta }}f(x-y)dy\Big |$ $T_{β}^{*} f (x) = sup_{ε > 0} | \int_{| y | \geq ε} \frac{Ω (y)}{{| y |}^{n - β}} f (x - y) d y |$ and non-tangential maximal Calderón–Zygmund type singular integral $T_{\Gamma _{\alpha },\beta }^{*} f(x):= \sup _{(y,\varepsilon )\in \Gamma _{\alpha }(x)} \Big |\int _{|t|\ge \varepsilon } \frac{\Omega (t)}{|t|^{n-\beta }}f(y-t)dt\Big |,$ $T_{Γ_{α}, β}^{*} f (x) : = sup_{(y, ε) \in Γ_{α} (x)} | \int_{| t | \geq ε} \frac{Ω (t)}{{| t |}^{n - β}} f (y - t) d t |,$ where $\Gamma _{\alpha }(x)=\{(y,\varepsilon )\in \mathbb {R}^{n}\times (0,\infty ): |x-y|<\alpha \varepsilon \}$ $Γ_{α} (x) = {(y, ε) \in R^{n} \times (0, \infty) : | x - y | < α ε}$ and $\alpha >0$ $α > 0$ is fixed. We establish the uniform $L^q$ $L^{q}$ estimate ( $1<q<\infty $ $1 < q < \infty$ ) and weak type (1, 1) estimate of $T_{\beta }^{*}$ $T_{β}^{*}$ and its non-tangential maximal version. Therefore, the strong type estimate and weak type estimate of the classical maximal Calderón–Zygmund type singular integrals can be recovered by letting $\beta \rightarrow 0$ $β \to 0$ .