<p>The object of investigation in this paper is the endpoint boundedness of the maximal singular integral <Equation ID="Equ93"> <EquationSource Format="TEX">$T_{\Omega }^{}f(x)=\sup \limits _{\epsilon>0}\Big |\int _{|y|>\epsilon } \frac{\Omega (y/|y|)}{|y|^{n}}f(x-y)dy\Big |,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>T</mi> <mrow> <mi mathvariant="normal">Ω</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="false">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> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </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> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> belongs to the block space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_q^{0,0}(\mathbb {S}^{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mi>q</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\int _{\mathbb {S}^{n-1}}\Omega (\theta )d\sigma (\theta )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </msub> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The weak type endpoint estimate of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L\log \log L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>log</mo> <mo>log</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> type for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$T_{\Omega }^{}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>T</mi> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is proved. This represents an essential improvement of a result (Honzík, Inter. Math. Res. Not. 2020). In addition, the weak type <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(1,\,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em" /> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> boundedness for rough maximal operator is obtained under the rough kernel in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$B_q^{0,0}(\mathbb {S}^{n-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mi>q</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which essentially improves and extends a recent result (Chen, Liu and Wu, J. Geom. Anal. 2025).</p>

A Weak Type Bound for Maximal Singular Integral Related to Block Spaces

期刊论文
发表日期：

2026年7月8日

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Feng Liu,
Simin Liu,
Xiao Zhang

摘要

The object of investigation in this paper is the endpoint boundedness of the maximal singular integral $T_{\Omega }^{*}f(x)=\sup \limits _{\epsilon>0}\Big |\int _{|y|>\epsilon } \frac{\Omega (y/|y|)}{|y|^{n}}f(x-y)dy\Big |,$ $T_{Ω}^{*} f (x) = sup_{ϵ > 0} | \int_{| y | > ϵ} \frac{Ω (y / | y |)}{{| y |}^{n}} f (x - y) d y |,$ where $\Omega $ $Ω$ belongs to the block space $B_q^{0,0}(\mathbb {S}^{n-1})$ $B_{q}^{0, 0} (S^{n - 1})$ with $q\in (1,\infty )$ $q \in (1, \infty)$ and satisfies $\int _{\mathbb {S}^{n-1}}\Omega (\theta )d\sigma (\theta )=0$ $\int_{S^{n - 1}} Ω (θ) d σ (θ) = 0$ . The weak type endpoint estimate of $L\log \log L$ $L log log L$ type for $T_{\Omega }^{*}$ $T_{Ω}^{*}$ is proved. This represents an essential improvement of a result (Honzík, Inter. Math. Res. Not. 2020). In addition, the weak type $(1,\,1)$ $(1, 1)$ boundedness for rough maximal operator is obtained under the rough kernel in $B_q^{0,0}(\mathbb {S}^{n-1})$ $B_{q}^{0, 0} (S^{n - 1})$ with $q\in (1,\infty )$ $q \in (1, \infty)$ , which essentially improves and extends a recent result (Chen, Liu and Wu, J. Geom. Anal. 2025).