<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> be a locally integrable function, <i>s</i> be a measurable function on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^+$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>. Then the generalized Hausdorff operator, associated to the parameter curves <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s(x,t):=s(t)x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, is defined by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {H}_{\Phi ,s}f(x)=\int _{\mathbb {R}^n}\frac{\Phi (y)}{|y|^n}f( s(|y|)x )dy. $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mrow> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </msub> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </msup> </mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> In this paper, we estabilish the boundedness of the generalized Hausdorff operators <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}_{\Phi ,s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mrow> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> on mixed radial-angular central Morrey spaces. Moreover, we also obtain sufficient and necessary conditions on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> which ensure the boundedness of their commutators on mixed radial-angular central Morrey spaces with symbols in mixed radial-angular central BMO spaces.</p>

Mixed Radial-Angular Integrabilities for Generalized Hausdorff Operators and Commutators

期刊论文
发表日期：

2026年2月2日

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Yaheng Liu,
Xiaomei Wu

摘要

Let $\Phi $ $Φ$ be a locally integrable function, s be a measurable function on $\mathbb {R}^+$ $R^{+}$ . Then the generalized Hausdorff operator, associated to the parameter curves $s(x,t):=s(t)x$ $s (x, t) : = s (t) x$ , is defined by $\mathcal {H}_{\Phi ,s}f(x)=\int _{\mathbb {R}^n}\frac{\Phi (y)}{|y|^n}f( s(|y|)x )dy. $ $H_{Φ, s} f (x) = \int_{R^{n}} \frac{Φ (y)}{{| y |}^{n}} f (s (| y |) x) d y .$ In this paper, we estabilish the boundedness of the generalized Hausdorff operators $\mathcal {H}_{\Phi ,s}$ $H_{Φ, s}$ on mixed radial-angular central Morrey spaces. Moreover, we also obtain sufficient and necessary conditions on $\Phi $ $Φ$ which ensure the boundedness of their commutators on mixed radial-angular central Morrey spaces with symbols in mixed radial-angular central BMO spaces.