<p>Consider the Dunkl Laplacian <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> associated with a root system <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and a nonnegative multiplicity function <i>k</i> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>. By following Stein (Proc Natl Acad Sci USA 73(7):2174–2175, 1976) and Strichartz (Trans Am Math Soc 148(2):461–471, 1970), we introduce and investigate a family of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-averaging operators parameterized by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. This family includes the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-spherical and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-volume mean operators as special cases. We prove that, for each order <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the averaging operator of order <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> satisfies a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Delta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-Pizzetti formula. In addition, we establish that this family of the <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-averaging operators provides various mean value characterizations of harmonic, polyharmonic, subharmonic and metaharmonic functions in the Dunkl setting. Furthermore, some of these characterizations yield new mean value properties for the usual classes of such functions associated with the standard Laplace operator.</p>

Mean value characterizations of harmonic, subharmonic and metaharmonic functions associated with the Dunkl Laplacian

期刊论文
发表日期：

2025年10月31日

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Chaabane Rejeb

摘要

Consider the Dunkl Laplacian $\Delta _k$ $Δ_{k}$ associated with a root system $\Phi$ $Φ$ in $\mathbb {R}^d$ $R^{d}$ and a nonnegative multiplicity function k on $\Phi$ $Φ$ . By following Stein (Proc Natl Acad Sci USA 73(7):2174–2175, 1976) and Strichartz (Trans Am Math Soc 148(2):461–471, 1970), we introduce and investigate a family of $\Delta _k$ $Δ_{k}$ -averaging operators parameterized by $\alpha \ge 0$ $α \geq 0$ . This family includes the $\Delta _k$ $Δ_{k}$ -spherical and $\Delta _k$ $Δ_{k}$ -volume mean operators as special cases. We prove that, for each order $\alpha \ge 0$ $α \geq 0$ , the averaging operator of order $\alpha$ $α$ satisfies a $\Delta _k$ $Δ_{k}$ -Pizzetti formula. In addition, we establish that this family of the $\alpha$ $α$ -averaging operators provides various mean value characterizations of harmonic, polyharmonic, subharmonic and metaharmonic functions in the Dunkl setting. Furthermore, some of these characterizations yield new mean value properties for the usual classes of such functions associated with the standard Laplace operator.