<p>The Dunkl operators are commuting differential-reflection operators on the Euclidean space <InlineEquation ID="IEq1"> <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> associated with a root system <i>R</i> and a multiplicity function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\kappa \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. There have been several works on the Hardy space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathscr {H}}_\kappa ^1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">H</mi> <mi>κ</mi> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> associated with the Dunkl operators. In this paper, we focus on the area integral characterization of the Hardy space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathscr {H}}_\kappa ^1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">H</mi> <mi>κ</mi> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation>. Specifically, we introduce a “global” Lusin-type area integral, which is defined by means of Dunkl’s generalized translation and the Dunkl operators. We shall show that for a generalized harmonic function on the upper half-space, its non-tangential maximal function and its Lusin-type area integral are globally equivalent in terms of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_\kappa ^1$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>κ</mi> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation>-norm.</p>

Area integrals and maximal functions of generalized harmonic functions associated with the Dunkl operators

期刊论文
发表日期：

2025年12月5日

查看全文

Jiaxi Jiu

摘要

The Dunkl operators are commuting differential-reflection operators on the Euclidean space ${\mathbb R}^d$ $R^{d}$ associated with a root system R and a multiplicity function $\kappa \ge 0$ $κ \geq 0$ . There have been several works on the Hardy space ${\mathscr {H}}_\kappa ^1$ $H_{κ}^{1}$ associated with the Dunkl operators. In this paper, we focus on the area integral characterization of the Hardy space ${\mathscr {H}}_\kappa ^1$ $H_{κ}^{1}$ . Specifically, we introduce a “global” Lusin-type area integral, which is defined by means of Dunkl’s generalized translation and the Dunkl operators. We shall show that for a generalized harmonic function on the upper half-space, its non-tangential maximal function and its Lusin-type area integral are globally equivalent in terms of $L_\kappa ^1$ $L_{κ}^{1}$ -norm.