For \(p\ge p_0:=2\lambda /(2\lambda +1)\) with \(\lambda >0\), the Hardy space \(H_\lambda ^p({\mathbb {R}}^2_+)\) associated with the Dunkl transform \({\mathscr {F}}_{\lambda }\) and the Dunkl operator D on the line \({\mathbb {R}}\), where \((D_xf)(x)=f'(x)+\frac{\lambda }{x} (f(x)-f(-x))\), is the set of functions \(F=u+iv\) on the half plane \({\mathbb {R}}^2_+=\{(x,y):\,x\in {\mathbb {R}}, y>0\}\), satisfying the generalized Cauchy–Riemann equations \(D_xu-\partial _yv=0\), \(\partial _yu+D_xv=0\), and \(\sup _{y>0}\int _{{\mathbb {R}}}| F(x,y)|^p|x|^{2\lambda }dx<+\infty \); and the real Hardy space \(H_{\lambda }^p({\mathbb {R}})\) on the line \({\mathbb {R}}\) is the collection of boundary functions of the real parts of functions \(F\in H_{\lambda }^p({\mathbb {R}}^2_+)\). In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential \(I_{\lambda }^{\alpha }\) associated to the Dunkl transform; and as the main result, we prove the equality \(D(I_{\lambda }^{1}f)=- {\mathscr {H}}_{\lambda }f\) for \(f\in H^1_{\lambda }({\mathbb {R}})\) in a weak sense, where \({\mathscr {H}}_{\lambda }\) is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for \(f\in H_\lambda ^1({\mathbb {R}})\).

对于\（p \ ge p_0：= 2 \ lambda /（2 \ lambda +1）\）与\（\ lambda> 0 \）的哈代空间\（H_ \ lambda ^ p（{\ mathbb {R}}与Dunkl变换\（{\ mathscr {F}} _ {\ lambda} \）和Dunkl运算符D在行\（{\ mathbb {R}} \\）上关联的^ 2 _ +）\），其中\（ （D_xf）（x）= f'（x）+ \ frac {\ lambda} {x}（f（x）-f（-x））\）是函数\（F = u + iv \ ）在半平面\（{\ mathbb {R}} ^ 2 _ + = \ {（x，y）：\，x \ in {\ mathbb {R}}，y> 0 \} \）中，满足广义Cauchy–Riemann方程\（D_xu- \ partial _yv = 0 \），\（\ partial _yu + D_xv = 0 \）和\（\ sup _ {y> 0} \ int _ {{\ mathbb {R}}} | F（x，y）| ^ p | x | ^ {2 \ lambda} dx <+ \ infty \） ; 而线\（{\ mathbb {R}} \\）上的真实Hardy空间\（H _ {\ lambda} ^ p（{\ mathbb {R}}）\）是的实部边界函数的集合函数\（F \ in H _ {\ lambda} ^ p（{\ mathbb {R}} ^ 2 _ +）\）。在本文中，我们在Hardy空间上建立了与Dunkl变换相关的Riesz势\（I _ {\ lambda} ^ {\ alpha} \）的Hardy-Littlewood-Sobolev型定理；和作为主要结果，证明了平等\（d（I _ {\拉姆达} ^ {1} F）= - {\ mathscr {H}} _ {\拉姆达} F到\）为\（H中˚F\ ^ 1 _ {\ lambda}（{\ mathbb {R}}）\）在较弱的意义上，其中\（{\ mathscr {H}} _ {\ lambda} \）是与Dunkl变换相关的广义Hilbert变换，它给出\（f \ in H_ \ lambda ^ 1（{\ mathbb {R}}）\）的特征。