For a normalized root system R in \({\mathbb {R}}^N\) and a multiplicity function \(k\ge 0\) let \({\mathbf {N}}=N+\sum _{\alpha \in R} k(\alpha )\). We denote by \(dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}\) the associated measure in \({\mathbb {R}}^N\). Let \(L=-\varDelta +V\), \(V\ge 0\), be the Dunkl–Schrödinger operator on \({\mathbb {R}}^N\). Assume that there exists \(q >\max (1,\frac{{\mathbf {N}}}{2})\) such that V belongs to the reverse Hölder class \(\mathrm{{RH}}^{q}(dw)\). We prove the Fefferman–Phong inequality for L. As an application, we conclude that the Hardy space \(H^1_{L}\), which is originally defined by means of the maximal function associated with the semigroup \(e^{-tL}\), admits an atomic decomposition with local atoms in the sense of Goldberg, where their localizations are adapted to V.

对于归一化的根系统ř在\（{\ mathbb {R}} ^ N \）和多个功能\（K \ GE 0 \）让\（{\ mathbf {N}} = N + \总和_ {\阿尔法\ in R} k（\ alpha）\）。我们用\（dw（{\ mathbf {x}}）= \ varPi _ {\ alpha \ in R} | \ langle {\ mathbf {x}}，\ alpha \ rangle | ^ {k（\ alpha}}来表示\，d {\ mathbf {x}} \）\（{\ mathbb {R}} ^ N \）中的相关度量。令\（L =-\ varDelta + V \），\（V \ ge 0 \）成为\（{\ mathbb {R}} ^ N \）上的Dunkl–Schrödinger运算符。假设存在\（q> \ max（1，\ frac {{\ mathbf {N}}} {2}）\），使得V属于反向Hölder类\（\ mathrm {{RH}} ^ {q}（dw）\）。我们证明L的Fefferman-Phong不等式。作为应用，我们得出结论，最初由与半群\（e ^ {-tL} \）相关的最大函数定义的Hardy空间\（H ^ 1_ {L} \）允许原子分解在戈德堡，其中它们的本地化适于感本地原子V。