AI 中文总结
该研究证明了有限反射群下邓克尔泊松半群的Fefferman--Stein好λ不等式，给出相关极大-面积估计与端点界，还通过腔提升得到等价形式，刻画了邓克尔泊松极大Hardy空间。
AI 中文摘要
我们证明了与有限反射群及非负重数函数相关的邓克尔泊松半群的Fefferman--Stein好λ不等式。对任意复值函数f∈C_c^∞(ℝ^N)，不要求其具有G-不变性，该不等式将轨道锥形非切向极大函数𝒩_P^β f与由全时空邓克尔 carré du champ（包括其反射差能量）构成的面积函数𝒮_P f进行比较。主要障碍在于，一般截断会产生无法被局部欧氏梯度乘积恒等式控制的壁差。好集E={x:𝒩_P^β f(x)≤λ}具有G-不变性；由泊松半群的等变性，a=φ(P_t 1_E)也具有G-不变性，因此截断的所有反射差均消失。泊松极大估计和尾部估计，结合P_t 1_{E^c}的L^2 Littlewood--Paley估计，可得到所需的分布不等式。其积分形式给出对所有0<p<2的极大-面积估计，以及轨道锥形和欧氏锥形固有面积函数的端点H^1→L^1界。对于全局光滑数据的腔提升，该不等式在基本腔上有等价形式，其中轨道锥变为欧氏锥，反射能量变为有限壁耦合。结合已知的半群平方函数刻画，这些界在L^1(dω)数据中刻画了邓克尔泊松极大Hardy空间。
英文摘要
We prove a Fefferman--Stein good-$λ$ inequality for the Dunkl Poisson semigroup associated with a finite reflection group and a non-negative multiplicity function. For arbitrary complex-valued $f\in C_c^\infty(\mathbb R^N)$, with no $G$-invariance assumption, it compares the orbit-conical non-tangential maximal function $\mathcal N_P^βf$ with the area function $\mathcal S_Pf$ formed from the full space-time Dunkl carré du champ, including its reflection-difference energy. The main obstruction is that a general cut-off creates wall differences not controlled by the local Euclidean-gradient product identity. The good set $E=\{x:\mathcal N_P^βf(x)\leλ\}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=φ(P_t \mathbf 1_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood--Paley estimate for $P_t \mathbf 1_{E^c}$, then yield the desired distribution inequality. Its integrated form gives maximal-to-area estimates for every $0<p<2$ and endpoint $H^1$-to-$L^1$ bounds for the orbit-conical and Euclidean-conical intrinsic area functions. For chamber lifts of globally smooth data, the inequality has an equivalent formulation on a fundamental chamber, where orbit cones become Euclidean cones and the reflection energy becomes a finite wall coupling. Combined with the known semigroup square-function characterization, these bounds characterize the Dunkl Poisson maximal Hardy space among $L^1(dω)$ data.
Comments37 pages
