A Fefferman--Stein inequality for the Dunkl Poisson semigroup and its chamber-lifted formulation

Abstract

We prove a Fefferman--Stein good-$\lambda$ 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 ${\mathcal{N}}_P^{\beta }f$ with the area function ${\mathcal{S}}_{P}f$ 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 $E=\{x:{\mathcal{N}}_P^{\beta }f(x)\le \lambda \}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=\phi (P_t{1}_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood--Paley estimate for $P_t{1}_{E^c}$, 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 $L^1(d\omega )$ data.

37 pages