Chamber lifting and non-radial Dunkl multipliers

Abstract

We study non-radial Dunkl multipliers via chamber lifting. For an arbitrary finite reflection group $G$, the chamber lifting records all reflected values of a function and conjugates a multiplier into a finite matrix-valued operator on the chamber. If the dyadic matrix entries admit off-diagonal kernels satisfying the chamber $L^2$ H\"ormander condition $\operatorname{CH}^2_{s,\eta}$ with $s>N_\kappa/2$, then the original multiplier is bounded on $L^p(\mathbb R^N,d\omega)$ for every $1 N_\kappa/2$ therefore imply $L^p$ boundedness, for all $1<p<\infty$, for a genuinely non-radial class of symbols. The order $N_\kappa/2$ is forced already by the rank-one Bessel transform. The same chamber theorem also applies to non-product examples once the matrix kernel condition is known, including the dihedral groups $I_2(q)$ and hence $A_2\simeq I_2(3)$ and $B_2\simeq I_2(4)$. The scalar Walsh--Sobolev verification is specific to $A_1^N$. In non-product groups such as $A_2$, $A_{N-1}$, and $B_N$, the product parity calculus is absent, so a scalar theorem of the same form would require additional transform estimates.