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 $L^2$ Hörmander condition $\operatorname{CH}^2_{s,η}$ with $s>N_κ/2$, then the original multiplier is bounded on $L^p(\mathbb R^N,dω)$ for every $1<p<\infty$. For the product reflection group $Σ_N=A_1^N\simeq\mathbb Z_2^N$ this chamber condition follows from scalar Sobolev conditions on the Walsh pieces of the multiplier. The tensor product of the one-dimensional even/odd Dunkl decompositions, together with the finite Walsh transform, identifies each lifted matrix entry with a Hankel multiplier acting between parity components. Wall separation and a scale-invariant $L^2$ Sobolev condition of order $σ>N_κ/2$ therefore imply $L^p$ boundedness, for all $1<p<\infty$, for a genuinely non-radial class of symbols. The order $N_κ/2$ 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.