Chamber lifting and non-radial Dunkl multipliers

We study non-radial Dunkl multipliers through a chamber representation. For an arbitrary finite reflection group G, chamber lifting records the reflected values of a function and conjugates a scalar multiplier to a finite matrix operator on a fundamental chamber. If its dyadic entries admit off-diagonal kernels satisfying the chamber L2 Hörmander condition CHs,η2 with s>Nκ/2, then the original multiplier is bounded on Lp(RN,dω) for every 1<p<∞. For the product group A1N≃Z2N, the finite Walsh transform and the tensor product of the rank-one even–odd decompositions identify the lifted entries with Hankel multipliers between parity components. We prove directly that wall-separated, scale-invariant W2σ control of the Walsh pieces, with σ>Nκ/2, gives the required chamber kernel estimates. The resulting Lp conclusion is already available in the product case from the classical full-symbol Hörmander theorem. The contribution here is the application of the chamber representation to the multiplier problem and the direct Walsh–Bessel verification of the resulting kernels, which together make explicit how a non-radial symbol mixes parity components. For non-product groups the chamber theorem remains conditional on the matrix kernel estimates; the scalar Walsh–Sobolev verification is specific to A1N.