Non-radial Dunkl multipliers at the $L^2$-Sobolev threshold

Abstract

We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^{\sigma }({\mathbb{R}}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma >\mathbf{N}/2$ and $\mathbf{N}$ the homogeneous dimension, imply $L^p(d\omega )$ boundedness for $1<p<\infty$ and weak type $(1,1)$. No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel $E_{\kappa }(i\xi ,x)$ and its first Euclidean derivatives in $x$, uniformly near arbitrary intersections of reflecting hyperplanes. Here $\xi$ lies in a fixed compact annulus. Writing $w_{\kappa }$ for the Dunkl weight and ${\Omega }_{\kappa }$ for its inhomogeneous counterpart, it gives the bound $C{\Omega }_{\kappa }(x{)}^{-1/2}{w}_{\kappa }(\xi {)}^{-1/2}$. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman--Weiss atomic $H^1$ space and the corresponding BMO space on a fixed chamber.