L^p-boundedness of generalized Dunkl translations

Determine whether the Dunkl-type generalized translation operators τ_y are bounded on L^p(R^N,w_k) for every 1 ≤ p < ∞ with p ≠ 2, beyond the rank-one case.

Background

The generalized translation τ_y is defined through the Dunkl transform and is known to extend continuously to L^2(R^N,w_k), with operator norm at most one. In rank one, explicit signed-hypergroup formulas yield boundedness on all L^p spaces with 1 ≤ p < ∞.

For general root systems and multiplicities, the paper records that the corresponding L^p-boundedness question remains unresolved. A partial result establishes positivity and L^p-boundedness for radial functions, but does not settle the full nonradial problem.