Poisson representations of Dunkl harmonic functions satisfying a Carleson condition

We prove that a smooth Dunkl harmonic function in the upper half-space whose Dunkl gradient satisfies a Carleson measure condition is the Poisson integral of a function in $\mathrm{B}\mathrm{M}\mathrm{O}(dw)$, defined using Euclidean balls. The result holds for arbitrary finite root systems and nonnegative multiplicities. The proof uses an estimate for increments from the origin, obtained from a finite-dimensional system indexed by the reflection group, together with the atomic characterization of the Dunkl Hardy space.