Dimension‐free square function estimates for Dunkl operators

Abstract

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood-Paley square function for Dunkl heat flows in ℝ 𝑑 is introduced by employing the full "gradient" induced by the corresponding carré du champ operator and then the 𝐿 𝑝 boundedness is studied for all 𝑝 ∈ (1, ∞). For 𝑝 ∈ (1, 2],we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the 𝐿 𝑝 boundedness, while for 𝑝 ∈ [2, ∞), we restrict to a particular case when the corresponding Weyl group is isomorphic to ℤ 𝑑 2 and apply a probabilistic method to prove the 𝐿 𝑝 boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry-Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.

Sharp Li–Yau inequalities for Dunkl harmonic oscillators

“…See e.g. [29], [14] and [19,Section 3] for some probabilistic aspects of the Dunkl theory and see e.g. [30] for more details on Lévy processes.…”