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The main purpose of this article is to establish theLp{L}^{p}Hardy’s identities and inequalities for Dunkl operator on any finite balls and the entire spaceRN{{\mathbb{R}}}^{N}. We also prove Hardy’s identities and inequalities on certain domains with distance function to the boundary∂Ω\partial \Omega. In particular, we use the notion of Bessel pairs introduced in Ghoussoub and Moradifam to extend Hardy’s identities for the classical gradients obtained by Lam et al., Duy et al., Flynn et al. to Dunkl gradients introduced by Dunkl. Our Hardy’s identities with explicit Bessel pairs significantly improve many existing Hardy’s inequalities for Dunkl operators.

L p Hardy's identities and inequalities for Dunkl operators