A Characterization of the Hardy Space Associated with the Dunkl Transform - Wei, Haihua; Liao, Jianquan; Li, Zhongkai (Complex analysis and operator theory, 2021)
For p ≥ p 0 : = 2 λ / ( 2 λ + 1 ) with λ > 0 , the Hardy space H λ p ( R + 2 ) associated with the Dunkl transform F λ and the Dunkl operator D on the line R , where ( D x f ) ( x ) = f ′ ( x ) + λ x ( f ( x ) - f ( - x ) ) , is the set of functions F = u + i v on the half plane R + 2 = { ( x , y ) : x ∈ R , y > 0 } , satisfying the generalized Cauchy–Riemann equations D x u - ∂ y v = 0 , ∂ y u + D x v = 0 , and sup y > 0 ∫ R | F ( x , y ) | p | x | 2 λ d x < + ∞ ; and the real Hardy space H λ p ( R ) on the line R is the collection of boundary functions of the real parts of functions F ∈ H λ p ( R + 2 ) . In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential I λ α associated to the Dunkl transform; and as the main result, we prove the equality D ( I λ 1 f ) = - H λ f for f ∈ H λ 1 ( R ) in a weak sense, where H λ is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for f ∈ H λ 1 ( R ) .