A Characterization of the Hardy Space Associated with the Dunkl Transform

摘要

For p >= p(0) := 2 lambda/(2 lambda + 1) with lambda > 0, the Hardy space H-lambda(p)(R-+(2)) associated with the Dunkl transform F-lambda and the Dunkl operator D on the line R, where (D(x)f)(x) = f '(x) + lambda/x(f(x) - f(-x)), is the set of functions F = u + iv on the half plane R-+(2) = {(x, y) : x is an element of R, y > 0}, satisfying the generalized Cauchy-Riemann equations D(x)u- partial derivative(y)v = 0, partial derivative(y)u + D(x)v = 0, and sup(y>0)integral(R)vertical bar F(x, y)vertical bar(p)vertical bar x vertical bar(2 lambda)dx < +infinity; and the real Hardy space H-lambda(p)(R) on the line R is the collection of boundary functions of the real parts of functions F is an element of H-lambda(p)(R + 2). In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential I-lambda(alpha) associated to the Dunkl transform; and as the main result, we prove the equality D(I(lambda)(1)f) = -H-lambda(f) for f is an element of H-lambda(1)(R) in a weak sense, where H-lambda is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for f is an element of H-lambda(1)(R).