Hardy-Littlewood-Sobolev-Type Inequality for the Fractional Littlewood-Paley g-Function in Jacobi Analysis

We introduce the fractional Littlewood-Paley g-function of order s, s > 0, noted g(s)(alpha,beta), associated with the Jacobi operator Delta(alpha,beta) on (0,infinity) as the operator g(s)(alpha,beta) (f)(x) = integral(infinity)(0) t(2s+1) vertical bar partial derivative u(alpha,beta) (f)/partial derivative t (x,t)vertical bar(2) dt)(1/2), where u(alpha,beta) (f) is the Poisson integral defined by u(alpha,beta) (f) = P-t(alpha,beta) *(alpha,beta) f (*(alpha,beta) being the convolution in the Jacobi setting). We establish the following Hardy-Littlewood-Sobolev-type inequality: For 0 < s < 2(alpha+ 1), 1 < p < 2(alpha+1)/s and 1/q = 1/p - s/2(alpha+1), there exists a constant C-alpha,C- s,C- p such that for all f is an element of L-p([0,+infinity[, d mu(alpha,beta)), parallel to g(s)(alpha,beta) f parallel to(q,mu) <= C-alpha,C- s,C- p parallel to f parallel to(p,mu) . Next, if p = 1, 0 < s < 2( alpha + 1) and q = alpha+1/2(alpha+1)-s, we prove that the operator g(s)(alpha,beta) is of weak type (1, q).