Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications

In this article we consider the fractional integral operator I-alpha on any Carnot group G (i. e. nilpotent stratified Lie group) in the weighted Lebesgue spaces L-p,L- rho(x)beta (G). We establish Stein-Weiss inequalities for I-alpha, and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator I-alpha from the spaces L-p,L- rho(x)beta (G) to L-q,L- rho(x)-gamma (G), and from the spaces L-1,L- rho(x)beta (G) to the weak spaces WLq, rho(x)-gamma (G) by using the Stein-Weiss inequalities. In the limiting case p = Q/alpha-beta-gamma, we prove that the modified fractional integral operator (I) over tilde (alpha) is bounded from the space L-p,L- rho(x)beta (G) to the weighted bounded mean oscillation (BMO) space BMO (rho(x)-gamma) (G), where Q is the homogeneous dimension of G. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev-Stein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As another application, we prove the boundedness of I-alpha from the weighted Besov spaces B-p theta,beta(s) (G) to B-q theta, - gamma(s)(G).