The Steın-Weıss Type Inequalıtıes For The B-Rıesz Potentıals

Abstract

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I(alpha,gamma) (B-Riesz potential operator) generated by the Laplace-Bessel differential operator Delta B in the weighted Lebesgue spaces L(p,vertical bar x vertical bar beta,gamma). We obtain necessary and sufficient conditions on the parameters for the boundedness of Ia,. from the spaces L(p,vertical bar x vertical bar beta,gamma) to L(q,vertical bar x vertical bar-lambda,gamma), and from the spaces L(1,vertical bar x vertical bar beta,gamma) to the weak spaces WL(q,vertical bar x vertical bar-lambda,gamma). In the limiting case p = Q/alpha we prove that the modified B-Riesz potential operator (I) over tilde (alpha,gamma) is bounded from the spaces L(p,vertical bar x vertical bar beta,gamma) to the weighted B-BMO spaces BMO(vertical bar x vertical bar-lambda,gamma). As applications, we get the boundedness of I(alpha,gamma) from the weighted B-Besov spaces B(p theta,vertical bar x vertical bar beta,gamma)(s) to the spaces B(q theta,vertical bar x vertical bar-lambda,gamma)(s). Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L(p,vertical bar x vertical bar beta,gamma) and weighted B-Besov spaces B(p theta,vertical bar x vertical bar beta,gamma)(s) by using the fundamental solution of the B-elliptic equation Delta(alpha/2)(B)