We establish two inequalities of Stein-Weiss type for the Riesz potential operator I-alpha,I-gamma (B-Riesz potential operator) generated by the Laplace-Bessel differential operator Delta B in the weighted Lebesgue spaces L-p,L-vertical bar x vertical bar beta,L-gamma. We obtain necessary and sufficient conditions on the parameters for the boundedness of Ia,. from the spaces L-p,L-vertical bar x vertical bar beta,L-gamma to L-q,L-vertical bar x vertical bar-lambda,L-gamma, and from the spaces L-1,L-vertical bar x vertical bar beta,L-gamma to the weak spaces WLq,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,L-vertical bar x vertical bar beta,L-gamma to the weighted B-BMO spaces BMO vertical bar x vertical bar-lambda,gamma.
As applications, we get the boundedness of I-alpha,I-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,L-vertical bar x vertical bar beta,L-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) |
A. D. Gadjiev
Vagıf GULIYEV
