The Stein-Weiss type inequalities for the B-Riesz potentials

Abstract

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I?,? (B-Riesz potential operator) generated by the Laplace-Bessel differential operator ?B in the weighted Lebesgue spaces Lp,|x|ß,?. We obtain necessary and sufficient conditions on the parameters for the boundedness of I?,? from the spaces Lp,|x|ß,? to Lq,|x|-?,?, and from the spaces L1,|x|ß,? to the weak spaces WLq,|x|-?,?. In the limiting case p=Q/? we prove that the modified B-Riesz potential operator I?,? is bounded from the spaces Lp,|x|ß,? to the weighted B-BMO spaces BMO|x|-?,?. As applications, we get the boundedness of I?,? from the weighted B-Besov spaces Bs p?,|x|ß,? to the spaces Bs q?,|x|-?,?. Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue Lp,|x|ß,? and weighted B-Besov spaces Bs p?,|x|ß,? by using the fundamental solution of the B-elliptic equation ??/2 B.