The Stein–weiss Type Inequalities for the B–riesz Potentials

We establish two inequalities of Stein-Weiss type for the Riesz potential operator Iα,γ (B−Riesz potential operator) generated by the Laplace-Bessel differential operator ΔB in the weighted Lebesgue spaces Lp,|x|β ,γ . We obtain necessary and sufficient conditions on the parameters for the boundedness of Iα,γ from the spaces Lp,|x|β ,γ to Lq,|x|−λ ,γ , and from the spaces L1,|x|β ,γ to the weak spaces WLq,|x|−λ ,γ . In the limiting case p = Q/α we prove that the modified B−Riesz potential operator ̃ Iα,γ is bounded from the spaces Lp,|x|β ,γ to the weighted B−BMO spaces BMO|x|−λ ,γ . As applications, we get the boundedness of Iα,γ from the weighted B -Besov spaces Bs pθ ,|x|β ,γ to the spaces B s qθ ,|x|−λ ,γ . Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue Lp,|x|β ,γ and weighted B -Besov spaces B s pθ ,|x|β ,γ by using the fundamental solution of the B -elliptic equation Δ B . Mathematics subject classification (2010): 42B20, 42B25, 42B35.