Two weighted inequalities for B-fractional integrals

Abstract

Abstract In this paper we prove a two weighted inequality for Riesz potentials I α , γ f $I_{\alpha,\gamma} f$ (B-fractional integrals) associated with the Laplace-Bessel differential operator Δ B = ∑ i = 1 n ∂ 2 ∂ x i 2 + ∑ j = 1 k γ j x j ∂ ∂ x j $\Delta_{B}=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} + \sum_{j=1}^{k} \frac{\gamma _{j}}{x_{j}}\frac{\partial}{\partial x_{j}}$ . This result is an analog of Heinig’s result (Indiana Univ. Math. J. 33(4):573-582, 1984) for the B-fractional integral. Further, the Stein-Weiss inequality for B-fractional integrals is proved as an application of this result.

Keywords

• Laplace-Bessel differential operator
• B-fractional integrals
• Stein-Weiss inequality