Abstract
In this paper we prove a two weighted inequality for Riesz potentials $I_{\alpha,\gamma} f$ (B-fractional integrals) associated with the Laplace-Bessel differential operator $\Delta_{B}=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} + \sum_{j=1}^{k} \frac{\gamma _{j}}{x_{j}}\frac{\partial}{\partial x_{j}}$ . 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.
Highlights
The Stein-Weiss inequality for B-fractional integrals is proved as an application of this result
The fractional integral operators play an important role in the theory of harmonic analysis, differentiation theory and PDE’s
Many mathematicians have dealt with the fractional integrals and related topics associated with the Laplace-Bessel differential operator B =
Summary
The Stein-Weiss inequality for B-fractional integrals is proved as an application of this result. The weighted Lebesgue space Lp,w,γ ≡ Lp,w,γ (Rnk,+), ≤ p < ∞, is the set of all classes of measurable functions f with finite norm f p,w,γ = In this paper we consider fractional (B-fractional) integrals in the weighted Lebesgue space Lp,w,γ (Rnk,+) associated with the generalized shift operator The B-fractional integral (or B-Riesz potential) is defined by
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