Hardy-Littlewood-Stein-Weiss type theorems for Riesz potentials and their commutators in Morrey spaces

In: Studia Universitatis Babeş-Bolyai. Mathematica, Band 68, Heft 3, S. 613-629

Abstract

"In this paper we consider weighted Morrey spaces $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn).$ We prove the Hardy-Littlewood-Stein-Weiss type $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)$ to $L_{q,\lambda,|\cdot|^{\mu}}(\Rn)$ theorems for Riesz potential $I^{\al}$ and its commutators $[b,I^{\alpha}]$ and $|b,I^{\alpha}|$, where $0<\alpha<n$, $0\leq\lambda <n-\alpha$, $1< p<\frac {n-\lambda}{\alpha}$, $-n+\lambda\le\gamma<n(p-1) +\lambda$, $\mu=\frac{q\gamma}{p}$, $\frac 1p-\frac 1q = \frac \alpha {n-\lambda}$, $b\in BMO(\Rn).$ As a result of these we obtain the conditions for the boundedness of the commutator $|b,I^{\al}|$ from Besov-Morrey spaces $B_{p,\theta,\lambda,|\cdot|^{\gamma}}^s (\Rn)$ to $B_{q,\theta,\lambda,|\cdot|^{\mu}}^s (\Rn).$ Furthermore, we consider the Schr\""{o}dinger operator $-\Delta + V$ on $\Rn$ and obtain weighted Morrey $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)$ estimates for the operators $V^{s} (-\Delta+V)^{-\beta}$ and $V^{s} \nabla (-\Delta+V)^{-\beta}.$ Finally we apply our results to various operators which are estimated from above by Riesz potentials."