Weighted Morrey spaces related to certain nonnegative potentials and Riesz transforms

Abstract

Let $\mathcal{L}=-\Delta +V$ be a Schrödinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}}^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_q$ for $q\ge d$. The Riesz transform associated with the operator $\mathcal{L}=-\Delta +V$ is denoted by $\mathcal{R}=\nabla {(-\Delta +V)}^{-1/2}$ and the dual Riesz transform is denoted by ${\mathcal{R}}^{\ast }=(-\Delta +V{)}^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class $R{H}_q$ for $q\ge d$. Then we will establish the boundedness properties of the operators $\mathcal{R}$ and its adjoint ${\mathcal{R}}^{\ast }$ on these new spaces. Furthermore, weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators $[b,\mathcal{R}]$ and $[b,{\mathcal{R}}^{\ast }]$ are also obtained. The classes of weights, the classes of symbol functions as well as weighted Morrey spaces discussed in this paper are larger than $A_p$, $\mathrm{BMO}({\mathbb{R}}^d)$ and $L^{p,\kappa }(w)$ corresponding to the classical Riesz transforms ($V\equiv 0$).

30 pages