Weighted Morrey spaces related to certain nonnegative potentials and Riesz transforms

TL;DR

This paper introduces new weighted Morrey spaces related to nonnegative potentials in Schrödinger operators and establishes boundedness and endpoint estimates for associated Riesz transforms and their commutators.

Contribution

It defines novel weighted Morrey spaces for Schrödinger operators with potentials in reverse H"older classes and proves boundedness of Riesz transforms and commutators on these spaces.

Findings

Boundedness of Riesz transforms on new Morrey spaces

Weighted endpoint estimates for commutators

Extension of classical results to larger weight and symbol classes

Abstract

Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator, where $\Delta$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_q$ for $q\geq d$. 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 H\"older class $RH_q$ for $q\geq d$. 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 ($V\equiv0$).