Abstract
LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).
Highlights
A nonnegative locally integrable function V (x) on Rd is said to belong to the reverse Holder class RHq for some exponent 1 < q < ∞, if there exists a positive constant C > 0 such that the following reverse Holder inequality
In [2], Bongioanni et al.obtained weighted strong (p, p), 1 < p < ∞, and weak L log L estimates for the commutators of the Riesz transform and its adjoint associated with the Schrodinger operator L = −∆ + V, where V
The higher order commutators formed by a BMOρ,∞(Rd) function b and the operators R and its adjoint R∗ are usually defined by
Summary
Holds for every ball B = B(x0, r) in Rd. For θ > 0, let us introduce the maximal operator that is given in terms of the critical radius function (1.1). It is well known that the following generalized Holder inequality in Orlicz spaces holds for any given ball B ⊂ Rd: 1 |B|
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