Abstract
Let ℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0<α<d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).
Highlights
For given V ∈ RHs with s ≥ d/2, we introduce the critical radius function ρ(x) ρ(x; V)
In view of Lemma 2, we define the reverse Holdertype class RHρq,θ that is given in terms of the critical radius function (2)
We will define several classes of weighted Morrey spaces related to certain nonnegative potentials satisfying appropriate reverse Holder inequality
Summary
In view of Lemma 2, we define the reverse Holdertype class RHρq,θ that is given in terms of the critical radius function (2). In eorem 4.4 of [7], Tang obtained weighted strong-type estimate for the commutator [b, Iα] of fractional integrals associated with Schrodinger operators, when b in a larger space than BMO(Rd), that is the space BMOρ,∞(Rd). He gave the following weighted result (see [11], eorem 3.5). We will define several classes of weighted Morrey spaces related to certain nonnegative potentials satisfying appropriate reverse Holder inequality. We use a ≈ b to denote the equivalence of a and b; that is, there exist two positive constants C1 and C2 independent of a and b such that C1a ≤ b ≤ C2a
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
