Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 67, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.67 BOUNDEDNESS ON GENERALIZED MORREY SPACES FOR THE SCHRÖDINGER OPERATOR WITH POTENTIAL IN A REVERSE HÖLDER CLASS GUIYUN WANG, SHENZHOU ZHENG Abstract. In this article, we prove boundedness for the Hessian of a Schrödinger operator with weak regularity on the coefficients, and potentials satisfying the reverse Hölder condition. This is done in in generalized Morrey spaces, and in vanishing generalized Morrey spaces. On the Schrödinger operator L = −aij(x)Dij + V (x) it is assumed that aij ∈ BMOθ(ρ) (a generalized Morrey space) and that V (x) ∈ B∗ n/2 (a reverse Hölder class). 1. Introduction This article presents local estimates in the framework of generalized Morrey spaces and vanishing generalized Morrey spaces for the Schrödinger operator with a weak assumption on the main coefficient and a singular potential satisfying the reverse Hölder class. More precisely, we consider the non-divergence Schrödinger operator with discontinuous coefficient Lu = −aij(x)Diju+ V (x)u for x ∈ Rn with n ≥ 3, (1.1) where ∇2u = (Diju)n×n is the Hessian matrix of u, and V (x) ∈ B∗n/2 is a non- negative singular potential belonging to the so-called reverse Hölder class defined below. Here, we assume that A = (aij(x))n×n is a measurable symmetric matrix with aij = aji defined on Rn and it satisfies the following uniform ellipticity and boundedness such that there exists a positive constant λ ∈ (0, 1] satisfying λ|ξ|2 ≤ aijξiξj ≤ λa−1|ξ|2, |aij | ≤ λ−1 for all ξ ∈ Rn. (1.2) The Calderón-Zygmund theory of second-order elliptic equations with discon- tinuous coefficients has been studied extensively in the last three decades. Interior and boundaryW 2,p-estimates were first established by Chiarenza, Frasca and Longo [8, 9] for nondivergence elliptic equations with VMO discontinuous coefficients, and they were extended to nondivergence parabolic equations by Bramanti and Cerutti [5]. Recently, Krylov [19] gave a unified approach to consider the Lp-solvability of elliptic and parabolic equations of divergence or nondivergence form with weak assumptions of the coefficients belonging to the VMO class in the spatial variables. 2020 Mathematics Subject Classification. 35J10, 42B35, 42B20. Key words and phrases. Schrödinger operators; reverse Hölder class; generalized Morrey space; vanishing generalized Morrey space; BMOθ(ρ) coefficients. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 29, 2023. Published October 13, 2023. 1 2 G. WANG, S. ZHENG EJDE-2023/67 This is achieved by pointwise estimates for the sharp maximal functions for the spatial derivatives of solutions by way of the famous Fefferman-Stein theorem. Re- cently, Bramanti, Brandolini, Harboure and Viviani [4] gave global W 2,p-estimates for the Schrödinger operator with VMO discontinuous coefficient and a potential V (x) satisfying a reverse Hölder class. We would like to point out that Byun and Wang [6] showed a global Lp regularity for elliptic equations with small BMO co- efficients in the Reifenberg flat domains. For divergence elliptic cases, Liang and Zheng [20] proved a global Orlicz estimate of gradients to a class of nonlinear ob- stacle problems with partially regular nonlinearities in nonsmooth domains, while Liang, Zheng and Feng [21] further gave a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for the variable power of gradient of solution pair to the generalized steady Stokes system over a bounded non-smooth domain. For nondivergence elliptic cases, Zhang and Zheng [31] proved weighted Lorentz estimates of the Hessian of solution to nondivergence linear elliptic equations with partially BMO coefficients, while Tian and Zheng [28] got global Lorentz estimates for a variable power of gradient to linear elliptic obstacle problems with small par- tially BMO coefficients over a nonsmooth domain. It is also worth noting that Bongioanni, Harboure and Salinas in [3] proved the Lp-boundedness for commuta- tors of Riesz transforms associated with Schrödinger operators with small BMOθ(ρ) coefficients which include the classical BMO functions. Guliyev and Softova [15] got global regularity in generalized Morrey spaces for the gradient of solutions of nondivergence elliptic equations with VMO coefficients. Guliyev, Omarova, and Ragusa [18] further derived the boundedness on local generalized Morrey spaces for Schrödinger type operators involved in certain nonnegative potentials. On the other hand, the Morrey spaces were first introduced by Morrey [23] to study a local behavior of solutions for elliptic differential equations of second-order. Later, many researchers studied Morrey spaces from various points of view. For examples, Fazio, Palagachev and Ragusa [12, 13] got an interior and global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with dis- continuous coefficients, respectively. Fan, Lu and Yang [11] also gave the regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients. Chen and Song [7] established the boundedness of the commutator for Riesz potential associated with Schrödinger operator on Morrey spaces. Recently, Tian and Zheng [29] provide another approach to Morrey regularity for linear el- liptic equations with partially BMO coefficients, and they in [30] further proved global Morrey regularity for nonlinear elliptic equations with controlled growth un- der weak assumption of partial BMO nonlinearities on Reifenberg domains. Zhang and Zheng [32] presented a local Morrey regularity for linear parabolic equations of divergence form under the assumption that the leading coefficient being indepen- dent of t and not necessarily symmetry. After studying Morrey spaces in detail, some researchers passed to generalized Morrey spaces, weighted Morrey spaces and generalized weighted Morrey spaces. Mizuhara in [22] introduced the generalized Morrey spaces and established the boundedness of some classical operators on gen- eralized Morrey spaces, which was later extended and studied by many authors. Note that Guliyev [14] introduced the generalized Morrey spaces Mp,ϕ(Rn) and EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 3 the weak generalized Morrey space WMp,ϕ(Rn) with normalized norms, respec- tively, as ‖f‖Mp,ϕ = sup x∈Rn,r>0 ϕ(x, r)−1|B(x, r)|−1/p‖f‖WLp(B(x,r)) <∞; and he obtained the boundedness of the maximal, potential and singular operators in the generalized Morrey spaces by imposing certain condition on ϕ(x, r). For more details regarding the boundedness of various classical linear operators on the generalized Morrey spaces we refer to Guliyev et al in [1, 15, 17, 18, 24]. It is well known that the Calderón-Zygmund theory plays an important role in applications of harmonic analysis and partial differential equations. In [2], Akbulut and Kuzu successfully used the idea and argument of Guliyev’s works for the boundedness on generalized Morrey spaces Mp,ϕ(Rn) for the Marcinkiewicz integrals associated to the Schrödinger operators L0 = −∆ + V with ∆ as a Laplacian. The present paper is actually inspired by the Lp-estimate for L0 = −∆ + V from Pan and Tang’s paper in [25], and the boundedness for the commutators associated with the Schrödinger operators L0 on local generalized Morrey spaces from Guliyev, Guliyev, Omarova and Ragusa’s research in [18]. In fact, our study for general Schrödinger operator L = −aij(x)Dij+V (x) will attract much attention due to its discontinuous coefficient aij(x) ∈ VMO(Ω). This leads to that a key point of our argument is in an effort how to handle the variable coefficient aij(x) as a perturbation of tha usual Schrödinger operators L′0 with constant coefficient in the sense of integral. The rest of this article is organized as follows. We devote Section 2 to the re- lated notations and statement of main results. In Section 3, we will give some auxiliary lemmas. In Section 4, we prove the the boundedness on generalized Mor- rey space and vanishing generalized Morrey space for Hessian of operators ∇2L−1, respectively. 2. Notation and main results To state our problem we first recall the definition of the reserve Hölder class Bq. Let V (x) be a locally Lq-integrable nonnegative function in Rn. We say that the potential V (x) belongs to the reverse Hölder class, denoted by V (x) ∈ Bq for 1 < q ≤ ∞; if there exists a positive constant C such that the reverse Hölder inequality holds:( 1 |B(x, r)| ∫ B(x,r) V q(y)dy )1/q ≤ C ( 1 |B(x, r)| ∫ B(x,r) V (y)dy ) (2.1) for all B(x, r) with centered at x ∈ Rn and the radius 0 < r < ∞. In particular, if V (x) is a nonnegative polynomial, then V ∈ B∞. As well known, while V ∈ Bq with q > 1, we obtain a higher integrability of V (x), which implies that there exists an ε > 0 such that V ∈ Bq+ε, where the constant ε depends only on n and C of (2.1). Furthermore, there exists the following double condition: for V (x) ∈ Bq it holds ∫ B(x,2r) V (y)dy ≤ C ∫ B(x,r) V (y)dy. (2.2) With the class of reverse Hölder for V (x) ∈ Bq in hand, we introduce the auxiliary function associated with the potentials V (x), by ρ(x) = 1 m(x, V ) := sup r>0 { r : 1 rn−2 ∫ B(x,r) V (y)dy ≤ 1 } . (2.3) 4 G. WANG, S. ZHENG EJDE-2023/67 In the following context, we consider that potential function V (x) satisfies the B∗n/2-condition, if there is a positive constant C independent of V (x) such that • V (x) ∈ Bn/2; • V (x) ≤ Cm2(x, V ), |∇V | ≤ Cm3(x, V ), and |∇2V | ≤ Cm4(x, V ). Before stating our main results, let us first recall some related notation and basic facts. It is necessary to impose some weaker regularity assumptions on the leading coefficients of Schröndinger operators. To this end, let us recall the concepts of BMO-space and BMOθ(ρ)-space. In this context, we denote the integral average over a ball B(x, r) by gB = 1 |B| ∫ B g(y)dy for a local integrable functions g(x). Definition 2.1 (classical BMO-space). We say that g ∈ BMO for a locally inte- grable function g(x) ∈ L1(Rn), if ‖g‖BMO := sup x∈Rn,r>0 1 |B(x, r)| ∫ B(x,r) |g(y)− gB |dy <∞. Definition 2.2 (BMOθ(ρ)-space). We say that g ∈ BMOθ(ρ) with θ ≥ 0 associated with the potentials V (x) ∈ Bq, if for a locally integrable function g(x) ∈ L1 loc(Rn) there holds 1 |B(x, r)| ∫ B(x,r) |g(y)− gB |dy ≤ C ( 1 + r ρ(x) )θ (2.4) for x ∈ Rn and r > 0, and the semi-norm of g ∈ BMOθ(ρ) is defined by [g]θ := sup x∈Rn,r>0 ( 1 + r ρ(x) )−θ 1 |B(x, r)| ∫ B(x,r) |g(y)− gB |dy <∞. As a direct consequence of BMOθ(ρ)-space, we obviously check that BMO ⊂ BMOθ(ρ) ⊂ BMOθ′(ρ) for 0 < θ ≤ θ′, see [25, 18]. We are now in a position to introduce the notations of generalized Campanato space and generalized Morrey space. Definition 2.3 (generalized Campanato space Λθν(ρ)). We say that g ∈ Λθν(ρ) with θ > 0 and 0 < ν < 1 associated with the potentials V (x), if a locally integrable function g(x) ∈ L1(Rn) satisfies 1 |B(x, r)|1+ν/n ∫ B(x,r) |g(y)− gB |dy ≤ C ( 1 + r ρ(x) )θ for all x ∈ Rn and r > 0. We denote the semi-norm of g ∈ Λθν(ρ) by [g]θν := sup x∈Rn,r>0 ∫ B(x,r) |g(y)− gB |dy |B(x, r)|1+ν/n ( 1 + r ρ(x) )θ <∞. Remark 2.4. We would like to remark that if θ = 0, then Λθν(ρ) is exactly the classical Campanato space; if ν = 0, then Λθν(ρ) is the generalized BMO space denoted by BMOθ(ρ) space; if θ = 0 and ν = 0, Λθν(ρ) is nothing but the so-called John-Nirenberg space, that is, the usual BMO space. Definition 2.5 (generalized Morrey space Mα,V p,ϕ ). Let ϕ(x, r) be a positive mea- surable function on Rn× (0,∞), and V ∈ Bq with q > 1. We say that g ∈Mα,V p,ϕ = Mα,V p,ϕ (Rn) for 1 ≤ p < ∞ and α ≥ 0 is the generalized Morrey space associated with the potentials V (x), if g ∈ Lploc(Rn) satisfies ‖g‖Mα,V p,ϕ := sup x∈Rn,r>0 ( 1 + r ρ(x) )α r−n/pϕ(x, r)−1‖g‖Lp(B(x,r)) <∞. (2.5) EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 5 Definition 2.6 (vanishing generalized Morrey space VMα,V p,ϕ ). We say that g ∈ VMα,V p,ϕ = VMα,V p,ϕ (Rn) be the vanishing generalized Morrey spaces associated with the potentials V (x), if g ∈Mα,V p,ϕ satisfies lim r→0 sup x∈Rn ( 1 + r ρ(x) )α r−n/pϕ(x, r)−1‖g‖Lp(B(x,r)) = 0. (2.6) As an immediate consequence of the above definitions, the generalized Morrey spaces and the vanishing generalized Morrey spaces associated with the potentials are Banach spaces with respect to their norm, respectively, see [22, 24]. Remark 2.7. (i) If α = 0 and ϕ(x, r) = r(λ−n)/p, then Mα,V p,ϕ (Rn) is actually the classical Morrey spaces denoted by Lp,λ(Rn), which was originally introduced by Morrey to study the local behavior of solutions to second order elliptic partial differential equations, see [23]. (ii) If α = 0, then Mα,V p,ϕ (Rn) is the generalized Morrey spaces denoted by Mp,ϕ(Rn), which was first introduced by Mizuhara and Nakai in [22, 24], and later Guliyev made further study for it, see [15, 17, 2]. (iii) If ϕ(x, r) = r(λ−n)/p, then Mα,V p,ϕ (Rn) is the Morrey spaces associated with the potentials, denoted by Lα,Vp,λ (Rn) that was introduced by Tang and Dong in [27]. (iv) Here, the generalized Morrey spaces Mα,V p,ϕ and the vanishing generalized Morrey spaces VMα,V p,ϕ associated with the singular potentials, respectively, were introduced by Guliyev to study the boundedness of some operators and their com- mutators, see [16]. We are now ready to present the main results of this paper, which is involved in the boundedness in the generalized Morrey spaces for the Hessian ∇2L−1 of solutions as follows. Theorem 2.8. Let aij(x) ∈ BMOθ(ρ) and V ∈ B∗n/2. For α ≥ 0 and 1 < p <∞, we assume that ϕ1, ϕ2 ∈ Ωα,Vp satisfies∫ ∞ r ess inft 0 is independent of x and r. If there exists a constant ε > 0 such that [aij ]θ < ε, for the solutions of Lu = f(x), then the Hessian ∇2L−1 is a bounded operator from the the generalized Morrey space Mα,V p,ϕ1 to the generalized Morrey space Mα,V p,ϕ2 . Moreover, for any 1 < p <∞ we have the estimate ‖∇2L−1f‖Mα,V p,ϕ2 ≤ C‖f‖Mα,V p,ϕ1 . (2.8) A more delicate result is stated in the vanishing generalized Morrey spaces. Let us consider it under the following assumptions on ϕ1 ∈ Ωα,Vp,1 , there still holds the boundedness of Schrödinger operators on the vanishing generalized Morrey spaces. Theorem 2.9. Let aij(x) ∈ BMOθ(ρ) and V ∈ B∗n/2. Under the same assumptions of Theorem 2.8 on ϕ1, ϕ2. If f ∈ VMα,V p,ϕ1 , then the operators ∇2L−1 is bounded from VMα,V p,ϕ1 to VMα,V q,ϕ2 for 1 < p <∞. Moreover, we have the estimate ‖∇2L−1f‖VMα,V q,ϕ2 ≤ C ‖f‖VMα,V p,ϕ1 . (2.9) 6 G. WANG, S. ZHENG EJDE-2023/67 We pointed out that our method in this paper is novel in some sense. It seems that we cannot obtain directly the regularity results for solutions of the Schrödinger equations with BMOθ(ρ) coefficients since the arguments depend heavily on the regularity of solutions of the elliptic equations with BMO coefficients. Our argu- ment is motivated by considering the operator ∇2(−∆ + V )−1 for V (x) ∈ Bn/2 as a standard Calderón-Zygmund operator so that its kernel K(x, y) of this operator possesses the estimate |K(x, y)| ≤ CN( 1 + |x−y| ρ(x) )N 1 |x− y|n (2.10) for all N ∈ N. We would like to mention that our regularity of solutions are worked in the generalized Morrey space which depends heavily on singular potential V (x). 3. Technical lemmas This section is devoted to some well-known facts about some fundamental in- equalities and technical lemmas that we will use later. Throughout this paper, C(n, λ, . . . ) stands for a universal positive constant depending only on prescribed quantities and possibly varying from line to line. However, the ones we need to emphasize will be denoted with special symbols, such as C1, C2, . . . . First of all, let us recall an inequality concerning the auxiliary function, and the relationships between the generalized BMO space and BMOθ(ρ) space. Lemma 3.1 ([26, Lemma 1.4]). Let V (x) ∈ Bq for q ≥ n 2 . Then, for the associated function ρ(x) there exist two positive constants k0 ≥ 1 and C0 > 0 such that 1 C0 ρ(x) ( 1 + |x− y| ρ(x) )−k0 ≤ ρ(y) ≤ C0 ρ(x) ( 1 + |x− y| ρ(x) ) k0 k0+1 . (3.1) In particular, if |x− y| ≤ Cρ(x), then ρ(x) ∼ ρ(y). Lemma 3.2 ([1]). Let x ∈ B(x0, r). Then for any k ∈ N there exists a positive constant C such that 1( 1 + 2kr ρ(x) )N ≤ C( 1 + 2kr ρ(x0) )N/(k0+1) , (3.2) where k0 is the constant as in Lemma 3.1. Lemma 3.3 ([3, Proposition 3]). Let g ∈ BMOθ(ρ). Then, for θ > 0 and 1 ≤ s < ∞ there exists a positive constant C such that( 1 |Br| ∫ Br |g(y)− gBr |sdy )1/s ≤ C [g]θ ( 1 + r ρ(x) )θ1 (3.3) for any Br = B(x, r) with x ∈ Rn, where θ1 = (k0 + 1)θ and k0 as in Lemma 3.1. Lemma 3.4 ([3, Lemma 1]). Let θ > 0 and 1 ≤ s < ∞. If g ∈ BMOθ(ρ), then there exists a positive constant C such that( 1 |2kBr| ∫ 2kBr |g(y)− gBr |sdy )1/s ≤ C [g]θ k ( 1 + 2kr ρ(x) )θ1 (3.4) for any Br = B(x, r) with x ∈ Rn and r > 0, where k ∈ N and θ1 = (k0 + 1)θ with k0 as in Lemma 3.1. EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 7 According to the definition of m(·, V ) in (2.3), we have the following decompo- sition lemma associated with m(·, V ) on Rn. Lemma 3.5 ([10, Lemma 2.3]). There exists a sequence of points {xk}∞k=1 in Rn such that the collection of balls Bk = B(xk, ρ(xk)), k ≥ 1 satisfies the following: (i) ∪kBk = Rn; (ii) for any k ∈ N, we conclude that card { j : 4Bj ⋂ 4Bk 6= φ } ≤ N with some positive integer N . We are now in a position to recall the following Hardy-Littlewood maximal functions and sharp maximal functions for g ∈ L1 loc(Rn). For given α > 0 we define Mρ,α(g)(x) = sup x∈B∈Bρ,α 1 |B| ∫ B |g(y)|dy, M ] ρ,α(g)(x) = sup x∈B∈Bρ,α 1 |B| ∫ B |g(y)− gB |dy, where Bρ,α = {B(y, r) : y ∈ Rn, r ≤ αρ(y)}. In what follows, we recall the relationship between Hardy-Littlewood functions Mρ,α and sharp coefficients M ] ρ,α, which is just the so-called famous Fefferman-Stein inequality. Lemma 3.6 ([3, Lemma 2]). Let {Bk}∞k=1 be the collection of balls as in Lemma 3.5, and g ∈ L1 loc(Rn). Then, for 1 < p < ∞ there exist positive constants C, β and γ such that∫ Rn |Mρ,β(g)(z)|pdz ≤ C ∫ Rn |M ] ρ,γ(g)(z)|pdz + C ∑ k |Bk| ( 1 |Bk| ∫ 2Bk |g(z)|dz )p . Let us now consider the Schrödinger operator L0 with constant coefficients. To this end, we let L0u(x) = −a0ijDiju(x) + V (x)u(x), where a0ij is an n× n symmetric constant matrix with uniformly elliptic condition (1.2). By a scaling argument this a0ijDij is actually Laplacian in the new coordinate system. Note that V (x) ∈ Bn/2 with V (x) ≤ cm2(x, V ). Then, for the operator L0 with constant coefficients we have the following regularity conclusion. Lemma 3.7 ([26, Remark 2.9]). If V (x) ∈ B∗n/2, then for 1 < p < ∞ there exists a positive constant C > 0 such that ‖∇2L−10 f‖Lp(Rn) ≤ C‖f‖Lp(Rn), (3.5) where C is independent of f . Let K(x, y) be the kernel function of the operator T0 = DijL −1 0 . Then, we have the following estimates for the kernel K(x, y). Lemma 3.8 ([25, Lemma 3.6]). If V (x) ∈ B∗n/2, then for every N ≥ 0 we have (i) there exists a constant CN such that |K(x, y)| ≤ CN ( 1 + |x−y| ρ(x) )−N |x− y|n . (3.6) 8 G. WANG, S. ZHENG EJDE-2023/67 (ii) there exists a constant CN such that |K(x, y)−K(x0, y)| ≤ CN |x− x0| ( 1 + |x0−y| ρ(x0) )−N |x− y|n+1 , (3.7) provided |x− x0| < 1 2 |x− y|. With Lemma 3.8 in hand, Pan and Tang concluded the following Lq-regularity of ∇2L−1. Lemma 3.9 ([25, Theorem 3.3]). Assume that u is the solutions of Lu = f(x) with V ∈ B∗n/2 and aij(x) ∈ BMOθ(ρ). Then there exist positive constants ε > 0 and C such that for all 1 < q <∞ it holds ‖∇2L−1f‖Lq(Rn) ≤ C‖f‖Lq(Rn) (3.8) provided that [aij ]θ < ε. 4. Boundedness on (vanishing) generalized Morrey spaces We devote this section to local bounded estimates for the Hessian of the operators L−1. First of all, let us show the following local Lp-estimate for the Hessian ∇2L−1. Theorem 4.1. For 1 < p < ∞, let u be the solutions of Lu = f(x) with aij(x) ∈ BMOθ(ρ) and V ∈ B∗n/2. If f ∈ Lploc(Rn), then there exists small constant ε > 0 such that ‖∇2L−1f‖Lp(B(x0,r)) ≤ Cr n/p ∫ ∞ 2r ‖f‖Lp(B(x0,t)) tn/p dt t (4.1) provided that [aij ]θ < ε, where C is independent of u, f . Proof. For any fixed x0 ∈ Rn, let Br = B(x0, r) and λBr = B(x0, λr) for any λ > 0. We now divide f(x) into two items as f(x) = f1(x) + f2(x), where f1(y) = f(y)χB(x0,2r)(y) with χB(x0,2r) being the characteristic function on B(x0, 2r). Then we obtain ‖∇2L−1f‖Lp(B(x0,r)) ≤ ‖∇ 2L−1f1‖Lp(B(x0,r)) + ‖∇2L−1f2‖Lp(B(x0,r)). (4.2) To estimate the first term we use the Lp-boundedness of ∇2L−1 in Lemma 3.9 and obtain ‖∇2L−1f1‖Lp(B(x0,r)) ≤ C‖f‖Lp(B(x0,2r)) ≤ Crn/p‖f‖Lp(B(x0,2r)) ∫ ∞ 2r dt t n p+1 ≤ Crn/p ∫ ∞ 2r ‖f‖Lp(B(x0,t)) tn/p dt t . (4.3) Next we estimate the second term on (4.2). To this end, for any x ∈ Br and y ∈ (2Br) c we see that 1 2 |x0− y| ≤ |x− y| ≤ 3 2 |x0− y|. Let us consider an operator L0 = −aijDi,j +V with a constant coefficient matrix aij = 1 |Br| ∫ Br aij(x)dx. Note that DijL −1f2 = Diju − DijL −1f1 leads to |DijL −1f2| ≤ |Diju| + |DijL −1f1|. With the estimate (4.3) for DijL −1f1 in hand, it suffice to only estimate the term Diju. To this end, we have Diju = DijL−10 L0u = DijL−10 L0(uχ2Br + uχ(2Br)c) EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 9 = DijL−10 (L0uχ2Br + L0uχ(2Br)c) = DijL−10 (L0uχ2Br + L0uχ(2Br)c − Luχ(2Br)c + Luχ(2Br)c) = DijL−10 (L0uχ2Br ) +DijL−10 [(L0u− Lu)χ(2Br)c ] +DijL−10 (Luχ(2Br)c). This leads to ‖Diju(x)‖Lp(B(x0,r)) ≤ (∫ Br |DijL−10 (L0uχ2Br )|pdx )1/p + (∫ Br |DijL−10 [(L0u− Lu)χ(2Br)c ]| pdx )1/p + (∫ Br |DijL−10 (Luχ(2Br)c)| pdx )1/p := I1 + I2 + I3. Let us first give an estimate of I1. By Lemma 3.7, Minkowski inequality, Hölder inequality, Lemma 3.4 we conclude that I1 = (∫ Br |DijL−10 (L0uχ2Br )|pdx )1/p ≤ C ( 1 |Br| ∫ 2Br |L0u(x)|pdx )1/p ≤ C ( 1 |2Br| ∫ 2Br |L0u(x)− Lu(x)|pdx )1/p + ( 1 |2Br| ∫ 2Br |Lu(x)|pdx )1/p ≤ C ( 1 |2Br| ∫ 2Br |aij(x)− aij |pv ′ dx ) 1 pv′ ( 1 |2Br| ∫ 2Br |Diju(x)|pvdx ) 1 pv + ( 1 |2Br| ∫ 2Br |Lu(x)|pdx )1/p ≤ C[aij ]θ‖Di,ju(x)‖Lp(B(x0,r)) + ‖f‖Lp(B(x0,2r)), where 1 v + 1 v′ = 1. We now estimate I2. We apply the boundedness of the kernel functions K(x, y) in Lemma 3.8 to show that I2 ≤ C |Br| (∫ Br |DijL−10 [(L0u− Lu)χ(2Br)c ]| pdx )1/p ≤ C |Br| (∫ Br (∫ (2Br)c |K(x, y)(L0u− Lu)|dy )p dx )1/p ≤ C |Br| (∫ Br ( ∞∑ k=1 ∫ 2kr<|y−x0|≤2k+1r ( 1 + |x−y| ρ(x) )−N |x− y|n |L0u− Lu|dy )p dx )1/p . By considering x ∈ Br and y ∈ (2Br) c we obtain that |x − y| ∼ |x0 − y|, which yields the following facts that I2 ≤ C 1 |Br| (∫ Br ( ∞∑ k=1 ∫ 2kr<|y−x0|≤2k+1r ( 1 + |x0−y| ρ(x0) )−N |x0 − y|n |L0u− Lu|dy )p dx )1/p ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N ∫ 2kr<|y−x0|≤2k+1r |L0u− Lu|dy 10 G. WANG, S. ZHENG EJDE-2023/67 ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N( 1 |2k+1Br| ∫ 2k+1Br |L0u− Lu|pdy )1/p . It follows from Hölder’s inequality and Lemma 3.4 that I2 ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N( 1 |2k+1Br| ∫ 2k+1Br |L0u− Lu|pdy )1/p ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N( 1 |2k+1Br| ∫ 2k+1Br |aij(x)− aij |pv ′ dx ) 1 pv′ × ( 1 |2k+1Br| ∫ 2k+1Br |Diju(y)|pvdy ) 1 pv ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N (k + 1)[aij ]θ ( 1 + 2k+1r ρ(x0) )θ1 × ( 1 |2k+1Br| ∫ 2k+1Br |Diju(y)|pvdy ) 1 pv ≤ C[aij ]θ‖Diju(x)‖Lp(B(x0,r)), where 1 v + 1 v′ = 1 and N > θ1. Finally, we show the estimate of I3. We use the boundedness of the kernel functions K(x, y) in Lemma 3.8 and |x− y| ∼ |x0 − y| to obtain I3 ≤ C |Br| (∫ Br |DijL−10 (Luχ(2Br)c)| pdx )1/p ≤ C |Br| (∫ Br (∫ (2Br)c |K(x, y)(Lu)|dy )p dx )1/p ≤ C |Br| (∫ Br ( ∞∑ k=1 ∫ 2kr<|y−x0|≤2k+1r ( 1 + |x−y| ρ(x) )−N |x− y|n |Lu(y)|dy )p dx )1/p ≤ C |Br| (∫ Br ( ∞∑ k=1 ∫ 2kr<|y−x0|≤2k+1r ( 1 + |x0−y| ρ(x0) )−N |x0 − y|n |Lu|dy )p dx )1/p ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N ∫ 2kr<|y−x0|≤2k+1r |Lu|dy ≤ C ∞∑ k=1 2−kn ( 1 + 2kr ρ(x0) )−N(∫ 2k+1B |Lu|pdy )1/p ≤ C‖f‖Lp(B(x0,2r)). Let us put the above estimates of I1, I2, I3 together and deduce that ‖Diju(x)‖Lp(B(x0,r)) ≤ C[aij ]θ‖Diju(x)‖Lp(B(x0,r)) + C‖f‖Lp(B(x0,2r)). By considering the small BMOθ(ρ) condition of aij with [aij ]θ < ε, we now take ε > 0 small enough that ε < 1 2Cn2 . Then we have ‖Diju(x)‖Lp(B(x0,r)) ≤ C‖f‖Lp(B(x0,2r)), which implies ‖DijL −1f2‖Lp(B(x0,r)) ≤ C‖f‖Lp(B(x0,2r)) EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 11 ≤ Crn/p‖f‖Lp(B(x0,2r)) ∫ ∞ 2r dt t n p+1 ≤ Crn/p ∫ ∞ 2r ‖f‖Lp(B(x0,t)) tn/p dt t , which yields ‖∇2L−1f2‖Lp(B(x0,r)) ≤ Cr n/p ∫ ∞ 2r ‖f‖Lp(B(x0,t)) tn/p dt t . (4.4) Let us combine (4.2),(4.3) and (4.4) to yield the desired inequality (4.1). � Proof of Theorem 2.8. By Lemma 3.6 we have 1 ess inft0 ( 1 + r ρ(x0) )α r−n/pϕ2(x0, r) −1‖[b, T ]f‖Lp(B(x0,r)) ≤ C sup x0∈Rn,r>0 ( 1 + r ρ(x0) )α r−n/pϕ2(x0, r) −1rn/p ∫ ∞ 2r ‖f‖Lp(B(x0,t)) tn/p dt t ≤ C‖f‖Mα,V p,ϕ1 . This completes the proof. � 12 G. WANG, S. ZHENG EJDE-2023/67 Proof of Theorem 2.9. By an argument similar to the one in Theorem 2.8, we con- clude the boundedness for ∇2L−1f in the vanishing generalized Morrey spaces. In fact, it suffices to only prove f ∈ VMα,V p,ϕ1 (Rn)⇒ ∇2L−1f ∈ VMα,V q,ϕ2 (Rn), (4.5) which yields lim r→0 sup x∈Rn ( 1 + r ρ(x) )α r−n/qϕ2(x, r)−1‖∇2L−1f‖Lq(B(x,r)) = 0. It suffices to only prove that for any ε > 0 we find sufficient small r > 0 such that sup x∈Rn ( 1 + r ρ(x) )α r−n/qϕ2(x, r)−1‖∇2L−1f‖Lq(B(x,r)) < ε. To this end, by the local Lq-boundedness of the operator ∇2L−1 as in Lemma (3.9) and taking δ0 > r with δ0 being specially determined later, then we obtain( 1 + r ρ(x) )α r−n/qϕ2(x, r)−1‖∇2L−1f‖Lq(B(x,r)) ≤ C0 ( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ ∞ 2r ‖f‖Lp(B(x0,t)) t n q dt t ≤ C0 ( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ ∞ r ‖f‖Lp(B(x0,t)) t n q dt t = C0 [( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ δ0 r ‖f‖Lp(B(x0,t)) t n q dt t + ( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ ∞ δ0 ‖f‖Lp(B(x0,t)) t n q dt t ] := C0 (A+B). (4.6) To estimate A, by condition (2.7), we have A := ( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ δ0 r ϕ1(x, t) 1 ϕ1(x, t) ‖f‖Lp(B(x0,t)) tn/p dt t ≤ C1ϕ2(x, r) ( 1 + r ρ(x) )α ϕ2(x, r)−1r− n p ϕ1(x, r)−1‖f‖Lp(B(x0,r)) = C1 ( 1 + r ρ(x) )α r− n p ϕ1(x, r)−1‖f‖Lp(B(x0,r)). Note that f ∈ VMα,V p,ϕ (Rn) for p > 1. We find a fixed δ0 > 0 such that if 0 < r < δ0 it holds sup x∈Rn ( 1 + r ρ(x) )α r−n/pϕ2(x, r)−1‖f‖Lp(B(x,r)) < ε 2C1 . It follows from (4.6) that sup x∈Rn C0A < ε 2 (4.7) for any 0 < r < δ0. For the estimate ofB, we also take r small enough. Then it follows from condition (2.7) that B := ( 1 + r ρ(x) )α ϕ2(x, r)−1 ∫ ∞ δ0 ϕ1(x, t) 1 ϕ1(x, t) ‖f‖Lp(B(x0,t)) tn/p dt t EJDE-2023/67 BOUNDEDNESS FOR SCHRÖDINGER OPERATOR WITH POTENTIAL 13 ≤ ϕ2(x, r)−1 ∫ ∞ δ0 ( 1 + t ρ(x) )α ϕ1(x, t) 1 ϕ1(x, t) ‖f‖Lp(B(x0,t)) tn/p dt t ≤ ϕ2(x, r)−1 ∫ ∞ δ0 ‖f‖VMα,V p,ϕ1 ϕ1(x, t) dt t1 ≤ cδ0ϕ2(x, r)−1‖f‖VMα,V p,ϕ1 ≤ cδ0ϕ2(x, r)−1 ( 1 + r ρ(x) )α ‖f‖VMα,V p,ϕ1 . 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[30] Tian, H; Zheng, S; Morrey regularity for nonlinear elliptic equations with partial BMO non- linearities under controlled growth, Nonlinear Anal., 180 (2019), 1–19. [31] Zhang, J.; Zheng, S.; Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients, Commun. Pure Appl. Anal., 16 (3) (2017), 899–914. [32] Zhang, J.; Zheng, S.; Optimal Morrey estimate for parabolic equations in divergence form via Green’s functions, Rocky Mountain J. Math., 48 (7) (2018), 2431–2457. Guiyun Wang Mathematics teaching and research group, Zhejiang Institute of Communications, Hangzhou 311112, China Email address: 154621582@qq.com Shenzhou Zheng (corresponding author) Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China Email address: shzhzheng@bjtu.edu.cn 1. Introduction 2. Notation and main results 3. Technical lemmas 4. Boundedness on (vanishing) generalized Morrey spaces Acknowledgements References