Variation inequalities related to Schrödinger operators on weighted Morrey spaces

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.


Introduction
Given a family of bounded operators T = {Tϵ} ϵ> acting between spaces of functions, one of the most signi cative problems in harmonic analysis is the existence of limits lim ϵ→ + Tϵ f and limϵ→∞Tϵ f , when f belongs to a certain space of functions. A question that arises naturally is what is the speed of convergence of the above limits. A classical way to measure the speed of convergence of {Tϵ} ϵ> is to study "square function" of the type ( ∞ i= |Tϵ i f − Tϵ i+ f | ) / , where ϵ i . Recently, other expressions have been considered, among which is the q -variation operator de ned by where the supremum is taken over all sequence {ϵ i } decreasing to zero. We denote Fq the space that includes all the functions φ : ( , ∞) → R, such that Then · Fq is a seminorm on Fq. It can be written that The variation for martingales and some families of operators have been studied in many recent papers on probability, ergodic theory, and harmonic analysis. We refer the readers to [1][2][3][4][5][6] and the references therein for more background information.
Recently, Betancor et al. [7] studied the bounded behaviors of variation operators for some Schrödinger type operators in Lebesgue spaces. Precisely, let n ≥ and L = −∆ + V be the Schrödinger operator de ned on R n associated with a xed non-negative potential V ∈ RHs (the reverse Hölder class) for s ≥ n/ , that is, there exists C > such that (1.4) for every ball B in R n . For = , · · · n, consider the -th Riesz transform in the L-context, which can be de ned by Here, for every x, y ∈ R n , x ≠ y, where Γ(x, y, τ) represents the fundamental solution for the operator L + iτ (see [7,8]). Betancor et al. [7] proved that when n/ ≤ s < n, for = , , · · · , n, the variation operators Vq(R L ,ε ) (resp., Vq(R L,* ,ε )) associated with the family of truncations {R L l,ε } ε> (resp., {R L,* ,ε } ε> ) are bounded from L p (R n ) into itself for < p < p (resp., p < p < ∞) with /p = /s− /n, and Vq(R L ,ε ) are of weak type ( , ); moreover, when s ≥ n, both Vq(R L ,ε ) and Vq(R L,* ,ε ) are bounded from L p (R n ) into itself for < p < ∞ and of weak type ( , ). More recently, Tang and Zhang [9] extend the results above to the weighted L p space. They established the weighted L p boundedness for Vq(R L ,ε ) and Vq(R L,* ,ε ) with the weight A γ,θ p class (see section 2 for the de nation), which includes the Mukenhoupt weight class. By di erent method, Zhang and Wu in [10] also obtained L p boundedness for Vq(R L ,ε ) and Vq(R L,* ,ε ) and the weighted weak type (1,1) estimation Vq(R L ,ε ) with the weight A γ,θ p class . In addition, for every V ∈ RH n/ , Shen [11] introduced the function γ, which is called as the critical radius and de ned as and plays key roles in the theory of harmonic analysis operators associated to L. In [12], Bongioann et al de ned the space BMO θ (γ), θ ≥ , as follows.
On the other hand, in order to extend the boundedness of Schrödinger type operators in Lebesgue spaces, Pan and Tang [13] introduced the following weighted Morrey spaces related to the non-negative potential V, denoted by L p,λ α,V ,ω (R n ).

De nition 1.2. Let k
where B = B(x , r) denote a ball with centered at x and radius r, γ(x ) is the critical radius at x de ned as in (1.5) and the weight function ω ∈ A γ,∞ p .
Based on the above, it is a natural and interesting question whether we can establish the L p,λ α,V ,ω (R n )boundedness of the variation operators aforementioned in Schrödinger setting. The main purpose of this paper is to answer this question. Our results can be formulated as follows: Theorem 1.3. Let = , · · · n, q > and V ∈ RHs. Assume α ∈ (−∞, +∞) and λ ∈ ( , ). Then (i) If n/ ≤ s < n and p is such that /p = /s − /n, then the variation operator Theorem 1.4. Let = , · · · , n. Assume that q > and V ∈ RHs. (i) If n/ ≤ s < n then for p = , η > and , holds for all balls B, where C is independent of x, r, η and f . Theorem 1.5. Let = , · · · , n, q > , b ∈ BMO γ θ with θ > and V ∈ RHs, Assume that α ∈ (−∞, +∞) and λ ∈ ( , ). (i) If n/ ≤ s < n and p is such that /p = /s − /n, then the variation operator Remark 1.6. In [8], it was proved that if V is a nonnegative polynomial, then V ∈ RHs for any < s < ∞. Therefore, as special cases of our results, the corresponding ones to the Hermite operator: H = −∆ + |x| hold. This can be regarded as the generalization of the corresponding results in [2,30]. The rest of this paper is organized as follows. In Section 2, we will recall some properties of the function γ and some basic facts concerning weights A γ p , which will play a crucial roles in our arguments. In Section 3, we will prove Theorem 1.3 and 1.4, the proof of Theorem 1.5 will be given in Section 4. Throughout this paper, the letter C always denotes a positive constant that is independent of main parameters involved but whose value may di er from line to line. We use For any t ∈ ( , ∞), we denote the ball B(x, tr) by tB. Given any p ∈ [ , ∞], p = p p− denote its conjugate index. In particular, it should be pointed out that these weighted Morrey spaces in De nition 1.2 are equivalent for di erent k > .

Preliminaries
In this section, we recall some known results, which will be used in our next proofs. We rst recall some properties of the auxiliary function γ(x), which will be used below. [8]) If V ∈ RH n/ , then there exist c and l ≥ such that for all x, y ∈ R n , According to [31], we recall a new class of weights A γ,∞ and A γ,θ is the set of those weights ω such that for every ball B = B(x, r). Clearly, the classes A γ,θ p are increasing with θ and for θ = , they are just the Munkenhoupt classes Ap. From [32], we know that the following properties for A γ,θ p hold.

Proof of Theorems 1.3 and 1.4
This section is devoted to the proof of Theorem 1.3 and 1.4. We rst recall several auxiliary results. Let R and R * be the kernel function of R L and R L,* , respectively. The following estimates for the kernel functions were established in [8] and will be very useful in the sequel.
Moreover, the last inequality also holds with γ(x) replaced by γ(z).
(ii) When s > n, the term involving V can be dropped from inequalities (3.1).
Proof of Theorem 1.3. To prove (i), without loss of generality, we may assume α < and ω ∈ A γ,∞ p/p . Pick any ball B := B(x , r) and write f (x) = f (x) + ∞ i= f i (x), where f = fχ B , f i = fχ i+ B\ i B for i ≥ . By the weighted L p -boundedness of Vq(R L ,ε )(f ) (see [9]). Hence, we have In the term last but one, we used that χ ε i+ <|x−y|<ε i Fq ≤ . Now it follows from Lemma 3.1 that .

(3.4)
Now, we will estimate the term A . Using (2.1) and Hölder's inequality, we can write If we choose ϱ such that ϱ + p + p = . Then by Hölder's inequality, we have Using the boundedness of the 1-th Euclidean fractional integral I : L s → L p with /p = /s − /n, we obtain that Recall that V ∈ Bs for some s > implies that V satis es the doubling condition,i.e., there exist constants µ ≥ and C such that, holds for every ball B and t > . Therefore Since ω ∈ A γ,θ p/p and p/ϱ = p/p − , we get Go back to (3.10), we have This together with (3.4) by choosing N which is big enough, we get For part (ii), we note that the adjoint R L ,ε of R L,* ,ε , when V ∈ Bs, with n/ ≤ s < n. We write By proceeding as the previous proofs (i), we can prove that We omit the details. The proof of part (iii) can be given analogously as in (i) and (ii), we leave the part to the part to the interested readers and complete the proof of therorem 1.3.
Proof of Theorem 1.4. As for the case p = , by replacing (3.2) with the corresponding weak estimate. By the Vq(R L l,ε )(f ) is bounded of weighted weak type (1,1) (see [10]), we have (3.11) According (3.3) and (3.1), we get (3.12) Note that for ω p ∈ A γ,∞ , we have ω p ∈ A γ,θ for some θ ≥ . In this case it is true that ω ∈ A γ,θ with θ = θ /p . Therefore, we obtain By Hölder's inequality and ω p ∈ A γ,θ , we have B I (Vχ i+ B )(y)ω(y)dy |f (z)|ω(z)dz where we choose N which is big enough and we have These together with (3.11) imply that which complete the proof of Theorem 1.4.

Proof of Theorem 1.5
In what follows, we will prove Theorem 1.5. The following property of BMO∞(γ) functions.
Proof of Theorem 1.5. We will only prove the results of Vq(R L,* b, ,ε )(f ) in part (i). Without loss of generality, we may assume that α < . Pick any x ∈ R n and r > , and write Applying Lemma 3.1, we can write Similarly, we can get we choose ϱ satis es ϱ + p + p = . According to Hölder's inequality, ω ∈ A p/p and (4.1), we have Now, Imitating the estimation of B , but we use Hölder's inequality with ν and t that p + p + ν + t = .