Regularity for the second order Riesz transforms associated with Schrödinger operators on weighted BMO type spaces

Nguyen Ngoc Trong1,∗, Nguyen Anh Dao2,∗∗, and L. X. Truong3,∗∗∗ 1Faculty of Mathematics and Computer Science, VUNHCM University of Science, Ho Chi Minh city, Vietnam. Department of Primary Education, HoChiMinh City University of Education, Vietnam 2Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam 3Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam


Introduction and preliminaries
The study of both function spaces and the boundedness of singular integral operators associated with Schrödinger operators arose from practical applications in some mathematical fields, such as harmonic analysis and partial differential equations, and has become an active area of research in the last few years.These type of questions have been already considered by many authors and they are very important questions which appeal to various techniques from partial differential equations and harmonic analysis.
In recent years, many authors have been interested in the problems of harmonic analysis associated with Schrödinger operators on R n , see [1-3, 5, 6, 12, 14-17, 19, 21, 25].More specifically, boundedness of singular integral versions on stratified Lie group was also attracted the attention, see [12,[14][15][16].Very recently, D. Yang developed the theory of localized non-weighted BMO spaces associated to admissible functions in spaces of homogeneous type and eastablished the resuts for the boundedness of the singular integrals on these space, see [22][23][24].for all ball B ⊂ R n .
It is well-known that if V ∈ RH q for some q > 1, then there exists > 0, such that V ∈ RH q+ .We introduce the definition of the reverse Hölder index of V as q 0 := sup{q : V ∈ RH q }.Under assumption V ∈ RH m we may conclude q 0 > m.
V(y)dy, hold for every x ∈ R n and 0 < r < ∞.For every 1 < p < ∞, it is easy to see that RH ∞ ⊂ RH p .
For a weight ω, we shall mean ω is a nonnegative measurable and locally integrable function on R n .For p ∈ (1, ∞), we say that ω belongs to the Muckenhoupt class A p if the following holds: there exists C > 0 such that for all balls B in R n .
For p = 1, we say that ω ∈ A 1 if there is a positive constant C such that for every ball B ⊂ R n , 1 |B| B ω(y)dy ≤ Cω(x) for a.e.x ∈ B.
We set A ∞ = ∪ p≥1 A p .For the ball B = B(x, r) and λ > 0, we denote by λB the ball B(x, λr).
For α ≥ 1, we say that the weight ω satisfies the doubling property with the doubling order α if there exists a constant C such that ω(λB) ≤ Cλ nα ω(B), for all balls B ⊂ R n and all λ > 1, where ω(E) = E ω(x)dx for any measurable subset E of R n .
We then denote by D α the set of all weights ω satisfying the doubling property with the doubling order α.It is important to note that if ω ∈ A p for p ∈ [1, ∞) then ω ∈ D p .Moreover, if V ∈ RH q , then V belongs to a certain A p class for some p ∈ [1, ∞) and hence V ∈ D p .See for example [20].
Associated to the potential V, the function of the critical radius is defined by for all x ∈ R n , see [16].
Let ρ 1 (x) be the function of the critical radius associated to |∇V|.The auxillary functions ρ(x), ρ 1 (x) play an important role in the problem of harmonic analysis related to Schrödinger operators.
Definition 1.1.Let 0 ≤ β < 1 and w be a weight.The space BMO β L (ω) is defined as the set of functions f for all the balls B ⊂ R n , with r < ρ(x), where f B = 1 for all balls B = B(x, r), with r ≥ ρ(x), where C is a positive constant.
A norm in the space BMO β L (ω) can be given by the maximum of the two infima of the constants in (1.2) and (1.3) respectively.This norm will be denoted by For brevity, if ω ≡ 1, then we will denote BMO β L , BMO β by BMO β L (ω), BMO β (ω) respectively .When β = 0, we write BMO L , BMO instead of BMO β L , BMO β respectively.
Inspired by the recent studies on the boundedness of the Riesz transforms in R n , see for example [1-4, 7, 8] and the references therein.In the case ω ≡ 1, it is well-known that if V ∈ RH q then the integral operators [19], where s, s depend only on q, n.The ranges of p in these results are optimal.This seminal results of Z. Shen play the crucial role in the boundedness of the Riesz transform on the other types functional space.When ω ≡ 1, and β = 0, J. Dong and Y. Liu [8] In this paper, we improve the results of J Dong and Y. Liu in [8] by studying the boundedness of the second order Riesz transform R * = L −1 ∇ 2 on the spaces BMO β L (ω), BMO β (ω).The next theorems are the main results of this paper.
The organization of this paper is as follows.In section 2, we give some preliminary results.Section 3 is devoted to the estimates on the size and smoothness of the kernels.Finally, the proof of the main theorems are also investigated in Section 4. Throughout this article, all the positive constants are signified as C although they may be different on the same line.Note that, ∇ means that we are taking all the partial derivaties with respect to the first variable.We write A ∼ B and A B if there exists some positive constants C, C such that C ≤ A B ≤ C and A ≤ C B, respectively.

Some preliminary results
Let V ∈ RH q where q > n/2.We begin with the results concerning the estimates on the potential V which are taken from [4, Lemma 1]  V(y)dy, for all balls B(x, r) in R n .
Lemma 2.5.There exist C > 0 and l 1 > 0 such that ITM Web of Conferences 20, 02005 (2018) https://doi.org/10.1051/itmconf/20182002005ICM 2018 See [19] for the proofs of Lemmas 2.1 -2.5.Lemma 2.6.Let V ∈ RH q with q > n/2 and > n q .Then, for any constant C 1 , there exists a constant C 2 such that The proof of this above lemma is similar to that of Lemma 1 in [4].The following results give other characterization of the space BMO β L (ω), the proof of this proposition is very similar with a small modification to that of [4, Proposition 2] and [18, Theorem 2.2], respectively. ) It follows from [18, Theorem 2.2], we have the following lemma.

and, moreover, this quantity gives an equivalent norm.
In what follows we denote I 1 = (−∆) −1/2 the classical fractional integral of order 1.To prove the main theorems, we need the following technical lemma.Lemma 2.8.Let V ∈ RH q with n/2 < q < n and ω ∈ RH s ∩ A p/s for some s > p , where , and any ball B(x, r), where for η and µ being the exponent of the doubling property satisfied by ω and V respectively.Proof.We first apply Hölder's inequality to estimate the right-hand side of (2.6) by To bound the first factor we apply again Hölder's inequality with exponent σ such that σp By Lemma 2.7, we come up to We then obtain that This implies that By combining the above estimates, we obtain (2.7) On the other hand, due to the boundedness of I 1 and V ∈ RH q , we have V(y)dy.
In the case 2 k r ≥ ρ(x 0 ), we apply the second part of Lemma 2.6 to get Combining the above estimates we arrive to (2.6).The case 2 k r < ρ(x), is handled similarly by using the first part of Lemma 2.6.
We have the following simple result.Lemma 2.9.For α > 0, we have B(x 0 ,r) Proof.Proof of this lemma is straightforward and we omit details here.

Some estimates for the kernels
Let Γ(g, h), Γ 0 (g, h) denote the fundamental solution for the operator −∆ + V, −∆ respectively.The following estimates of the fundamental solution for the Schrödinger operator on the nilpotent Lie group have been proved in [16].Lemma 3.1.Let l > 0 be an integer.
1. Suppose V ∈ RH n/2 .Then there exists C l > 0 such that for x y, 2. Suppose V ∈ RH n .Then there exists C l > 0 such that for x y, where ∇, ∇ 2 means that we are taking all the partial derivaties with respect to the first and second variable, respectively.
Next, we recall the imbedding theorem of Morrey.The following results give the estimate and the connection between the fundamental solutions of the Schrödinger operator L and the Laplacian −∆.The proofs of these lemmas are similar to that in [12,13].
Lemma 3.5.Suppose that V ∈ RH q for some q > n and |∇V| ∈ RH q 1 for some q 1 > n 2 .There exists a constant C l > 0 for each l > 0, where δ = 1 − n/t for t > n.
We will prove the following lemma.
Lemma 3.8.Suppose V ∈ RH q for q > n.Then, for any integer k > 0, This completes the our proof.
We denote by K * and K * the associated kernels of R * = ∇ 2 (−∆) and R * , respectively.The following result will play an important role in the sequel.Lemma 3.9.Let V ∈ RH q with q > n such that |∇V| ∈ RH q 1 with q 1 > n/2 and 0 < δ < min{1 − n q , 2 − n q 1 }.Suppose that ρ ρ 1 1.Then, there exists a positive constant C such that Moreover, using the fact that ∆(∇Γ 0 ) = 0, we have To show this, we split R n into 4 regions: We have We only estimate J 1 here.The same proof works for J 2 .
Finally, noticing |x − u| ∼ |u − z| if u ∈ E 4 , we ensures that Then E 4 = F 1 ∪ F 2 .Hence, we can get estimate where we have used the definition of ρ(x) in the last inequality.Taking into account of n + δ ≥ 2 − n/q, for sufficiently large k leads us to

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where we have used that V belongs to D µ for some µ ≥ 1 in the third inequality.
Therefore, we imply that For J 1 , we majorize the difference related to Γ by the sum of the absolute values of each term and estimate each integral separately.Since both are similar we work out one of them.First we notice that |x − z| > 2|x − y| implies |z − u| > 1 4 |x − z| for u ∈ E 1 .Then, using Lemma 3.1, we deduce that where in the last inequality we have used Lemma 2.6 with = 2 − δ and r = |x − y| < 2ρ(x).
For the remaining regions we will use Lemma 3.6.To estimate J 2 we use Lemma 3.6 to get where in the last inequality we have used Lemma 2.6 with = 2 − δ and r = 1 2 |x − z|.
To deal with J 3 we notice E 3 ⊂ B(z, 3|x − z|).Using again Lemma 3.6 we arrive to We set , where E 1 4 = {u : 2|x − z| ≤ |x − u| ≤ ρ 1 (x)}.Applying Hölder's inequality the above integral over E 1  4 is bounded by where in the last inequality we have used the reverse Hölder condition on |∇V| and the definition of ρ 1 .
To estimate the integral on E 2 4 , we have

|∇V|.
Since |∇V| satisfies a doubling condition and we can choose k large enough, we have where we have use that |∇V| belongs to D µ for some µ ≥ 1 and n − 1 + δ ≥ 2 − n/q 1 .Now using the estimates in E 1 4 and E 2 4 reminding that |x − z| ≤ ρ 1 (x), we obtain The proof is completed.

Proofs of the main results
First, we prove the following theorem.
By the boundedness of R * and the John-Nirenberg's inequality, we have On the other hand, observe that Applying Proposition 1.7 in [9], we come up to Note that |x − z| ≥ 2|x − y| for x, y ∈ B and z ∈ (5B) c .Therefore, we have ITM Web of Conferences 20, 02005 (2018) https://doi.org/10.1051/itmconf/20182002005ICM 2018 The last series converges provided that α < 1 + 1 − β n .Combining these, we obtain the desired result.
First, we prove (4.10).Let B = B(x 0 , ρ(x 0 )), set g = f − f B .We write g = g 1 + g 2 , with For g 2 we estimate the size of K * using Lemma 3.3.Now, note that for x ∈ B and We have Splitting the integral into dyadic annuli and using the doubling property, I 1 is bounded by and the last sum is finite choosing k big enough.Now, using that for x ∈ B and z ∈ R n \ 2B, we have that We may use Lemma 2.8 and ω ∈ D α , |∇V| ∈ D µ to obtain the bound Choosing k large enough to make the series convergent and note ρ ρ 1 1 we arrive to the desired estimate.Now we take care of the oscillation of R * on a ball B = B(x 0 , r) with r < ρ(x 0 ).Then, we deduce that
Then, we get that I ≤ I 1 + I 2 + I 3 , where Now we proceed to estimate I 2 .We apply Lemma 3.9 with β < δ < min{1 − n/q, 2 − n/q 1 } For every x, y ∈ B and for every z ∈ (5B) c , noting that |x−z| ≥ 2|x−y|, ρ(x 0 ) ∼ ρ(x), |x−z| ∼ |x 0 − z| and |x 0 − z| < 5ρ(x 0 ) 1, Let j 0 be the integer part of log 2 (5ρ(x 0 )/r).Then, we have ITM Web of Conferences 20, 02005 (2018) https://doi.org/10.1051/itmconf/20182002005ICM 2018 Using the inequality 1 + log 2 (t) ≤ Ct ε/2 for t > 1/8, with ε small enough, we get We only have to take care of I 2,2 by I 2,3 is the same as I 2,1 .Now, using that for x ∈ B and z Breaking the integral in z dyadically and setting j 0 such that 2 We deduce from Lemma 2.8 that Using the inequality 1 + log 2 (t) ≤ Ct ε/2 for t > 1/8, with ε small enough, we obtain I 2,2 f BMO β (ω) ω(B)|B| β/n .Now we take care of I 3 .We use the smoothness of each kernel separately.For K * we use Calderón-Zygmund condition and for K * we use Lemma 3.5.We have Using the inequality j ≤ 2 j ln2 with small enough, we arrive at Choosing k large enough to make the series convergent we get and the last factor is bounded since r < ρ 1 (x 0 ).
where we have used r < ρ(x 0 ) ρ 1 (x 0 ) 1 in the last inequality.The proof is completed.
with the supremum taken over all balls B and f B = 1 |B| B f (y)dy.