Discrete Carleman estimates and three balls inequalities

We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schr\"odinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.


Introduction
In this article, we provide robust quantitative unique continuation results for discrete magnetic Schrödinger operators P h of the form where f : (hZ) d → R, D h ±,j f (n) := ±(f (n ± he j ) − f (n)) denotes the (unscaled) left/right difference operator on scale h, B k : (hZ) d → R d is a (uniformly in h) bounded tensor field, modelling, for instance, magnetic interactions and where the potential V : (hZ) d → R is assumed to be uniformly bounded (independently of h). The operator ∆ d := The operators considered in (1) correspond to discrete versions of the continuous magnetic Schrödinger operator. While many features of the continuous and the discrete operators are shared, if correspondingly adapted (e.g. regularity estimates), there are striking differences in the validity of the unique continuation property in these settings. In fact, even for the case of the model operator, the discrete Laplacian, it is well-known that while in the continuum the (weak) unique continuation property holds as a direct consequence of the analyticity of the solutions, this fails in general in the discrete setting [GM14]. Indeed, in [GM14] the authors show that it is possible to construct non-trivial harmonic polynomials vanishing on a large, prescribed square. In spite of these differences, it is expected that as the lattice spacing decreases, h → 0, the properties of continuous harmonic functions are recovered. That this is in fact the case for the setting of the discrete Laplacian was proved in [GM14,GM13,LM15], where propagation of smallness estimates with correction terms were proved for the discrete Laplacian. For similar phenomena for related operators we refer to [FBV17,JLMP18] and the references therein.
Most of the cited propagation of smallness results from the literature however strongly relied on the specific properties of the constant coefficient Laplacian, e.g. by using methods from complex analysis. It is the purpose of this article to provide quantitative unique continuation estimates and three spheres inequalities for a large class of Schrödinger operators by means of robust Carleman estimates. We emphasize that in addition to the intrinsic interest in the quantitative unique continuation properties of discrete elliptic equations, important applications of these quantitative unique continuation estimates involve inverse and control theoretic problems (see for instance [BHR10,EDG11]).
Here for r > 0 we define B r = B r (0) ∩ (hZ) d , with h ∈ (0, h 0 ) denoting the lattice spacing, and all L 2 norms are L 2 norms on the lattice (hZ) d .
Due to the restriction on the upper bound of τ ≤ δ 0 h − 1 2 , this logarithmic convexity estimate does not immediately yield a three balls inequality as in the continuum. It however implies a three balls estimate with a corresponding correction term: Theorem 2. There exist α ∈ (0, 1), c 0 > 0 h 0 > 0 and C > 1 such that for h ∈ (0, h 0 ) and u : (hZ) d → R with P h u = 0 in B 4 we have This estimate thus quantitatively connects the discrete situation in which the unique continuation property fails to its continuous counterpart. It provides quantitative evidence of the fact that as h → 0, the propagation of smallness properties of the associated elliptic operator is recovered. We remark that the scaling behaviour of the form e −c0h − 1 2 in h ∈ (0, h 0 ) is known to be optimal as the dimension tends to infinity (see [LM15, Theorem 1.13]).
We remark that our results (and arguments) remain valid if instead of the differential equation (1) we consider the differential inequality Further, it is possible to deduce propagation of smallness estimates for some controlled hdependent growth of V and B j (see Remark 4.2) which however, of course, do not pass to the limit as h → 0.

Main ideas.
Similarly as in [BHR10,EDG11] and contrary to the results in [GM14,LM15], both of our results rely on a robust L 2 Carleman estimate. More precisely, as our key auxiliary result we prove the following Carleman estimate with a weight which is a slightly convexified version of the limiting Carleman weight ψ(x) = −τ log(|x|) and which we choose as, for example, in [KRS16]: for a certain constant c ps > 0. Then, there exist h 0 > 0, C > 1 and τ 0 > 1 (which are independent of u) such that for all h ∈ (0, h 0 ) and τ ∈ (τ 0 , Here , where e j is the unit vector in the j-th direction, denotes the symmetric discrete difference operator. Remark 1.1. We remark that the choice of the symmetric discrete derivative D s in (4) does not play a substantial role. With only minor changes it is also possible to replace it by D h + or D h − . We refer to the beginning of Section 2 for the precise definitions.
While building on similar ideas as in its continuous counterpart (see for instance [KT01,AKS62]), our Carleman estimate is restricted to a certain range of values of τ which is a consequence of the discreteness of the problem. Similar restrictions had been observed in [BHR10,EDG11] in the context of Carleman estimates for control theoretic and inverse problems. In deriving this estimate, we localize to suitable scales on which we freeze coefficients and compare our discrete problem to the continuum setting.
1.3. Outline of the article. The remainder of the article is organized as follows: In Section 2 we compute the conjugated discrete operator and its expansion into its symmetric, antisymmetric parts and their commutator. In the main part, in Section 3, we derive the main Carleman estimate of Theorem 3. Building on this, in Section 4 we deduce the results of Theorems 1 and 2. Last but not least, in Section 5, we comment on rescaled versions of the main estimates.
1.4. Remarks on the notational conventions. Concerning notation, with the letters c, C, . . . we denote structural constants that depend only on the dimension and on parameters that are not relevant. Their values might vary from one occurrence to another, and in most of the cases we will not track the explicit dependence. For the Fourier transform of a function f we will both use the notation F f andf .

The Conjugated Laplacian and the Commutator
From now on, D j ± will stand for the forward/backward operators D h ±,j from Section 1 and D j s will denote the symmetric discrete derivatives in the j-th direction. All operators are understood to be taken with step size h. Moreover, D h ± := d j=1 D j ± and D s := d j=1 D j s . We remark that the symmetric difference operator is associated with the Fourier multiplier 2 d j=1 sin(hξ j ).
Heading towards the proof of the Carleman inequality of Theorem 3, we introduce the conjugated Laplacian where the symmetric and anti-symmetric operators are We compute the commutator of this to be Now, using trigonometric identities, these can be simplified to read Indeed, for instance, for A j,k we obtain The arguments for the other contributions are similar. We next seek to investigate the commutator in more detail.
Remark 2.1. In the one-dimensional situation the commutator can be simplified significantly: Indeed, if we study the commutator [S, A]f, f , the case j = k is quite simple and leads to The main term is a discrete version of 4φ jj |f j | 2 + 4φ jj φ j φ j |f | 2 − φ jjjj |f | 2 . Note that the main term of the higher dimensional continuous commutator is more complicated and is of the form In the general case, we can rewrite the contributions of [S, A]f, f in the following way (where with slight abuse of notation, we refrain from spelling out the sums in Z d and the sum in j, k): The interest of writing the general term in this form is that we seek to bring the commutator into a form which is as close as possible to the form of the commutator in the continuous setting which reads 4∇φ · ∇ 2 φ∇φf 2 + 4∇f · ∇ 2 φ∇f − ∆ 2 φf 2 . To this end, we note that the first four terms in (6) are closely related to the part 4φ jk f j f k −φ jjkk |f | 2 and the last four terms to 4φ jk φ j φ k |f | 2 correspondingly.
We will use the expression (6) as the starting point of our commutator estimates in the following sections.

Proof of the Carleman Estimate from Theorem 3
Before turning to the proof of Theorem 3 let us recall an auxiliary result showing the strong pseudoconvexity (in the continuous sense) of the weight function φ(x): Lemma 3.1. Let φ(x) := τ ϕ(|x|), where for some constant c ps > 0 (7) ϕ(t) = − log t + c ps log t arctan(log t) − 1 2 log(1 + log 2 t) .
In the sequel, we present several auxiliary results which allow us to steadily transform the discrete conjugated operator into an operator that closely resembles the continuum version of the conjugated Laplacian. Recall that we define the discrete Laplacian in direction j ∈ {1, . . . , d} as As a first step towards the desired Carleman estimate, we localize the problem to scales of order ǫ −1 0 τ − 1 2 , where ǫ 0 > 0 is a small constant which will be chosen below (see the proof of Theorem 3): Here, S φ and A φ are the operators from (5) and the operator sin( D j h) is defined as a Fourier We remark that here and in the sequel, for brevity of notation, we abbreviate the L 2 norm on the lattice without adding subindeces, i.e. f := f L 2 ((hZ)) d .
Proof of Lemma 3.2. As the estimates for S φ and for A φ are analogous, we mainly focus on the argument for S φ . The first bound in the estimate for S φ in (8) is a direct consequence of Minkowski's inequality. In order to observe the second estimate for S φ in (8), we spell out the contributions S φ f k (n). We begin by rewriting While we seek to keep the first contribution in this expansion to recombine it to h −2 ∆ d,h,j f after summing over the partition of unity, we only provide estimates on the remaining contributions. To this end, we note that Using the same reasoning for the term h −2 (f (n + e j h) − f (n − e j h))(ψ k (n − e j h) − ψ k (n)), and combining these estimates, we thus infer that where we used that for y ∈ B 2 \ B 1 2 we have that As a consequence, combining the estimates from (9) and (10) yields This concludes the argument for the localization estimate for S φ .
The arguments for A φ and L φ are analogous. Indeed, for A φ we note that Estimating the terms of L φ by using the bounds for A φ and S φ then implies the result.
As a next auxiliary step, we expand the trigonometric identities which then allows for easier manipulations of the contributions in the sequel.
Then, for S φ and A φ as in (5), Here, as above, the operator sin( D j h) is defined as a Fourier multiplier, i.e. F (sin( Similarly as above, we here drop the subscript in the L 2 scalar product and simply write (·, ·) := (·, ·) L 2 ((hZ) d ) .
Proof of Lemma 3.3. The results follow by expanding the expressions for S j , A j . More precisely, we first approximate all discrete derivatives of φ and the corresponding nonlinear functions and then estimate the resulting errors.
We first discuss the symmetric part of the operator. For instance, we expand Here y ∈ R d is an intermediate value such that y ∈ [n, n + he j ]. Thus, the symmetric part becomes where S j f (n) is as in our statement and E Sj f ≤ C(hτ 2 + τ 4 h 2 ) (|∇ϕ| 2 + |∇ϕ| 4 + |∇ 2 ϕ| 2 )f , with n ∈ (hZ) d , φ(n) = τ ϕ(n) with ϕ a bounded function (on the relevant domain). Choosing τ ∈ (0, δ 0 h − 1 2 ) with δ 0 sufficiently small, we may assume that τ 2 h + τ 4 h 2 ≤ C (or even τ 2 h ≪ 1), hence the error E Sj f in the symmetric part is of zeroth order in τ and an L 2 contribution in f , i.e. E S φ f ≤ C f . Therefore, in the sequel, we will estimate For the antisymmetric part we argue analogously. We thus expand where y is an intermediate value in [n, n + he j ]. Thus, the antisymmetric part becomes Finally, we turn to the commutator which is given by For the first four contributions in (15), we expand Thus, the first four contributions in (15) can be written as Noting that yields the first part in the expression which is claimed for f (n)C jk f (n) in the lemma.
For the second four terms in (15), we similarly expand as in (16) and (cosh(φ(n + he j + he k ) − φ(n)) − 1) Hence the second set of four terms from (15) can be rewritten as Combining the errors in the two expansions of the commutator exactly yields the claimed estimate.
As a next step, we freeze coefficients in the operators S φ , A φ and C jk when acting on functions supported in sets of the size ǫ −1 0 τ − 1 2 for ǫ 0 > 0 sufficiently small and τ > 0 sufficiently large, both of which are to be determined below (see the proof of Theorem 3).
Assume that φ is as in Theorem 3 and that 1 < τ ≤ δ 0 h − 1 2 for a sufficiently small constant δ 0 > 0. Letn ∈ R d be a point which is in the interior of supp(f ) and set Then, Proof. Using the triangle inequality and the support condition, we estimate As the arguments for A φ and for C jk are analogous, we do not discuss the details.
Finally, as a last auxiliary step before combining all the above ingredients into the proof of Theorem 3, we prove a lower bound for the operators with the frozen variables.
Proposition 3.5. LetS φ ,Ā φ andC jk be as in Lemma 3.4. Then there exist C low > 0, c 0 > 0, h 0 ∈ (0, 1) (small) and τ 0 > 1 such that for all τ ∈ (τ 0 , Proof. Using that the operators under consideration all have constant coefficients, we may perform a Fourier transform and infer that (20) In order to prove the positivity of this expression, we will choose c 0 > 0 so small, that outside of a sufficiently small neighbourhood of the union of the (joint) characteristic sets of the Fourier symbols p r,j (ξ) := −4h −2 sin 2 (hξ j /2) + (∂ j φ(n)) 2 cos(ξ j h) the third term in (20) is controlled by these. In order to observe that this is possible, we first study the contributions p r,j and p i,j separately. We first consider the terms p r,j and p r (ξ) := d j=1 p r,j (ξ) associated with the symmetric operator. We begin by observing that the first summand in (21) p r (ξ) = j |∂ j φ(n)| 2 cos(ξ j h) + 2 j cos(ξ j h) − 1 h 2 is bounded from above by Cτ 2 . For the second summand, we deduce that, since | cos(x)| ∈ (0, 1) and for ξ j ∈ h −1 (−π, π), we have where R(ξ j h) is the remainder term in the Taylor approximation. Hence, Combining these two observations, we note that there exists a constant C 1 > 0 such that if |ξ| ≥ C 1 τ , the expression in (21) can be estimated from below by Here the constant c hf > 0 is independent of τ and ξ. In the sequel, this will motivate a distinction between the two regimes |ξ| ≥ C 1 τ and |ξ| ≤ C 1 τ . We further note that if the constant c 0 > 0 in (20) is sufficiently small, then the a priori not necessarily signed Fourier multipliers associated with contributions in the third and fourth line in (20) may be absorbed into the lower bound in (22). Motivated by the estimate (22), we call the region {|ξ| ≥ C 1 τ } the high frequency elliptic region. By the above considerations the claimed lower bound (19) always holds in this region. It thus remains to study the region complementary to this, i.e. the region in which |ξ| ≤ C 1 τ . In this region, we expand the symbols in hξ j (noting that h|ξ| ≤ C 1 τ δ 2 0 τ −2 = C 1 δ 2 0 τ −1 which is small for τ > 1 and δ 0 > 0 small). For the symmetric part we obtain for some constant C > 0 which depends on C 1 > 0 For the antisymmetric part in turn we infer for denote the joint characteristic sets of the symmetric and antisymmetric parts of the operator. Further define to be a γ 0 τ neighbourhood of the joint characteristic set C τ with γ 0 > 0 small (to be determined below). With this notation fixed, we prove that for |ξ| ≤ C 0 τ outside of N τ,C there exists some constant c lf,1 > 0 (depending on γ 0 ) independent of τ > 0 such that (26) p 2 r (ξ) + p 2 i (ξ) ≥ c lf,1 (τ 4 + |ξ| 4 ). Indeed, this is true for the leading order approximations (|∇φ(n)| 2 − |ξ| 2 ) 2 + 4(∇φ(n) · ξ) 2 , and transfers to the full symbols p 2 r (ξ) + p 2 i (ξ) since the error estimates in (23), (24) are of order Ch 2 τ 4 ≤ Cδ 0 if τ ∈ (1, δ 0 h − 1 2 ). Thus, if δ 0 is sufficiently small (depending on γ 0 ), these error contributions can be absorbed. Again, if the constant c 0 > 0 is sufficiently small, we may absorb the contributions originating from the not necessarily signed Fourier symbols of the operators in the third and fourth line in (20) into the lower bound (26).
We next seek to argue that by continuity a similar lower bound also holds on N τ,C . To this end, note that for ξ ∈ N τ,C we have ξ = τ ξ 0 for some ξ 0 ∈ (h −1 T) d with |ξ 0 | ∈ (C 0,1 , C 0,2 ), where the constants C 0,1 , C 0,2 > 0 only depend on γ 0 and the dimension d and, in particular, are independent of τ > 1 and h > 0. Thus, for ξ ∈ N τ,C and ξ 0 = τ −1 ξ we have that by homogeneitỹ is independent of τ . Since for ξ ∈ C τ the pseudoconvexity condition for φ implies thatq(ξ) ≥ c cf,1 > 0, by continuity, it remains true thatq(ξ) ≥ c cf,1 /2 in the neighbourhood N τ,C if γ 0 > 0 is sufficiently small (but independent of τ > 1). By the scaling of q(ξ) we thus infer that for ξ ∈ N τ,C and δ 0 > 0 sufficiently small we have Thus, in total, we have obtained that for all ξ ∈ (h −1 T) d By the Parseval identity, this implies that which yields the claim of the Proposition.
With all of these auxiliary results in hand, we now address the proof of Theorem 3.
Proof of Theorem 3. The proof of Theorem 3 follows by combining all the previous estimates. We first rewrite the desired estimate in terms of the functions f := e φ u for which we seek to prove and for which we note that, the action of D s on e φ u yields terms D s e φ that can be absorbed in the first term with u ). We now argue in two steps, first reducing the estimate to a bound for the localized functions and then proving the estimate for these.
Step 1: Localization. As a first step, we note that it suffices to prove the estimate for the localized functions f k from Lemma 3.2. Indeed, assuming that the estimate (28) is proven for f k , an application of Minkowski's inequality and the error estimates from Lemma 3.2 yield Now choosing ǫ 0 ≤ 1 10C loc and recalling that τ h 1 2 ≤ δ 0 for some δ 0 ∈ (0, 1), we may absorb the contribution on the right hand side of (29) into its left hand side (in particular we note that τ 2 τ 1 2 h ≤ δ 2 0 τ 1 2 by our assumptions on the relation between τ and h). This then yields the estimate (28). The estimate (4) follows from this by possibly choosing the constants in the terms which involve derivatives on the left hand side of (28) smaller, carrying out the product rule and absorbing the L 2 errors into the L 2 contribution on the left hand side of (28).
Step 2. Proof of (28) for the localized functions. It thus suffices to prove (28) for f = f k . To this end, we observe that for f k = f ψ k with supp(f ) ⊂ B 2 \ B 1/2 , ψ k as in Lemma 3.2 and c 0 ∈ (0, 1) to be chosen below, where by Lemma 3.3 Here we have used that τ ≥ 1 and Fixing h 0 > 0 such that Ch 0 ≤ C low 10 , where C low is the constant from Proposition 3.5, we will be able to treat the contributions in (31) as error contributions in the following arguments.
Exploiting the bounds from Lemma 3.4, we may further estimate where by the estimates from Lemma 3.4 Finally, invoking Proposition 3.5, we infer that We now fix τ 0 > 1 so large and h 0 > 0 so small that Further, we possibly decrease the value of h 0 > 0 even more and choose it so small that δ 0 h − 1 2 0 ≥ 100τ 0 > 100, which in particular implies that for all h ∈ (0, h 0 ) the interval (τ 0 , δ 0 h − 1 2 ) is non-empty. With these choices, it follows that for τ ∈ (τ 0 , δ 0 h − 1 2 ), we may absorb the error contributions E 1 and E 2 from (31) and (32) into the positive right hand side contributions in (33). Therefore, we obtain that Dividing by τ > τ 0 implies the desired result.

Proofs of Theorems 1 and 2
In this section we provide the proofs of the results of Theorems 1 and 2.
4.1. Derivation of Theorem 2 from Theorem 1. We first show how Theorem 1 implies Theorem 2.
Proof of Theorem 2. Let us assume that Theorem 1 holds. First, let us take the value τ * such that . It is easy to check that with this value of τ * it holds Given u satisfying (2), we can assume that τ 0 < τ * , and we are in one of the following two cases: • If τ * ∈ (τ 0 , δ 0 h − 1 2 ), then plugging this into the right hand side of (2) yields, for τ = τ * , that C(e c1τ u L 2 (B 1/2 ) + e −c2τ u L 2 (B2) ) = 2C u c 2 c 1 +c 2 We hence obtain that Thus, since (2) holds for all τ ∈ (τ 0 , δ 0 h − 1 2 ) and by using (34), we have Combining both cases implies (2) with α = c2 c1+c2 and c 0 = δ0 2 c 2 . 4.2. Derivation of Theorem 1 from the Carleman estimate of Theorem 3. In this section, we deduce Theorem 2 from Theorem 3. As an auxiliary result we deduce a Caccioppoli inequality for more general second order difference equations. In particular this applies to the difference Schrödinger equation (1).
Lemma 4.1 (Caccioppoli). Let a jk : (hZ) d → R d×d be symmetric, bounded and uniformly elliptic with ellipticity constant λ ∈ (0, 1), i.e. assume that for all ξ ∈ R d \ {0} we have Let 0 < 10h < r 1 < r 1 + 100h < r 2 . Then there exists a constant C > 1 depending on Here H 1 loc,h ((hZ) d ) and H 1 ((hZ) d ) denote the local and global H 1 spaces on the lattice. Proof of Lemma 4.1. The result follows along the same lines as the continuous Caccioppoli inequality; we only present the proof for completeness. As for general r 1 , r 2 the proof is analogous, we only discuss the details in the case r 1 = 1, r 2 = 2 and 0 < h ≤ h 0 for h 0 ≪ 1 sufficiently small.
Noting that a ij ≤ 1 2 λ −1 (this follows from the ellipticity condition when choosing appropriate ξ) we obtain that Combining this with the bounds for B j and V , we obtain Here the first contribution in (39) originates from the first right hand side contribution in (37). We may absorb it from the right hand side of (39) into the left hand side of (39). Using Young's inequality for the contribution allows us to also absorb the gradient term in this contribution into the left hand side of (39). Due to the bounds on η, this concludes the proof of the Caccioppoli estimate.
Proof of Theorem 1. The proof of Theorem 1 from the Carleman estimate in Theorem 3 follows from a standard cut-off argument. For completeness, we present the details. Let u : (hZ) d → R such that P h u(n) = 0 for all n ∈ B 4 . Fix ε > 0 to be small enough and assume that h 0 > 0 is sufficiently small. We consider the function w(n) = θ(n)u(n), with 0 ≤ θ(x) ≤ 1 a C ∞ (R d ) cut-off function defined as Using the equation for u, we then write Applying the Carleman estimate (4) from Theorem 3, using Remark 1.1 and the triangle inequality, we obtain L ∞ , B 2 L ∞ } allows us to absorb the first two contributions from the right hand side of (40) into the left hand side of (40). We thus obtain the bound We next deal with the errors on the right hand side of (41). On the one hand, we have On the other hand, for T d,j with j ∈ {1, 2}, where we used the Caccioppoli estimate from Lemma 4.1. Moreover, τ 3 e φ u 2 L 2 ≥ τ 3 e φ f 2 Remark 4.2. We remark that as a feature of the discrete setting, to a certain degree we can also deal with more singular potentials. Tracking the argument from above (in particular the passage from (40) to (41)), we note that if V and B only satisfy the bounds with µ 0 ≤ C Carl 10 δ 0 , we can deduce that for some constantsc 1 ,c 2 > 0 (independent of h) u L(B1) ≤ C(ec 1 h − 1 2 u L 2 (B 1 2 ) + e −c2h − 1 2 u L 2 (B2) ).
We also remark that while yielding quantitative propagation of smallness type estimates, as expected these estimates do not pass to the limit h → 0. Further, the h dependence in the exponentials can be adapted to the size of the potentials (with different bounds in the exponents of the logarithmic convexity estimates depending on the bounds on V , B).

Remarks on Scaling
Having established (3), we note that to a certain degree -although this is substantially weaker than in the continuous setting -it is possible to rescale this estimate. We discuss this in the case of the Laplacian (for more general operators similar observations remain valid). To this end, we make the following observation: Lemma 5.1. Let u : B 4 → R be such that ∆ d,h u = 0 in B R ⊂ (hZ) d . Then, for any m ∈ N such that hm ≤ 2, we also have ∆ d,mh u = 0 in B R/m ⊂ (mhZ) d (i.e. with respect to the lattice (mhZ) d ).
Summing and noting that the corresponding contributions in the brackets yield the Laplacian on (hZ) d implies the claim for m = 2.
Assuming the induction hypothesis for any m, i.e., d j=1 (f (x + mhe j ) + f (x − mhe j ) − 2f (x)) = 0 for x ∈ (mhZ) d , we prove the statement for m + 1. We have The conclusion follows from the cases m = 1 (after translation) and the inductive steps for m and m − 1.
Proof. We consider the function u m (x) := u(m −1 x) with x ∈ (hZ) d . By the considerations from Lemma 5.1 this is also harmonic on (hZ) d . Thus, we may apply Theorem 2. Rescaling z = m −1 x then implies the claim.
Remark 5.3. We remark that, of course, apart from rescalings also translations are always possible due to the translation invariance of the operator at hand.