Discrete Carleman estimates and three balls inequalities

Abstract

We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger 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.

1 Introduction

In this article, we provide robust quantitative unique continuation results for discrete magnetic Schrödinger operators $P_h$ of the form

$$\begin{aligned} P_h f(n):= h^{-2}\Delta _d f(n) + h^{-1}\sum \limits _{j=1}^{d} B_j(n) D_{+,j}^h f(n) + V(n) f(n), \end{aligned}$$

(1)

where $f: (h\mathbb {Z})^d \rightarrow \mathbb {R}$, $D_{\pm ,j}^h f(n):= \pm (f(n \pm h e_j)- f(n))$ denotes the (unscaled) left/right difference operator on scale h, $B: (h\mathbb {Z})^d \rightarrow \mathbb {R}^d$ is a (uniformly in h) bounded tensor field, modelling, for instance, magnetic interactions and where the potential $V: (h \mathbb {Z})^d \rightarrow \mathbb {R}$ is assumed to be uniformly bounded (independently of h). The operator $\Delta _{d}:= \sum \nolimits _{j=1}^d \Delta _{d,j,h} = \sum \nolimits _{j=1}^d D_{-,j}^h D_{+,j}^h$ is the (not normalized) discrete Laplacian on the lattice $(h \mathbb {Z})^d$.

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 [6]. Indeed, in [6] 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, $h \rightarrow 0$, 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 [5, 6, 10], where propagation of smallness estimates with correction terms were proved for the discrete Laplacian. For similar phenomena for related operators we refer to [4, 7] 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 [2, 3]).

1.1 Main results

Let us describe our main results. As a first main result, we seek to prove a discrete analogue (with correction term) of a logarithmic convexity inequality. More precisely, for u with $P_h u = 0$ and $P_h$ being the Schrödinger operator from (1) we provide the following bounds:

Theorem 1

There exist constants $h_0,\delta _0\in (0,1)$, $c_1, c_2 >0$ and $\tau _0>1$ such that for all $\tau \in (\tau _0, \delta _0 h^{-1})$, $h \in (0,h_0)$ and $u: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ with $P_h u = 0$ in $B_4$ it holds

$$\begin{aligned} \Vert u\Vert _{L^2(B_1)} \le C(e^{c_1 \tau } \Vert u\Vert _{L^2(B_{1/2})} + e^{- c_2 \tau }\Vert u\Vert _{L^2(B_2)}). \end{aligned}$$

(2)

Here for $r > 0$ we define $B_{r}=B_r(0) \cap (h \mathbb {Z})^d$, with $h\in (0,h_0)$ denoting the lattice spacing, and all $L^2$ norms are $L^2$ norms on the lattice $(h \mathbb {Z})^d$.

Due to the restriction on the upper bound of $\tau \le \delta _0 h^{-1}$, 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 $\alpha \in (0,1)$, $c_0>0$, $h_0\in (0,1)$ and $C>1$ such that for $h\in (0,h_0)$ and $u: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ with $P_h u = 0$ in $B_4$ we have

$$\begin{aligned} \Vert u\Vert _{L^2(B_1)} \le C(\Vert u\Vert _{L^2(B_{1/2})}^{\alpha }\Vert u\Vert _{L^2(B_2)}^{1-\alpha } + e^{- c_0 h^{-1}}\Vert u\Vert _{L^2(B_2)}). \end{aligned}$$

(3)

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 $h \rightarrow 0$, the propagation of smallness property of the associated elliptic operator is recovered. We remark that the scaling behaviour of the form $e^{-c_0 h^{-1}}$ in $h \in (0,h_0)$ had earlier been proven for the special case of the Laplacian (see [6, Theorem 1]) and is known to be optimal (see the discussion in [6, Section 4]). A similar, asymptotically optimal three balls estimate with Gaussian weight and with error terms is given in [10], see also [10, Corollary 1.14].

We remark that our results (and arguments) remain valid if instead of the differential Eq. (1) we consider the differential inequality

$$\begin{aligned} |h^{-2}\Delta _{d} f(n)| \le C\Big ( h^{-1}\sum \limits _{j=1}^{d} |D_{+,j}^h f(n)| + |f(n)| \Big ) \text{ for } \text{ some } 0<C<\infty . \end{aligned}$$

Further, it is possible to deduce propagation of smallness estimates for some controlled h-dependent growth of V and $B_{j}$ (see Remark 4.2) which however, of course, do not pass to the limit as $h\rightarrow 0$.

1.2 Main ideas

Similarly as in [2, 3] and contrary to the results in [6, 10], both of our results rely on a robust $L^2$ Carleman estimate. While [2] however relies on Carleman estimates with weights which have strong (pseudo)convexity properties, proving a three balls inequality requires working with (close to) limiting Carleman weights. 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 $\psi (x) = -\tau \log (|x|)$ and which we choose as, for example, in [8]:

Theorem 3

Let $u: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ be such that $h^{-2}\Delta _{d} u = g$ in $B_4$ with $\text {supp}(u) \subset B_2 \setminus B_{1/2}$ and $g\in L^2(B_4)$. Let $\phi (x):=\tau \varphi (|x|)$, where

$$\begin{aligned} \varphi (t)= -\log t + c_{ps} \Big ( \log t \arctan (\log t) - \frac{1}{2} \log (1+\log ^2t) \Big ) \end{aligned}$$

for a certain small constant $c_{ps}>0$. Then, there exist $h_0,\delta _0\in (0,1)$ with $h_0<\delta _0$, $C >1$ and $\tau _0>1$ (which are independent of u) such that for all $h\in (0,h_0)$ and $\tau \in (\tau _0, \delta _0 h^{-1})$ we have

$$\begin{aligned} \tau ^3 \Vert e^{\phi } u\Vert _{L^2}^2 + \tau \Vert e^{\phi } h^{-1} D_s u \Vert _{L^2}^2 + \tau ^{-1} \Vert e^{\phi } h^{-2} D_s^2 u \Vert _{L^2}^2 \le C \Vert e^{\phi } g \Vert _{L^2}^2. \end{aligned}$$

(4)

Here $D_s u(n) := \frac{1}{2}\sum \nolimits _{j=1}^{d} (u(n + h e_j) - u(n-h e_j))$, 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 Sect. 2 for the precise definitions.

The constraints on the size of the constant $c_{ps}>0$ are specified in the proof of Lemma 3.1.

Comparing our estimate with the previous Carleman estimates for discrete operators from [2, 3], we emphasize that in proving three balls inequalities and doubling properties, it is no longer possible to use strongly convex Carleman weights as in [2]. As a consequence, the derivation of positivity for the commutator becomes more intricate. In this sense, our estimate is closer in spirit to the estimates from [3], in which the authors construct discrete complex geometric optics solutions and which thus requires working with limiting Carleman weights. However, contrary to [3] working in a “unique continuation setting”, we can not rely on “plane wave” Carleman weights but have to use the more singular (almost) logarithmic weights which we only convexify very slightly. On a technical level this also leads to more complex commutator contributions. Hence, we are confronted with a situation in which only very little pseudoconvexity persists and in which the algebraic, discrete computations become rather involved. In order to overcome this, as one of the main ingredients of our proof, we relate the discrete quantities to their continuous counterparts for which the underlying pseudoconvexity structures become more transparent.

While building on similar ideas as in its continuous counterpart (see for instance [1, 9]), our Carleman estimate is restricted to a certain range of values of $\tau$ which is a purely discrete phenomenon. Similar restrictions had earlier been observed in [2, 3] in the context of Carleman estimates for control theoretic and inverse problems. From a technical perspective, as a key step in deriving the main Carleman estimate from Theorem 3, we localize to suitable scales on which we freeze coefficients and compare our discrete problem to the continuum setting. This is partly inspired by unique continuation results and Carleman estimates for (low) regularity variable coefficient operators [9]. As in [9] (and in contrast to the continuum constant coefficient or continuum higher regularity settings) in this discrete framework we lack tools such as “natural” polar/geodesic coordinates (see, for instance, [1]) which – after a diagonalization of the spherical contributions—would allow us to work with an essentially one-dimensional problem. Hence, as in [9] we localize to scales on which the coefficients are essentially constant and then exploit the more direct arguments in this set-up. Contrary to the setting in [9], in our problem these scales are determined not by the coefficient regularity but by the interplay with the discrete lattice spacing.

1.3 Outline of the article

The remainder of the article is organized as follows: In Sect. 2 we compute the conjugated discrete operator and its expansion into its symmetric, antisymmetric parts and their commutator. In the main part, in Sect. 3, we derive the main Carleman estimate of Theorem 3. Building on this, in Sect. 4 we deduce the results of Theorems 1 and 2. Last but not least, in Sect. 5, we comment on rescaled versions of the main estimates.

1.4 Remarks on the notational conventions

Concerning notation, with the letters $c, C,\ldots$ 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 use the notation $\widehat{f}$.

2 The conjugated Laplacian and the commutator

From now on, $D_{\pm }^j$ will stand for the forward/backward operators $D_{\pm ,j}^h$ from Sect. 1 and $D_{s}^j$ will denote the symmetric discrete derivatives in the j-th direction, i.e. $D_s^j u(n):= \frac{1}{2}(u(n+h e_j)- u({n}- h e_j))$. All operators are understood to be taken with step size h. Moreover, $D_{\pm }^h:= \sum \nolimits _{j=1}^d D_{\pm }^j$ and $D_{s}:= \sum \nolimits _{j=1}^d D_{s}^j$. We remark that the symmetric difference operator is associated with the Fourier multiplier $i\sum \nolimits _{j=1}^{d} \sin (h \xi _j)$, where i denotes the complex unit and where $\xi _j \in h^{-1}[-\pi ,\pi ]$. Here we have used the fact that the Fourier transform maps functions on the lattice $(h\mathbb {Z})^d$ to $2\pi h^{-1}$-periodic functions and have identified the torus of length $2\pi h^{-1}$ with the interval $h^{-1}[-\pi ,\pi ]^d$. We will make use of this convention throughout the article; in particular, we will only study Fourier transformed lattice functions on the interval $h^{-1}[-\pi ,\pi ]^d$.

Heading towards the proof of the Carleman inequality of Theorem 3, we introduce the conjugated Laplacian

$$\begin{aligned} L_{\phi ,h} f(n):= h^{-2}e^{\phi } \Delta _d e^{-\phi } f(n) = \sum \limits _{j=1}^{d} \big [ S_j f(n) + A_j f(n) \big ]=:(S_{\phi }f(n)+A_{\phi }f(n)), \end{aligned}$$

(5)

where the symmetric and anti-symmetric operators are

$$\begin{aligned} S_jf(n)&=h^{-2}((\cosh (D_+^j\phi (n)))f(n+ h e_j)+ (\cosh (D_{-}^j\phi (n)))f(n- h e_j)-2f(n)),\\ A_jf(n)&=h^{-2}((\sinh (D_{-}^j\phi (n)))f(n- h e_j)- (\sinh (D_+^j\phi (n)))f(n+he_j)). \end{aligned}$$

We compute the commutator of this to be

$$\begin{aligned} {[S_{\phi },A_{\phi }]}f(n)&= \sum _{j,k}[S_j,A_k]f(n)=\sum _{j,k}\{S_jA_k-A_k S_j\}f(n) \\&=\frac{1}{2}\sum _{j,k}\{S_jA_k-A_k S_j\}f(n) +\frac{1}{2}\sum _{j,k}\{S_kA_j-A_j S_k\}f(n)\\&=:\sum _{j,k}h^{-4}T_{j,k}f(n), \end{aligned}$$

where

$$\begin{aligned} T_{j,k} f(n)&:= h^4[S_j, A_k]f(n) =A_{j,k} f(n+ h e_j+ h e_k)\\&\quad +B_{j,k} f(n- h e_j- h e_k)+ C_{j,k}f(n+ h e_j- h e_k) + E_{j,k}f(n- h e_j+ h e_k), \end{aligned}$$

with

$$\begin{aligned} A_{j,k}&= -\cosh (D^j_+ \phi (n)) \sinh (D_+^k \phi (n+ h e_j)) + \sinh (D_+^k \phi (n))\cosh (D^j_+ \phi (n+ h e_k)),\\ B_{j,k}&= \cosh (D^j_-\phi (n))\sinh (D^k_- \phi (n-h e_j)) - \sinh (D^k_- \phi (n))\cosh (D^j_-\phi (n- h e_k)),\\ C_{j,k}&= \cosh (D^j_+\phi (n))\sinh (D^k_- \phi (n+ h e_j))-\sinh (D^k_- \phi (n))\cosh (D^j_+ \phi (n- h e_k)),\\ E_{j,k}&= -\cosh (D^j_- \phi (n))\sinh (D^k_+\phi (n- h e_j)) + \sinh (D^k_+ \phi (n)) \cosh (D^j_-\phi (n+ h e_k)). \end{aligned}$$

Now, using trigonometric identities, these can be simplified to read

$$\begin{aligned} A_{j,k}&=-\sinh (D_+^jD_+^k\phi (n))\cosh (D_+^j\phi (n)-D_+^k\phi (n)),\\ B_{j,k}&=-\sinh (D_-^jD_-^k\phi (n))\cosh (D_-^j\phi (n)-D_-^k\phi (n)),\\ C_{j,k}&=\sinh (D_+^jD_-^k\phi (n))\cosh (D_+^j\phi (n)+D_-^k\phi (n)),\\ E_{j,k}&=\sinh (D_-^jD_+^k\phi (n))\cosh (D_-^j\phi (n)+D_+^k\phi (n)). \end{aligned}$$

Indeed, for instance, for $A_{j,k}$ we obtain

$$\begin{aligned} A_{j,k}&=-\cosh (D^j_+ \phi (n)) \sinh (D_+^k \phi (n+ h e_j)) + \sinh (D_+^k \phi (n))\cosh (D^j_+ \phi (n+ h e_k))\\&= - \cosh (D^j_+ \phi (n)) \sinh (D^k_+ D^j_+ \phi (n) + D^k_+ \phi (n))\\&\quad + \cosh (D^j_+ D^k_+ \phi (n) + D^j_+ \phi (n)) \sinh (D^k_+ \phi (n))\\&= -\cosh (D^j_+ \phi (n))\big [ \sinh (D^k_+ D^j_+ \phi (n)) \cosh (D^k_+ \phi (n)) + \sinh (D^k_+ \phi (n))\cosh (D^k_+ D^j_+ \phi (n)) \big ]\\&\quad + \sinh (D^k_+\phi (n))\big [ \cosh (D^j_+ D^k_+ \phi (n))\cosh (D^j_+ \phi (n)) + \sinh (D^j_+ D^k_+ \phi (n)) \sinh (D^j_+ \phi (n)) \big ]\\&= \sinh (D^j_+ D^k_+ \phi (n))\big ( \sinh (D^j_+ \phi (n))\sinh (D^k_+\phi (n)) - \cosh (D^k_+\phi (n))\cosh (D^j_+\phi (n)) \big )\\&= - \sinh (D^j_+ D^k_+ \phi (n)) \cosh (D^j_+\phi (n) - D^k_+ \phi (n)). \end{aligned}$$

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 term $\langle [S,A]f,f\rangle$, the case $j=k=1$ is quite simple and leads to

$$\begin{aligned}&\sum _{ n\in \mathbb {Z} }\Big \{4\sinh (\Delta _d \phi (n))|D_s f(n)|^2-\Delta _d\sinh (\Delta _d \phi (n))|f(n)|^2\\&\qquad \quad +2\sinh (\Delta _d \phi (n))(\cosh (2D_s\phi (n))-1)|f(n)|^2\Big \}. \end{aligned}$$

The main term is a discrete version of $4\phi _{jj}|f_j|^2+4\phi _{jj} \phi _j\phi _j|f|^2- \phi _{jjjj}|f|^2$, where the subindices refer to differentiation in the corresponding direction. Note that the main term of the higher dimensional continuous commutator is more complicated and is of the form

$$\begin{aligned} 4\phi _{jk}f_j \overline{f_k}+4\phi _{jk} \phi _j\phi _k|f|^2- \phi _{jjkk}|f|^2. \end{aligned}$$

In the general case, we can rewrite the contributions of $h^4 \langle [S,A]f,f\rangle$ in the following way (where with slight abuse of notation, we refrain from spelling out the sums in $(h\mathbb {Z})^d$ and the sum in j, k):

$$\begin{aligned} \begin{aligned}&\sinh (D_{++}^{j,k}\phi (n))f(n+ h e_j)\overline{f(n+ h e_k)}+\sinh (D_{--}^{j,k}\phi (n))f(n- h e_j)\overline{f(n- h e_k)}\\&\quad -\sinh (D_{+-}^{j,k}\phi (n))f(n+ h e_j)\overline{f(n- he_k)}-\sinh (D_{-+}^{j,k}\phi (n))f(n- h e_j)\overline{f(n+ h e_k)}\\&\quad +\sinh (D_{++}^{j,k}\phi (n))\big (\cosh (\phi (n+ h e_j+ h e_k)-\phi (n))-1\big )f(n+ h e_j)\overline{f(n+ h e_k)}\\&\quad +\sinh (D_{--}^{j,k}\phi (n))\big (\cosh (\phi (n- h e_j- h e_k)-\phi (n))-1\big )f(n- h e_j)\overline{f(n- h e_k)}\\&\quad -\sinh (D_{+-}^{j,k}\phi (n))\big (\cosh (\phi (n+h e_j- h e_k)-\phi (n))-1\big )f(n+ h e_j)\overline{f(n- h e_k)}\\&\quad -\sinh (D_{-+}^{j,k}\phi (n))\big (\cosh (\phi (n- h e_j+ h e_k)-\phi (n))-1\big )f(n- h e_j)\overline{f(n+ h e_k)}. \end{aligned} \end{aligned}$$

(6)

The interest of writing the general term in this form is that we seek to bring the commutator term into a form which is as close as possible to the form of the commutator in the continuous setting which reads

$$\begin{aligned} 4( \nabla \phi \cdot \nabla ^2 \phi \nabla \phi ) f^2 + 4 \nabla f \cdot \nabla ^2 \phi \nabla f - \Delta ^2 \phi f^2. \end{aligned}$$

To this end, we note that the first four terms in (6) are closely related to the part $4\phi _{jk}f_j \overline{f_k}- \phi _{jjkk}|f|^2$ and the last four terms to $4\phi _{jk} \phi _j\phi _k|f|^2$ correspondingly.

We will use the expression (6) as the starting point of our commutator estimates in the following sections.

3 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 $\phi (x)$:

Lemma 3.1

Let $\phi (x):=\tau \varphi (|x|)$, where for some small constant $c_{ps}>0$

$$\begin{aligned} \varphi (t)= -\log t + c_{ps} \Big ( (\log t) \arctan (\log t) - \frac{1}{2} \log (1+(\log t)^2) \Big ). \end{aligned}$$

(7)

Then $\phi (x)$ is strongly pseudoconvex with respect to the Laplacian and with respect to the domain $B_4 \setminus B_1$ in the sense that there exists a constant $C>0$ (which is independent of $\tau$) such that in $B_4 \setminus B_1$ we have

$$\begin{aligned} \nabla \phi \cdot \nabla ^2 \phi \nabla \phi + \xi \cdot \nabla ^2 \phi \, \xi \ge C\tau ^3 >0 \text{ on } \{|\xi |^2 = |\nabla \phi |^2, \ \nabla \phi \cdot \xi = 0\}. \end{aligned}$$

Proof

In order to prove pseudoconvexity, we seek to prove that

$$\begin{aligned} \nabla \phi \cdot \nabla ^2 \phi \nabla \phi + \xi \cdot \nabla ^2 \phi \,\xi \ge C \tau ^3 >0 \text{ on } \{|\xi |^2 = |\nabla \phi |^2 ,\ \nabla \phi \cdot \xi = 0\}. \end{aligned}$$

Without loss of generality, we consider the case $\tau =1$ only; the general case follows then by rescaling. Now, if $\phi (x) = \varphi (|x|)$ we have that

$$\begin{aligned} \nabla ^2 \phi (x) = \frac{\varphi '(|x|)}{|x|} {\text {Id}} + \Big ( \varphi ''(|x|)-\frac{\varphi '(|x|)}{|x|}\Big )\frac{x}{|x|}\otimes \frac{x}{|x|}. \end{aligned}$$

Moreover, $\nabla \phi (x)$ is an eigenvector of $\nabla ^2 \phi (x)$ with eigenvalue $\lambda (x)= \varphi ''(|x|)$ and

$$\begin{aligned} \nabla \phi (x) \cdot \nabla ^2 \phi (x) \nabla \phi (x) = \varphi ''(|x|)(\varphi '(|x|))^2. \end{aligned}$$

Furthermore, the only other eigenvalue (of multiplicity $d-1$) of $\nabla ^2 \varphi (|x|)$ is given by $\mu (|x|) = \frac{\varphi '(|x|)}{|x|}$. Due to the constraint that $\{|\xi |^2 = |\nabla \phi |^2, \ \nabla \phi \cdot \xi = 0\}$, we therefore infer that

$$\begin{aligned} \nabla \phi (x) \cdot \nabla ^2 \phi (x) \nabla \phi (x) + \xi \cdot \nabla ^2 \phi (x) \xi = (\varphi '(|x|))^2 \Big (\varphi ''(|x|)+\frac{\varphi '(|x|)}{|x|} \Big ). \end{aligned}$$

For $\phi (x)= -\tau \log (|x|)$ this vanishes (as it is a limiting Carleman weight) but for (7) one obtains that on the critical set

$$\begin{aligned} \nabla \phi \cdot \nabla ^2 \phi \nabla \phi + \xi \cdot \nabla ^2 \phi \,\xi = (\varphi '(|x|))^2 \Big (\varphi ''(|x|)+\frac{\varphi '(|x|)}{|x|} \Big ) = c_{ps}\frac{(-1 + c_{ps} \arctan (\log (|x|)))^2}{|x|^4(1+\log ^2(|x|))}, \end{aligned}$$

which is positive for some sufficiently small $c_{ps}>0$. $\square$

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 $j\in \{1,\dots , d\}$ as

$$\begin{aligned} \Delta _{d,h,j}f(n) = f(n + h e_j) + f(n - h e_j) - 2 f(n). \end{aligned}$$

As a first step towards the desired Carleman estimate, we localize the problem to scales of order $\epsilon _0^{-1} \tau ^{-\frac{1}{2}}$, where $\epsilon _0>0$ is a small constant which will be chosen below (see the proof of Theorem 3):

Lemma 3.2

Let $\phi$ be as in Lemma 3.1. Let $S_{\phi }$, $A_{\phi }$ be as in (5). Let $\{\psi _k(x)\}_{k\in \mathbb {Z}}$ be a partition of unity subordinate to an open cover of $B_2 \setminus B_{\frac{1}{2}}$ which is localized to scales of order $\epsilon _0^{-1} \tau ^{-\frac{1}{2}}$ for some $\epsilon _0 \in (0,1)$ small. Suppose further that $\tau \in (1,\delta _0 h^{-1})$, where $h \in (0, h_0)$ for $\delta _0, h_0 \in (0,1)$ is chosen to be small and such that $h_0<\delta _0$. Let $f_k(n):= (f\psi _k)(n)$. Then,

$$\begin{aligned} \begin{aligned} \Vert S_{\phi } f\Vert&\le \sum \limits _{k} \Vert S_{\phi } f_k\Vert \le C \Vert S_{\phi } f\Vert + C \tau ^{\frac{1}{2}}\epsilon _0 \sum \limits _{j=1}^d \Vert h^{-1}D_s^jf\Vert + C (\tau \epsilon _0 + \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0) \Vert f\Vert ,\\ \Vert A_{\phi } f\Vert&\le \sum \limits _{k} \Vert A_{\phi } f_k\Vert \le C \Vert A_{\phi } f\Vert + C \tau ^{\frac{3}{2}} \epsilon _0 \Vert f\Vert ,\\ \Vert L_{\phi } f\Vert&\le \sum \limits _{k} \Vert L_{\phi } f_k\Vert \le C \Vert L_{\phi } f\Vert + C \tau ^{\frac{1}{2}}\epsilon _0 \sum \limits _{j=1}^d \Vert h^{-1}D_s^jf\Vert \\&\qquad + C (\tau \epsilon _0 +\tau ^{\frac{3}{2}} \epsilon _0 + \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0) \Vert f\Vert . \end{aligned} \end{aligned}$$

(8)

We remark that the condition $h_0<\delta _0$ is imposed in order to ensure that for all $h\in (0,h_0)$, we have $\delta _0h^{-1}>1$. Here and in the sequel, for brevity of notation, we write the $L^2$ norm on the lattice without adding subindices, i.e. $\Vert f\Vert := \Vert f\Vert _{L^2((h\mathbb {Z}))^d}$.

Proof of Lemma 3.2

As the estimates for $S_{\phi }$ and for $A_{\phi }$ are analogous, we mainly focus on the argument for $S_{\phi }$. The first bound in the estimate for $S_{\phi }$ in (8) is a direct consequence of Minkowski’s inequality. In order to observe the second estimate for $S_{\phi }$ in (8), we spell out the contributions coming from $S_{\phi } f_k(n)$. We begin by rewriting

$$\begin{aligned} S_j f_k(n)= & {} h^{-2}(\cosh (D^j_+ \phi (n))-1) f_k(n + h e_j)\\&+ h^{-2}(\cosh (D^j_- \phi (n))-1) f_k(n - h e_j) + h^{-2} \Delta _{d,h,j} f_k(n). \end{aligned}$$

Hence, inserting the function $f_k(n)$ for $h^{-2} \Delta _{d,h,j} f_k(n)$, we obtain

$$\begin{aligned}&h^{-2} \Delta _{d,h,j}f_k(n)\\&\quad = h^{-2}[f(n+h e_j) + f(n- h e_j) - 2 f(n)] \psi _k(n)\\&\qquad + h^{-2}[f(n+ h e_j)(\psi _k(n + h e_j)-\psi _k(n)) + f(n-h e_j)(\psi _k(n-h e_j)-\psi _k(n))]\\&\quad = \psi _k(n) (h^{-2} \Delta _{d,h,j} f(n))\\&\qquad + h^{-2}(f(n + h e_j) - f(n-h e_j))(\psi _k(n + h e_j)-\psi _k(n-h e_j))\\&\qquad + h^{-2} (f(n+h e_j)-f(n-h e_j))(\psi _k(n-h e_j)-\psi _k(n))\\&\qquad + f(n-h e_j)h^{-2}( \Delta _{d,h,j}\psi _k)(n). \end{aligned}$$

While we seek to keep the first contribution in this expansion to recombine it to $h^{-2} \Delta _{d,h,j} f$ after summing over the partition of unity, we only provide estimates on the remaining contributions. To this end, denoting by $2h{\text {supp}}( \psi _k )$ a 2h-neighbourhood of the support of $\psi _k$, we observe that

$$\begin{aligned}&|h^{-2}(f(n + h e_j) - f(n-h e_j))(\psi _k(n + h e_j)-\psi _k(n-he_j))|\\&\quad \le C|h^{-1}(f(n + h e_j) - f(n-h e_j))| |\nabla \psi _k( y ) |\\&\quad \le C \epsilon _0 \tau ^{\frac{1}{2}} |h^{-1}(f(n + h e_j) - f(n-h e_j))| \chi _{2h{\text {supp}}( \psi _k )}( n ), \end{aligned}$$

as, at an intermediate point y with $y \in [n-he_j, n+he_j]$,

$$\begin{aligned} |\nabla \psi _k(y)| \le C \epsilon _0 \tau ^{\frac{1}{2}} \chi _{2h{\text {supp}}( \psi _k )}(n),\quad \text { for all } k \end{aligned}$$

and

$$\begin{aligned} |f(n-h e_j)h^{-2}( \Delta _{d,h,j}\psi _k)(n)| \le C|f(n-h e_j)||D^2 \psi _k| \le C \tau \epsilon _0^2|f(n-h e_j)| \chi _{2h{\text {supp}}( \psi _k )}(n ). \end{aligned}$$

Using the same reasoning for the term $h^{-2} (f(n+h e_j)-f(n-h e_j))(\psi _k(n-h e_j)-\psi _k(n))$, and combining these estimates, we thus infer that

$$\begin{aligned} \begin{aligned}&|h^{-2} \Delta _{d,h,j}f_k(n) - \psi _k(n) h^{-2} \Delta _{d,h,j}f(n)|\le C\tau \epsilon _0^2|f(n-h e_j)| \chi _{2h{\text {supp}}( \psi _k )}(n) \\&+ C \epsilon _0 \tau ^{\frac{1}{2}} |h^{-1}(f(n + h e_j) - f(n-h e_j))| \chi _{2h{\text {supp}}( \psi _k )}(n). \end{aligned} \end{aligned}$$

(9)

Similarly, for $h^{-2} (\cosh (D^j_+ \phi (n))-1) f_k(n + h e_j)$ we obtain

$$\begin{aligned}&h^{-2} (\cosh (D^j_+ \phi (n))-1) f_k(n + h e_j) = \psi _k(n) h^{-2} (\cosh (D^j_+ \phi (n))-1) f(n + h e_j)\\&\quad + h^{-2} (\cosh (D^j_+ \phi (n))-1) f(n + h e_j)(\psi _k(n+ h e_j)- \psi _k(n)). \end{aligned}$$

Estimating

$$\begin{aligned} |\cosh (D^j_+ \phi (n))-1| \le \frac{|D^j_+ \phi (y)|^2}{2} \le \tau ^2 h^2 \frac{|\nabla \varphi (\widetilde{y})|^2}{2}, \end{aligned}$$

where we used that for $y \in B_{2}\setminus B_{\frac{1}{2}}$ we have that

$$\begin{aligned} |D^j_+\phi (n)| = h |\partial _j \phi (y)| = h \tau |\partial _j \varphi (y)| \le C \delta _0 \end{aligned}$$

is small, allowing for a Taylor expansion of the hyperbolic cosine. Here ${\widetilde{y}}$, $y \in [n, n+ h e_j]$ are intermediate values. We hence obtain

$$\begin{aligned} \begin{aligned}&C h^{-1} |(\cosh (D^j_+ \phi (n))-1) f(n + h e_j) (\psi _k(n + he_j) - \psi _k(n))| h^{-1}\\&\le C \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0|f(n+he_j)| \chi _{2h{\text {supp}}( \psi _k )}(n) . \end{aligned} \end{aligned}$$

(10)

As a consequence, combining the estimates from (9) and (10) yields

$$\begin{aligned} \sum \limits _k \Vert S_{\phi } f_k\Vert&\le C \sum \limits _{k} \Vert \psi _k S_{\phi } f\Vert + \sum \limits _{k}\Big ( C \tau ^{\frac{1}{2}}\epsilon _0 \sum \limits _{j=1}^d \Vert \chi _{2h{\text {supp}}( \psi _k )} h^{-1}D_s^jf\Vert \\&\qquad + C( \tau \epsilon _0 + \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0 ) \Vert f \chi _{2h{\text {supp}}( \psi _k )}\Vert \Big )\\&\le C \Vert S_{\phi } f \Vert + C \tau ^{\frac{1}{2}}\epsilon _0 \sum \limits _{j=1}^d \Vert h^{-1}D_s^jf\Vert + C( \tau \epsilon _0 + \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0 ) \Vert f\Vert . \end{aligned}$$

This concludes the argument for the localization estimate for $S_{\phi }$.

The arguments for $A_{\phi }$ and $L_{\phi }$ are analogous. Indeed, for $A_{\phi }$ we note that, for an intermediate value y,

$$\begin{aligned} |A_j(\psi _k f)(n)-\psi _k A_jf (n)|\le |-h^{-1}\partial _j\phi (y) f(n+he_j)(\psi _k(n+he_j)-\psi _k(n))\\+h^{-1}\partial _j\phi (y)f(n-he_j)(\psi _k(n-he_j)-\psi _k(n))|, \end{aligned}$$

which yields

$$\begin{aligned} \sum \limits _{k} \Vert A_{\phi } f_k\Vert \le C \Vert A_{\phi } f\Vert + C \tau ^{3/2} \epsilon _0 \Vert f\Vert . \end{aligned}$$

Estimating the terms of $L_{\phi }$ by using the bounds for $A_{\phi }$ and $S_{\phi }$ then implies the result. $\square$

As a next auxiliary step, we expand the trigonometric identities which then allows for easier manipulations of the contributions in the sequel.

Lemma 3.3

Let $\phi$ be as in Lemma 3.1. Let

$$\begin{aligned} \begin{aligned} S_j f(n)&= h^{-2}\big (\cosh (D^j_+ \phi (n)) f(n+ h e_j) + \cosh (D_-^j\phi (n))f(n- h e_j) - 2f(n) \big ),\\ A_j f(n)&= -h^{-2}\sinh (D^j_+ \phi (n)) f(n + h e_j) + h^{-2}\sinh (D^j_- \phi (n)) f(n- h e_j), \end{aligned} \end{aligned}$$

and $[S_j, A_k] f(n)$ be the quantities from Sect. 2. Let further

$$\begin{aligned} \begin{aligned} \widetilde{S}_j f(n)&:= h^{-2}\Big (\Big (\frac{h^2 (\partial _j \phi (n))^2}{2}+1 \Big ) f(n+ h e_j) + \Big (\frac{h^2 (\partial _j \phi (n))^2}{2} +1 \Big ) f(n-h e_j) - 2 f(n) \Big )\\&= h^{-2}\Delta _{d,h,j} f(n) + \frac{(\partial _j \phi (n))^2}{2}\big ( f(n + h e_j) + f(n - h e_j) \big ),\\ \widetilde{A}_j f(n)&:= -h^{-1}(\partial _j \phi (n))( f(n+ h e_j) - f(n-h e_j)),\\ \mathcal {C}_{jk}^{f,f}(n)&:= (\partial _{jk}\phi (n))\left( h^{-1}(f(n+he_j)-f(n-h e_j))\overline{h^{-1}(f(n+h e_k)-f(n- he_k))} \right) \\&\qquad + \frac{1}{2}\partial _{jk}\phi (n)\big (|\partial _j \phi (n)+\partial _k \phi (n)|^2(f(n+he_j)\overline{f(n+h e_k)} + f(n-h e_j) \overline{f(n-he_k)}) \\&\qquad - |\partial _j \phi (n) - \partial _k \phi (n)|^2(f(n+he_j)\overline{f(n-he_k)} + f(n-he_j)\overline{f(n+h e_k)}) \big ). \end{aligned} \end{aligned}$$

Let $\tau \in (1, h^{-1} \delta _0)$, where $h\in (0,h_0)$ with $\delta _0 \in (0,1)$ (to be chosen below, see the proof of Theorem 3) and $h_0<\delta _0$.

Then, for $S_{\phi }$ and $A_{\phi }$ as in (5), $\widetilde{S}_{\phi } f(n):= \sum \nolimits _{j=1}^{d} \widetilde{S}_{j} f(n)$ and $\widetilde{A}_{\phi } f(n) := \sum \nolimits _{j=1}^{d} \widetilde{A}_{j} f(n)$, and $f\in L^2(B_2 \setminus B_{\frac{1}{2}})$ with $\text {supp}(f) \subset B_2 \setminus B_{\frac{1}{2}}$, we have

$$\begin{aligned} \Vert S_{\phi } f\Vert ^2&\ge \Vert \widetilde{S}_{\phi } f\Vert ^2 - 2 C(\delta _0^2 \tau ^2 + \delta _0^4 \tau ^4) \Vert f\Vert ^2,\\ \Vert A_{\phi } f\Vert ^2&\ge \Vert \widetilde{A}_{\phi } f\Vert ^2 - 2 C(\tau ^2+\delta _0^2\tau ^4) \Vert f\Vert ^2,\\ ([S_{\phi }, A_{\phi }]f, f)&\ge \sum _{n\in (h\mathbb {Z})^d}\sum \limits _{j,k=1}^{d}\mathcal {C}_{jk}^{f,f}(n) - 2C(\tau ^2+\delta _0^2\tau ^3) \Vert f \Vert ^2 - 2C \sum \limits _{j=1}^{d} \Vert h^{-1}D_s^j f\Vert ^2. \end{aligned}$$

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 $\phi$ and the corresponding nonlinear functions and then estimate the resulting errors.

Step 1::

The symmetric part. We first discuss the symmetric part of the operator. For instance, we expand

$$\begin{aligned} \begin{aligned}&\cosh (D^j_+ \phi (n))\\&\quad = \cosh (h\partial _j \phi (n) + O(h^2|\nabla ^2 \phi (y)|))\\&\quad = 1 + \frac{1}{2}\big |h \partial _j \phi (n) + O(h^2|\nabla ^2 \phi (y)|)\big |^2 + O\big ((|h \nabla \phi (y)|+ |h^2 \nabla ^2 \phi (y)|)^4\big )\\&\quad = 1 + \frac{1}{2}h^2|\partial _j \phi (n)|^2 + O(h^3(| \nabla ^2 \phi (y)|^2+|\nabla \phi (y)|^2) \\&\qquad + h^4|\nabla \phi (y)|^4 + h^8 |\nabla ^2 \phi (y)|^{4}). \end{aligned} \end{aligned}$$

Here $y \in \mathbb {R}^d$ are intermediate values, not necessarily the same, such that $y \in [n,n+he_j]$. Thus, the symmetric part becomes

$$\begin{aligned} S_j f(n) =\widetilde{S}_{j} f(n) - E_{S_{j}}f(n), \end{aligned}$$

where $\widetilde{S}_j f(n)$ is as in our statement and

$$\begin{aligned} \Vert E_{S_{j}} f\Vert \le C(h \tau ^2 + \tau ^4h^2)\Vert (|\nabla \varphi |^2 + |\nabla \varphi |^4+|\nabla ^2 \varphi |^2 +|\nabla ^2 \varphi |^4) f\Vert , \end{aligned}$$

with $n \in (h \mathbb {Z})^d$, $\phi (n)=\tau \varphi (n)$ with $\varphi$ a bounded function (on the relevant domain). Choosing $\tau \in (1, \delta _0 h^{-1})$ with $\delta _0$ sufficiently small, we may assume that $h \tau ^2 + \tau ^4 h^2 \le C(\delta _0 \tau + \delta _0^2 \tau ^2 )$, hence the error $\Vert E_{S_{j}} f\Vert$ in the symmetric part is an $L^2$ contribution and, combining this with the explicit form of $\varphi$, satisfies the estimate $\Vert E_{S_{j}} f\Vert \le C \delta _0 (\tau + \delta _0 \tau ^2) \Vert f\Vert$. Therefore, in the sequel, we will estimate

$$\begin{aligned} \Vert S_{\phi }f\Vert ^2 \ge \Vert \widetilde{S}_{\phi } f\Vert ^2 - 2 \sum \limits _{j=1}^{d}\Vert E_{S_j} f\Vert ^2. \end{aligned}$$

Step 2::

The antisymmetric part. For the antisymmetric part we argue analogously. We thus expand

$$\begin{aligned} \sinh (D^j_+ \phi (n))&= \sinh (h \partial _j \phi (n) + O(h^2 |\nabla ^2 \phi (y)|))\\&= h \partial _j \phi (n) + O(h^2 |\nabla ^2 \phi (y)|)+O\big ((h |\nabla \phi (y)| + O(h^2 |\nabla ^2 \phi (y)|))^3\big )\\&= h \partial _j \phi (n) +O(h^2 |\nabla ^2 \phi (y)|+h^3|\nabla \phi (y)|^3+h^6|\nabla ^2\phi (y)|^3), \end{aligned}$$

where y are intermediate values in $[n,n+he_j]$. Thus, the antisymmetric part becomes

$$\begin{aligned} A_j f(n) = \widetilde{A}_j f(n) - E_{A_j} f(n) \end{aligned}$$

with

$$\begin{aligned} \widetilde{A}_{j}f(n)&=-h^{-1}(\partial _j \phi (n)) (f(n+ h e_j) - f(n-h e_j)),\\ \Vert E_{A_j}f\Vert&\le C (\tau +\delta _0\tau ^2) \Vert (|\nabla ^2 \varphi |+ |\nabla \varphi |^3+|\nabla ^2\varphi |^3)f\Vert \\&\le C (\tau +\delta _0\tau ^2) \Vert f\Vert , \end{aligned}$$

for which we have used the bounds for $\varphi$ in $B_2 \setminus B_{\frac{1}{2}}$.

Step 3::

The commutator. Finally, we turn to the commutator which is given by

$$\begin{aligned} \begin{aligned}&{([S_{\phi }, A_{\phi }]f, f) =}\sum _{n\in (h\mathbb {Z})^d}\sum _{j,k=1}^df(n)\overline{[S_j, A_k] f(n)} \\&\quad = \sum _{n\in (h\mathbb {Z})^d}\sum _{j,k=1}^dh^{-4}\Big ( \sinh (D_{++}^{j,k}\phi (n))f(n+ he_j)\overline{f(n+ h e_k)}\\&\qquad +\sinh (D_{--}^{j,k}\phi (n))f(n- h e_j)\overline{f(n-h e_k)} \\&\qquad -\sinh (D_{+-}^{j,k}\phi (n))f(n+ h e_j)\overline{f(n-h e_k)}-\sinh (D_{-+}^{j,k}\phi (n))f(n-h e_j)\overline{f(n+ h e_k)} \\&\qquad +\sinh (D_{++}^{j,k}\phi (n))\big (\cosh (\phi (n+ he_j+ h e_k)-\phi (n))-1\big )f(n+ he_j)\overline{f(n+ h e_k)}\\&\qquad +\sinh (D_{--}^{j,k}\phi (n))\big (\cosh (\phi (n-h e_j- h e_k)-\phi (n))-1\big )f(n- h e_j)\overline{f(n- h e_k)}\\&\qquad -\sinh (D_{+-}^{j,k}\phi (n))\big (\cosh (\phi (n+h e_j- h e_k)-\phi (n))-1\big )f(n+ h e_j)\overline{f(n- h e_k)}\\&\qquad -\sinh (D_{-+}^{j,k}\phi (n))\big (\cosh (\phi (n- h e_j+ h e_k)-\phi (n))-1\big )f(n- h e_j)\overline{f(n+ h e_k)} \Big ). \end{aligned} \end{aligned}$$

(11)

For the first four contributions in (11), we expand, for each $n\in (h\mathbb {Z})^d$ and fixed $j,k\in \{1,\dots ,d\}$,

$$\begin{aligned} \begin{aligned} \sinh (D^{j,k}_{++}\phi (n))&= h^2 \partial _{jk}\phi (n) + \frac{1}{2} h^3(\partial _k \partial _j^2 + \partial _{k}^2 \partial _j)\phi (n) \\&\quad + O(h^4 |\nabla ^4 \phi (y)| + h^6\max \{|\nabla ^2 \phi (y)|, |\nabla ^3 \phi (y)|, |\nabla ^4 \phi (y)|\}^3),\\ \sinh (D^{j,k}_{--}\phi (n))&= h^2 \partial _{jk}\phi (n) - \frac{1}{2} h^3(\partial _k \partial _j^2 + \partial _k^2 \partial _j)\phi (n) \\&\quad + O(h^4 |\nabla ^4 \phi (y)| + h^6\max \{|\nabla ^2 \phi (y)|, |\nabla ^3 \phi (y)|, |\nabla ^4 \phi (y)|\}^3),\\ \sinh (D^{j,k}_{+-}\phi (n))&= h^2 \partial _{jk}\phi (n) - \frac{1}{2} h^3(\partial _j \partial _k^2 - \partial _j^2 \partial _k)\phi (n)\\&\quad + O(h^4 |\nabla ^4 \phi (y)| + h^6\max \{|\nabla ^2 \phi (y)|, |\nabla ^3 \phi (y)|, |\nabla ^4 \phi (y)|\}^3)\\ \sinh (D^{j,k}_{-+}\phi (n))&= h^2 \partial _{jk}\phi (n) + \frac{1}{2} h^3(\partial _j \partial _k^2 - \partial _j^2 \partial _k)\phi (n) \\&\quad + O(h^4 |\nabla ^4 \phi (y)| + h^6\max \{|\nabla ^2 \phi (y)|, |\nabla ^3 \phi (y)|, |\nabla ^4 \phi (y)|\}^3) \end{aligned} \end{aligned}$$

(12)

with y intermediate points. Here we have carried out Taylor expansions of both the functions $D^{j,k}_{\pm \pm }\phi (n)$ and of $\sinh (\cdot )$. Thus, the first four contributions in (11) can be written as

$$\begin{aligned} \begin{aligned}&h^{-2} (\partial _{jk}\phi (n)) \big (f(n+he_j)\overline{f(n+h e_k)} + f(n-he_j)\overline{f(n-h e_k)}\\&\qquad -f(n+he_j)\overline{f(n-h e_k)}-f(n-h e_j)\overline{f(n+h e_k)} \big ) + E_1(n,j,k) + E_2(n,j,k), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} E_1(n,j,k)&:= \frac{1}{2}\partial _k \partial _j^2 \phi (n) f(n+ h e_j)[h^{-1}\overline{(f(n+he_k)-f(n-he_k))}]\\&\quad + \frac{1}{2} \partial _k^2 \partial _j \phi (n)\overline{f(n + h e_k)}[h^{-1}(f(n+ h e_j)-f(n-he_j))]\\&\quad + \frac{1}{2} \partial _k^2 \partial _j \phi (n) [h^{-1}(f(n+he_j)-f(n-he_j)) \overline{f(n-he_k)}]\\&\quad + \frac{1}{2}\partial _k \partial _j^2 \phi (n) [h^{-1}\overline{(f(n+he_k)-f(n-he_k)))}f(n-h e_j)], \end{aligned}$$

and

$$\begin{aligned} |E_2(n,j,k)|&\le C (\tau \max \{|\nabla ^4 \varphi (y)|,\delta _0^2|\nabla ^2\varphi (y)|^3, \delta _0^2 |\nabla ^3 \varphi (y)|^3,\delta _0^2 |\nabla ^4 \varphi (y)|^3)\})\times \\&\quad \times (|f(n+he_j)|^2 + |f(n-he_j)|^2 + |f(n + h e_k)|^2 + |f(n- he_k)|^2). \end{aligned}$$

Note that

$$\begin{aligned}&h^{-2}\big (f(n+he_j)\overline{f(n+h e_k)} + f(n-he_j)\overline{f(n-h e_k)} \\&\qquad -f(n+he_j)\overline{f(n-h e_k)}-f(n-h e_j)\overline{f(n+h e_k)} \big )\\&\quad = \big (h^{-1}(f(n+he_j)-f(n-h e_j))\overline{h^{-1}(f(n+h e_k)-f(n- he_k))} \big ), \end{aligned}$$

which yields the first part in the expression which is claimed for $\mathcal {C}_{jk}^{f,f}$ in the lemma. Moreover, the error $E_1$ can be bounded by the Cauchy-Schwarz inequality:

$$\begin{aligned} \sum _{j,k=1}^{d}\sum _{n \in (h\mathbb {Z})^d} | E_1(n,j,k)|&\le C\Vert |\nabla ^3\phi | f\Vert ^2+C\sum _{j=1}^{d}\Vert h^{-1}D_s^jf\Vert ^2\\&\le C \tau ^{2}\Vert |\nabla ^3\varphi | f\Vert ^2+C\sum _{j=1}^{d}\Vert h^{-1}D_s^jf\Vert ^2. \end{aligned}$$

Here we have used that there exists a constant $C>1$ such that $\max \limits _{x\in \overline{B_{2}}\setminus B_{1/2}}|\nabla ^3 \phi (x)| \le C |\nabla ^3 \phi (y)|$ for any $y\in \overline{B_{2}}\setminus B_{1/2}$. For the second four terms in (11), we similarly expand as in (12)

$$\begin{aligned}&(\cosh (\phi (n+he_j + h e_k)-\phi (n))-1)\\&\quad = \cosh (D^j_+ \phi (n+h e_k)+D^k_+ \phi (n))-1\\&\quad = \cosh (h \partial _j \phi (n)+h \partial _k \phi (n) + O(h^2|\nabla ^2 \phi (y)|))\\&\quad = 1 + \frac{1}{2} h^2 |\partial _j \phi (n)+\partial _k \phi (n)|^2 + E_3(n) -1\\&\quad = \frac{1}{2}h^2 |\partial _j \phi (n)+\partial _k \phi (n)|^2 + E_3(n), \end{aligned}$$

where

$$\begin{aligned} |E_3(n)|&\le C(h^3|\nabla ^2 \phi (y)|^{2} + |\nabla \phi (y)|^2)+Ch^4\max \{|\nabla \phi (y)|, |\nabla ^2 \phi (y)|\}^4 \\&\le C\left[ h^{3} \tau ^2 \max \{|\nabla \varphi |,|\nabla ^2 \varphi |\}^2 + h^4\tau ^4 \max \{|\nabla \varphi |,|\nabla ^2 \varphi |\}^4\right] . \end{aligned}$$

A similar expansion holds for the term involving $(\cosh (\phi (n-he_j - h e_k)-\phi (n))-1)$, while

$$\begin{aligned} (\cosh (\phi (n-he_j + h e_k)-\phi (n))-1) = \frac{1}{2}h^2 |\partial _j \phi (n)-\partial _k \phi (n)|^2 + E_3(n) \end{aligned}$$

with similar expansion for the term $(\cosh (\phi (n+he_j - h e_k)-\phi (n))-1)$. Hence, expanding the contribution of the $\sinh$ just with one main term $h^2\partial _{jk}\phi (n)$, i.e.

$$\begin{aligned} \sinh (D^{j,k}_{+,-}\phi (n)) = h^2 \partial _{jk}\phi (n) + O(h^6 |\nabla ^2 \phi (y)|^3 + h^9 |\nabla ^3 \phi (y)|^3), \end{aligned}$$

a multiplication of these expansions turns the second set of four terms from (11) into

$$\begin{aligned}&\frac{1}{2}\partial _{jk}\phi (n)\big (|\partial _j \phi (n)+\partial _k \phi (n)|^2(f(n+he_j)\overline{f(n+h e_k)} + f(n-h e_j) \overline{f(n-he_k)}) \\&\qquad - |\partial _j \phi (n) - \partial _k \phi (n)|^2(f(n+he_j)\overline{f(n-he_k)} + f(n-he_j)\overline{f(n+h e_k)}) \big ) \\&\qquad + E_4(n,j,k), \end{aligned}$$

where

$$\begin{aligned} |E_4(n,j,k)|&\le C \left[ h \tau ^3 \max \{|\nabla \varphi | ,|\nabla ^2 \varphi |,|\nabla ^3\varphi | \}^3+ h^2\tau ^5 \max \{|\nabla \varphi | ,|\nabla ^2 \varphi |,|\nabla ^3\varphi | \}^5\right. \\ {}&\left. \quad +h^6\tau ^7 \max \{|\nabla \varphi | ,|\nabla ^2 \varphi |,|\nabla ^3\varphi | \}^7 \right] (|f(n\pm he_j)|^2 + |f(n \pm h e_k)|^2), \\&\le C \left[ \delta _0 \tau ^2 + \delta _0^2 \tau ^3 \right] (|f(n\pm he_j)|^2 + |f(n \pm h e_k)|^2), \end{aligned}$$

where we used Young’s inequality $ab \le \dfrac{a^{p}}{p} + \dfrac{b^q}{q}$ for conjugate exponents p and q. Combining the estimates for $E_1,\ E_2$ and $E_4$ we arrive at the claimed estimate for the commutator by taking into account that $h,\delta _0\in (0,1)$ and $\tau h <\delta _0$.

$\square$

As a next step, we freeze coefficients in the operators $\widetilde{S}_{\phi }$, $\widetilde{A}_{\phi }$ and $\mathcal {C}_{jk}^{f,f}$ when acting on functions supported in sets of the size $\epsilon _0^{-1} \tau ^{-\frac{1}{2}}$ for $\epsilon _0>0$ sufficiently small and $\tau >1$ sufficiently large, both of which are to be determined below (see the proof of Theorem 3).

Lemma 3.4

Let $f \in C^{\infty }_c(B_2 \setminus \overline{B_{\frac{1}{2}}})$ be such that $\text {supp}(f)$ is of the size $\epsilon _0^{-1} \tau ^{-\frac{1}{2}}$. Assume that $\phi$ is as in Theorem 3 and that $1 < \tau \le \delta _0 h^{-1}$ for a sufficiently small constant $\delta _0>0$. Let $\bar{n} \in \mathbb {R}^d$ be a point which is in the interior of $\text {supp}(f)$ and set

$$\begin{aligned} \bar{S}_{\phi }^j f(n)&:= h^{-2} \Delta _{d,h,j} f(n) + \frac{(\partial _j \phi (\bar{n}))^2}{2}(f(n + e_j h) + f(n- e_j h)),\\ \bar{A}_{\phi }^j f(n)&:= - h^{-1} (\partial _j \phi (\bar{n}))(f(n + h e_j) - f(n - h e_j)),\\ \bar{\mathcal {C}}_{jk}^{f,f}(n)&:= (\partial _{jk} \phi (\bar{n}))\big ( h^{-1}(f(n + h e_j) - f(n - h e_j)) \overline{h^{-1} (f(n + h e_k) - f(n - h e_k))} \big )\\&\quad +\dfrac{1}{2} (\partial _{jk}\phi (\bar{n}))\big ( |\partial _j \phi (\bar{n}) + \partial _k \phi (\bar{n})|^2 (f(n + h e_j) \overline{f(n + h e_k)} + f(n - h e_j)\overline{f(n- h e_k)}) \\&\quad - |\partial _j \phi (\bar{n}) - \partial _k \phi (\bar{n})|^2(f(n + h e_j)\overline{f(n - h e_k)} + f(n - h e_j)\overline{f(n + h e_k)}) \big ). \end{aligned}$$

Then,

$$\begin{aligned} |\Vert \widetilde{S}_{\phi } f\Vert - \Vert \bar{S}_{\phi } f\Vert |&\le C \tau ^{\frac{3}{2}} \epsilon _0^{-1} \Vert f\Vert , \\ |\Vert \widetilde{A}_{\phi } f\Vert - \Vert \bar{A}_{\phi } f\Vert |&\le C\tau ^{\frac{1}{2}} \epsilon _0^{-1} \sum \limits _{j=1}^{d} h^{-1}\Vert D_s^j f\Vert , \\ \sum _{n\in (h\mathbb {Z})^d}\sum _{j,k}|\mathcal {C}_{jk}^{f,f} - \bar{\mathcal {C}}_{jk}^{f,f}|&\le C \tau ^{\frac{1}{2}} \epsilon _0^{-1} \sum \limits _{j=1}^d \Vert h^{-1}D_s^jf\Vert ^2 + C\tau ^{\frac{5}{2}} \epsilon _0^{-1} \Vert f\Vert ^2. \end{aligned}$$

Proof

Using the triangle inequality and the support condition, we estimate

$$\begin{aligned} |\Vert \widetilde{S}_{\phi } f\Vert - \Vert \bar{S}_{\phi } f\Vert |&\le C \Vert (\widetilde{S}_{\phi } - \bar{S}_{\phi }) f\Vert \le C ( \Vert ((\partial _j \phi (n))^2 - (\partial _j \phi (\bar{n}))^2) f(n + h e_j)\Vert \\&\quad + \Vert ((\partial _j \phi (n))^2 - (\partial _j \phi (\bar{n}))^2) f(n - h e_j)\Vert )\\&\le C \tau ^2 \sup \limits _{n \in \text {supp}(f)}|n- \bar{n}|\Vert f\Vert \le C \tau ^{\frac{3}{2}} \epsilon _0^{-1} \Vert f\Vert . \end{aligned}$$

As the arguments for $\widetilde{A}_{\phi }$ and for $\mathcal {C}_{jk}^{f,f}$ are analogous, we do not discuss the details. $\square$

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

Let $\bar{S}_{\phi }$, $\bar{A}_{\phi }$ and $\bar{\mathcal {C}}_{jk}^{f,f}$ be as in Lemma 3.4. Then there exist $C_{low }>0$, $c_0>0$, $h_0,\delta _0 \in (0,1)$ (small) and $\tau _0>1$ such that for all $\tau \in ( \tau _0, \delta _0 h^{-1})$ (large), $h \in (0,h_0)$ and for all $f\in C_c^{\infty }(B_2 \setminus B_{\frac{1}{2}})$ we have

$$\begin{aligned}&\Vert \bar{S}_{\phi } f \Vert ^2 + \Vert \bar{A}_{\phi } f \Vert ^2 + c_0 \tau \sum _{n\in (h\mathbb {Z})^d}\sum \limits _{j,k=1}^{d} \bar{\mathcal {C}}_{jk}^{f,f}(n)\nonumber \\&\qquad \ge C_{low } \Big (\tau ^4 \Vert f\Vert ^2 + \tau ^2 h^{-2} \sum \limits _{j=1}^{d}\Vert D^j_s f\Vert ^2 + h^{-4} \sum \limits _{j=1}^{d}\Vert (D^j_s)^2 f\Vert ^2\Big ). \end{aligned}$$

(13)

Proof

Using that the operators under consideration all have constant coefficients, we may perform a Fourier transform and infer that

$$\begin{aligned} \begin{aligned}&\Vert \bar{S}_{\phi } f \Vert ^2 + \Vert \bar{A}_{\phi } f \Vert ^2 + c_0\tau \sum _{n\in (h\mathbb {Z})^d}\sum _{j,k=1}^d\bar{\mathcal {C}}_{jk}^{f,f}\\&= \sum \limits _{j = 1}^d \Vert \big [ -4 h^{-2} \sin ^2(h \xi _j/2 ) + (\partial _j \phi (\bar{n}))^2 \cos (\xi _j h) \big ] \widehat{f} \Vert ^2 + \sum \limits _{j=1}^d \Vert 2h^{-1}(\partial _j \phi (\bar{n}))\sin (\xi _j h) \widehat{f}\Vert ^2 \\&\quad + c_0\tau \sum \limits _{j,k=1}^d \left[ 4 ( h^{-1}\sin (\xi _j h) (\partial _{jk}\phi (\bar{n})) h^{-1}\sin (\xi _k h)\widehat{f}, \widehat{f}) \right. \\&\quad \left. + \left( \left( \partial _{jk}\phi (\bar{n})\right) \left( |\partial _j \phi (\bar{n}) + \partial _k \phi (\bar{n})|^2 \cos (h \xi _j - h \xi _k) - |\partial _j \phi (\bar{n}) - \partial _k \phi (\bar{n})|^2 \cos (h \xi _j + h \xi _k) \right) \widehat{f}, \widehat{f}\right) \right] . \end{aligned} \end{aligned}$$

(14)

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

$$\begin{aligned} p_{r,j}(\xi )&:= - 4 h^{-2} \sin ^2(h \xi _j/2 ) + (\partial _j \phi (\bar{n}))^2 \cos (\xi _j h) \\&= 2 h^{-2}(\cos (h \xi _j)-1) + (\partial _j \phi (\bar{n}))^2 \cos (\xi _j h),\\ p_{i,j}(\xi )&:= 2 \partial _j \phi (\bar{n}) h^{-1} \sin (\xi _j h), \end{aligned}$$

the third term in (14) 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 $p_r(\xi ):= \sum \nolimits _{j=1}^{d} p_{r,j}(\xi )$ associated with the symmetric operator. We begin by observing that the first summand in

$$\begin{aligned} p_r(\xi )=\sum \limits _{j=1}^{d}|\partial _j \phi (\bar{n})|^2 \cos (\xi _j h) +2\sum \limits _{j=1}^{d} \frac{\cos (\xi _j h)-1}{h^2} \end{aligned}$$

(15)

is bounded from above by $C \tau ^2$. For the second summand, we deduce that (using our convention of identifying $2\pi h^{-1}$-periodic functions with functions on the torus $h^{-1}(-\pi ,\pi )$), since $|\cos (x)|\in (0,1)$ and for $\xi _j\in h^{-1}(-\pi , \pi )$, we have

$$\begin{aligned} |\cos (\xi _jh)-1|=\Big |\frac{(\xi _jh)^2}{2}+R(\xi _jh)\Big |\ge (\xi _jh)^2\Big (\frac{1}{2}-\frac{\pi ^2}{24}\Big )\ge \frac{1}{16}(\xi _jh)^2, \end{aligned}$$

where $R(\xi _jh)$ is the remainder term in the Taylor approximation. Hence,

$$\begin{aligned} \Big | \sum \limits _{j=1}^d \frac{\cos (\xi _j h) - 1}{h^2} \Big | = h^{-2} \sum \limits _{j=1}^d |\cos (\xi _j h) - 1| \ge \frac{1}{16} h^{-2} \sum \limits _{j=1}^d |\xi _j h|^2 \ge \frac{1}{16}|\xi |^2. \end{aligned}$$

(16)

Combining these two observations, we note that there exists a constant $C_1>0$ such that if $|\xi |\ge C_1 \tau$, the expression in (15) can be estimated from below by

$$\begin{aligned}&p_r(\xi )^2\ge \Big |\sum \limits _{j=1}^{d} \frac{\cos (\xi _j h)-1}{h^2}\Big |^2\ge 3 c_{hf}|\xi |^4\nonumber \\&\quad \ge c_{hf}(|\xi |^4+\tau ^2|\xi |^2 + \tau ^4) \nonumber \\&\quad \ge c_{hf}\Big (\sum \limits _{j=1}^{d} h^{-4}\sin ^4(h\xi _j)+\tau ^2\sum \limits _{j=1}^{d} h^{-2}\sin ^2(h\xi _j) + \tau ^4\Big ). \end{aligned}$$

(17)

Here the constant $c_{hf}>0$ is independent of $\tau$ and $\xi$. In the sequel, this will motivate a distinction between the two regimes $|\xi |\ge C_1 \tau$ and $|\xi |\le C_1 \tau$. We further note that if the constant $c_0>0$ in (14) is sufficiently small, then the a priori not necessarily signed Fourier multipliers associated with contributions in the third and fourth line in (14) may be absorbed into the lower bound in (16). Motivated by the estimate (16), we call the region $\{|\xi |\ge C_1 \tau \}$ the high frequency elliptic region. By the above considerations the claimed lower bound (13) always holds in this region.

It thus remains to study the region complementary to this, i.e. the region in which $|\xi |\le C_1 \tau$. In this region, we expand the symbols in $h\xi _j$ (noting that $h|\xi | \le C_1 \tau \delta _0 \tau ^{-1} = C_1 \delta _0$ which is small for $\delta _0>0$ small). For the symmetric part we obtain for some constant $C>0$ which depends on $C_1>0$

$$\begin{aligned} \begin{aligned}&\big | p_r(\xi )- \sum \limits _{j=1}^{d} \big ( |\partial _j \phi ({\bar{n}})|^2- |\xi _j|^2 \big ) \big |\\&\quad \le C \sum \limits _{j=1}^{d}\big ( |\partial _j\phi ({\bar{n}})|^2|\xi _j h|^2 + h^{-2}|\xi _j h|^4 \big )\\&\quad \le C \tau ^2 h^2 |\xi |^2 |\nabla \varphi ({\bar{n}})|^2 + h^2 |\xi |^4 \le C (\tau ^4 h^2 |\nabla \varphi ({\bar{n}})|^2 + h^2 \tau ^4). \end{aligned} \end{aligned}$$

(18)

For the antisymmetric part in turn we infer for $p_i:=\sum _{j=1}^dp_{i,j}$,

$$\begin{aligned} \begin{aligned} \big |p_{i}(\xi )- 2\sum _{j=1}^d \partial _j \phi ({\bar{n}})\xi _j\big |\le C \tau |\nabla \varphi ({\bar{n}})|h^{-1}\sum _{j=1}^{d}|h\xi _j|^3 \le C h^2 \tau ^4 |\nabla \varphi ({\bar{n}})|. \end{aligned} \end{aligned}$$

(19)

Let now

$$\begin{aligned} \mathcal {C}_{\tau }:=\{\tau ^2 |\nabla \varphi (\bar{n})|^2 = |\xi |^2\}\cap \{\tau \nabla \varphi (\bar{n})\cdot \xi =0\}, \end{aligned}$$

(20)

denote the joint characteristic sets of the symmetric and antisymmetric parts of the operator. Further define

$$\begin{aligned} \mathcal {N}_{\tau , \mathcal {C}}:= \{ \xi \in (h^{-1} \mathbb {T})^d: \ \text {dist}(\xi , \mathcal {C}_{\tau }) \le \gamma _0 \tau \} \end{aligned}$$

to be a $\gamma _0 \tau$ neighbourhood of the joint characteristic set $\mathcal {C}_{\tau }$ with $\gamma _0>0$ small (to be determined below). With this notation fixed, we prove that for $|\xi |\le C_1 \tau$ outside of $\mathcal {N}_{\tau , \mathcal {C}}$ there exists some constant $c_{lf,1}>0$ (depending on $\gamma _0$) independent of $\tau >0$ such that

$$\begin{aligned} p_r^2(\xi ) + p_i^2(\xi ) \ge c_{lf,1}(\tau ^4 + |\xi |^4). \end{aligned}$$

(21)

Indeed, this is true for the leading order approximations

$$\begin{aligned} (|\nabla \phi (\bar{n})|^2 - |\xi |^2)^2 + 4(\nabla \phi (\bar{n})\cdot \xi )^2, \end{aligned}$$

and transfers to the full symbols $p_r^2(\xi ) + p_i^2(\xi )$ since the error estimates in (18), (19) are of order $C h^2 \tau ^4 \le C \delta _0^2 \tau ^2$ if $\tau \in (1,\delta _0 h^{-1})$. Thus, if $\delta _0$ is sufficiently small (depending on $\gamma _0$), these error contributions can be absorbed into the right hand side of (21). 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 (14) into the lower bound (21).

It remains to study the behaviour of the Fourier symbols associated to the operators from (13) in the neighbourhood $\mathcal {N}_{\tau , \mathcal {C}}$ of the joint characteristic set (20). To this end, we also carry out an expansion of the symbol associated with the operators in the third and fourth line of (14) (which originates from the commutator) and obtain the symbol $q(\xi ) = q_1(\xi ) + q_2(\xi )$ with

$$\begin{aligned} q_1(\xi )&=\sum _{j,k=1}^d 4\tau \partial _{jk}\phi (\bar{n}) \xi _j \xi _k +\tau h^{2}O(|\nabla ^2\phi (\bar{n})| |\xi |^4) \\&= \sum _{j,k=1}^d4\tau ^2 \partial _{jk}\varphi (\bar{n}) \xi _j \xi _k +\tau ^6 h^2O(|\nabla ^2\varphi (\bar{n})|), \end{aligned}$$

and

$$\begin{aligned} q_2(\xi )&= \sum _{j,k=1}^d 4 \tau ^4 \partial _{jk}\varphi (\bar{n})\partial _j \varphi (\bar{n})\partial _k \varphi (\bar{n}) + \tau ^4 O(|\nabla \varphi (\bar{n})|^2 |\nabla ^2 \varphi (\bar{n})| |h\xi |^2) \\&= \sum _{j,k=1}^d4 \tau ^4 \partial _{jk}\varphi (\bar{n})(\partial _j \varphi (\bar{n}))(\partial _k \varphi (\bar{n}))+ \tau ^6 h^{2} O(|\nabla \varphi (\bar{n})|^2 |\nabla ^2 \varphi (\bar{n})|). \end{aligned}$$

Using that $\tau \in (1,\delta _0 h^{-1})$, we thus obtain that

$$\begin{aligned} q(\xi ) = 4 (\tau ^2 \xi \cdot \nabla ^2 \varphi (\bar{n}) \xi + \tau ^4 \nabla \varphi (\bar{n})\cdot \nabla ^2 \varphi (\bar{n}) \nabla \varphi (\bar{n}) ) + O(C \delta _0^2 \tau ^4 ). \end{aligned}$$

Now by the pseudoconvexity conditions on $\phi$ for $\bar{n}\in B_2 \setminus B_{\frac{1}{2}}$ (see Lemma 3.1), we infer that for $\xi$ in the characteristic set (20) there exist constants $c_{cf,1}, c_{cf}>0$ which are independent of $\tau$ and h such that

$$\begin{aligned} q(\xi ) \ge c_{cf,1}(\tau ^4 + |\xi |^2 \tau ^2+|\xi |^4) - C \delta _0^2 \tau ^4 \ge c_{cf}(\tau ^4 + |\xi |^2 \tau ^2+|\xi |^4). \end{aligned}$$

We next seek to argue that by continuity a similar lower bound also holds on $\mathcal {N}_{\tau , \mathcal {C}}$. To this end, note that for $\xi \in \mathcal {N}_{\tau , \mathcal {C}}$ we have $\xi = \tau \xi _0$ for some $\xi _0 \in (h^{-1} \mathbb {T})^d$ with $|\xi _0| \in (C_{0,1}, C_{0,2})$, where the constants $C_{0,1}, C_{0,2}>0$ only depend on $\gamma _0$ and the dimension d and, in particular, are independent of $\tau >1$ and $h>0$. Thus, for $\xi \in \mathcal {N}_{\tau , \mathcal {C}}$ and $\xi _0 = \tau ^{-1} \xi$ we have that by homogeneity

$$\begin{aligned} \widetilde{q}(\xi ):= \tau ^{-4}q(\xi )=\ \xi _0 \cdot \nabla ^2 \varphi (\bar{n}) \xi _0 + \nabla \varphi (\bar{n})\cdot \nabla ^2 \varphi (\bar{n}) \nabla \varphi (\bar{n}) + O(C \delta _0^2 ) \end{aligned}$$

is independent of $\tau$. Since for $\xi \in \mathcal {C}_{\tau }$ the pseudoconvexity condition for $\phi$ implies that $\widetilde{q}(\xi ) \ge c_{cf,1}>0$, by continuity, it remains true that $\widetilde{q}(\xi )\ge c_{cf,1}/2$ in the neighbourhood $\mathcal {N}_{\tau , \mathcal {C}}$ if $\gamma _0>0$ is sufficiently small (but independent of $\tau >1$). By the scaling of $q(\xi )$ we thus infer that for $\xi \in \mathcal {N}_{\tau , \mathcal {C}}$ and $\delta _0>0$ sufficiently small we have

$$\begin{aligned} q(\xi ) \ge \frac{c_{cf,1}}{2}(\tau ^4 + |\xi ^2| \tau ^2) - C \delta _0^2 \tau ^4 \ge \frac{c_{cf}}{4}(\tau ^4 + |\xi |^2 \tau ^2+|\xi |^4). \end{aligned}$$

(22)

Thus, in total, by (17), (21) and (22), we have obtained that for all $\xi \in (h^{-1} \mathbb {T})^d$

$$\begin{aligned}&p_r^2(\xi ) + p_i^2(\xi ) + q(\xi )\\&\qquad \ge \min \{c_{cf}/4, c_{lf,1}, c_{hf}\}\Big (\tau ^4 + \tau ^2 h^{-2} \sum \limits _{j=1}^d \sin ^2(h \xi _j) + h^{-4}\sum \limits _{j=1}^d \sin ^4(h \xi _j) \Big ). \end{aligned}$$

By the Parseval identity, this implies that

$$\begin{aligned} \begin{aligned}&\Vert \bar{S}_{\phi } f \Vert ^2 + \Vert \bar{A}_{\phi } f \Vert ^2 + c_0\tau \sum _{n\in (h\mathbb {Z})^d}\sum _{j,k=1}^d\bar{\mathcal {C}}_{jk}^{f,f} \\&\quad \ge C_{\text {low}} (\tau ^4 \Vert f\Vert ^2 + h^{-4}\sum \limits _{j=1}^{d}\Vert (D^j_s)^2 f\Vert ^2 + \tau ^2 h^{-2}\sum \limits _{j=1}^{d}\Vert D^j_s f\Vert ^2) , \end{aligned} \end{aligned}$$

which yields the claim of the Proposition. $\square$

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 $f := e^{\phi }u$ for which we seek to prove

$$\begin{aligned} \tau ^{\frac{3}{2}}\Vert f\Vert + \tau ^{\frac{1}{2}} \Vert h^{-1} D_s f\Vert + \tau ^{-\frac{1}{2}} \Vert h^{-2} D^2_s f\Vert \le C \Vert L_{\phi } f\Vert \end{aligned}$$

(and for which we note that the action of $D_s$ on $e^{\phi }u$ yields terms $D_s e^{\phi }$ that can be absorbed in the first term with $\Vert e^{\phi }u\Vert$). 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

$$\begin{aligned} \tau ^{\frac{3}{2}}\Vert f\Vert + \tau ^{\frac{1}{2}} \Vert h^{-1} D_s f\Vert + \tau ^{-\frac{1}{2}} \Vert h^{-2} D^2_s f\Vert \le C \Vert L_{\phi } f\Vert \end{aligned}$$

(23)

for the localized functions $f_k$ from Lemma 3.2. Indeed, assuming that the estimate (23) is proven for $f_k$, an application of Minkowski’s inequality and the error estimates from Lemma 3.2 yield

$$\begin{aligned} \begin{aligned}&\tau ^{\frac{3}{2}}\Vert f\Vert + \tau ^{\frac{1}{2}} \Vert h^{-1} D_s f\Vert + \tau ^{-\frac{1}{2}} \Vert h^{-2} D^2_s f\Vert \\&\quad \le \tau ^{\frac{3}{2}}\sum _{k} \Vert f_k \Vert + \tau ^{\frac{1}{2}} \sum _{k}\Vert h^{-1} D_s f_k\Vert + \tau ^{-\frac{1}{2}} \sum _{k} \Vert h^{-2} D^2_s f_k\Vert \\&\quad \le C \sum \limits _k \Vert L_{\phi } f_k\Vert \le C \Vert L_{\phi } f\Vert + C_{\text {loc}} \tau ^{\frac{1}{2}} \epsilon _0 \sum \limits _{j=1}^{d}\Vert h^{-1}D^j_s f\Vert \\&\qquad + C_{\text {loc}} (\tau \epsilon _0 +\tau ^{\frac{3}{2}} \epsilon _0+ \tau ^2 \tau ^{\frac{1}{2}} h \epsilon _0) \Vert f\Vert . \end{aligned} \end{aligned}$$

(24)

Now choosing

$$\begin{aligned} \epsilon _0 \le \frac{1}{10C_{\text {loc}}} \end{aligned}$$

(25)

and recalling that $\tau h \le \delta _0$ for some $\delta _0 \in (0,1)$, we may absorb the contribution on the right hand side of (24) into its left hand side (in particular we note that $\tau ^2 \tau ^{\frac{1}{2}} h \le \delta _0 \tau ^{\frac{3}{2}}$ by our assumptions on the relation between $\tau$ and h). This then yields the estimate (23). The estimate (4) follows from this by possibly choosing the constants in the terms which involve derivatives on the left hand side of (23) smaller, carrying out the product rule and absorbing the $L^2$ errors into the $L^2$ contribution on the left hand side of (23).

Step 2.:

Proof of (23) for the localized functions. It thus suffices to prove (23) for $f= f_k$. To this end, we observe that for $f_k = f \psi _k$ with $\text {supp}(f) \subset B_2 \setminus B_{1/2}$, $\psi _k$ as in Lemma 3.2, $c_0 \in (0,1)$ as in Proposition 3.5 and $\tau _0>1$ such that $\tau _0 c_0 \ge 1$,

$$\begin{aligned} \begin{aligned} \tau \Vert L_{\phi } f_k \Vert ^2&= \tau \Vert S_{\phi } f_k \Vert ^2 + \tau \Vert A_{\phi } f_k \Vert ^2 + \tau (f_k ,[S_{\phi }, A_{\phi } ] f_k )\\&\ge \Vert S_{\phi } f_k \Vert ^2 + \Vert A_{\phi } f_k \Vert ^2 + \tau c_0 (f_k ,[S_{\phi }, A_{\phi } ] f_k )\\&= \Vert \widetilde{S}_{\phi } f_k \Vert ^2 + \Vert \widetilde{A}_{\phi } f_k \Vert ^2 + \tau c_0 \sum _{n \in (h \mathbb {Z})^d}\sum \limits _{j_1, j_2=1}^{d}\mathcal {C}_{j_1j_2}^{f_k,f_k}- E_1, \end{aligned} \end{aligned}$$

where by Lemma 3.3, taking into account that $0<\delta _0<1<\tau$, we get

$$\begin{aligned} |E_1| \le C (\delta _0^2\tau ^4+\tau ^{3}) \Vert f_k\Vert ^2 + C \tau \sum \limits _{j=1}^d \Vert h^{-1}D^j_s f_k \Vert ^2. \end{aligned}$$

(26)

Choosing $\delta _0>0$ such that $C \delta _0 \le \frac{C_{\text {low}}}{10}$, where $C_{\text {low}}$ is the constant from Proposition 3.5, we will be able to treat the contributions in (26) as error contributions in the following arguments. Exploiting the bounds from Lemma 3.4, we may further estimate

$$\begin{aligned}&\Vert \widetilde{S}_{\phi } f_k \Vert ^2 + \Vert \widetilde{A}_{\phi } f_k \Vert ^2 +\tau c_0 \sum _{n\in (h\mathbb {Z})^d}\sum \limits _{j_1, j_2=1}^{d}\mathcal {C}_{j_1j_2}^{f_k,f_k} - E_1 \\&\quad \ge \Vert \bar{S}_{\phi } f_k \Vert ^2 +\Vert \bar{A}_{\phi } f_k \Vert ^2 + \tau c_0 \sum _{n\in (h\mathbb {Z})^d}\sum \limits _{j_1, j_2=1}^{d}\bar{\mathcal {C}}_{j_1j_2}^{f_k,f_k}- E_1 - E_2 , \end{aligned}$$

where by the estimates from Lemma 3.4

$$\begin{aligned} |E_2| \le C(\tau ^{3}\epsilon _0^{-2} + \tau ^{\frac{7}{2}}\epsilon _0^{-1}) \Vert f_k \Vert ^2 + C(\tau ^{\frac{3}{2}} \epsilon _0^{-1} + \tau \epsilon _0^{-2})\sum \limits _{j=1}^d \Vert h^{-1}D^j_sf_k\Vert ^2. \end{aligned}$$

(27)

Finally, invoking Proposition 3.5, we infer that

$$\begin{aligned} \begin{aligned}&\Vert \bar{S}_{\phi } f_k \Vert ^2 + \Vert \bar{A}_{\phi } f_k \Vert ^2 + \tau c_0 \sum _{n\in (h\mathbb {Z})^d}\sum \limits _{j_1, j_2=1}^{d}\bar{\mathcal {C}}_{j_1j_2}^{f_k,f_k} - E_1 - E_2\\&\quad \ge C_{\text {low}}\Big (\tau ^4 \Vert f_k \Vert ^2 + h^{-4}\sum _{j=1}^d\Vert (D^j_s)^2 f_k \Vert ^2 + \tau ^2 h^{-2}\sum _{j=1}^d\Vert D^j_s f_k\Vert ^2\Big ) - E_1 - E_2. \end{aligned} \end{aligned}$$

(28)

Recalling the condition for $\epsilon _0>0$ from (25), we now choose $\epsilon _0 = \frac{1}{20 C_{\text {loc}}}$ and fix $\tau _0>1$ so large and $\delta _0>0$ so small that

$$\begin{aligned}&C\max \{\tau _0^3,\delta _0^2 \tau _0^4, \tau _0^{\frac{7}{2}}\epsilon _0^{-1}, \tau _0^{3} \epsilon _0^{-2}\} \le C_{\text {low}}\frac{\tau _0^4}{10}\quad \text{ and } \\&C\max \{\tau _0^{\frac{3}{2}} \epsilon _0^{-1}, \tau _0 \epsilon _0^{-2} \}\le C_{\text {low}}\frac{\tau _0^2}{10}. \end{aligned}$$

Further, we choose the value of $h_0> 0$ so small that $\delta _0 h_0^{-1} \ge 100 \tau _0> 100$, which in particular implies that for all $h\in (0,h_0)$ the interval $(\tau _0,\delta _0 h^{-1})$ is non-empty. With these choices, it follows that for $\tau \in (\tau _0,\delta _0 h^{-1})$, we may absorb the error contributions $E_1$ and $E_2$ from (26) and (27) into the positive right hand side contributions in (28). Therefore, we obtain that

$$\begin{aligned} \tau \Vert L_{\phi } f_k\Vert ^2 \ge \frac{C_{\text {low}}}{2} \Big ( \tau ^4 \Vert f_k \Vert ^2 + h^{-4}\sum _{j=1}^d\Vert (D^j_s)^2 f_k \Vert ^2 + \tau ^2 h^{-2}\sum _{j=1}^d\Vert D^j_s f_k \Vert ^2 \Big ). \end{aligned}$$

Dividing by $\tau >\tau _0$ implies the desired result. $\square$

4 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 $\tau ^*$ such that $(c_1 + c_2) \tau ^* =\log \frac{\Vert u\Vert _{L^2(B_2)}}{\Vert u\Vert _{L^2(B_{1/2})}}$. It is easy to check that with this value of $\tau ^*$ it holds

$$\begin{aligned} e^{c_1\tau ^*} \Vert u\Vert _{L^2(B_{1/2})} = e^{-c_2 \tau ^*}\Vert u\Vert _{L^2(B_2)}. \end{aligned}$$

Given u satisfying (2), we can assume that $\tau _0<\tau ^*$, and we are in one of the following two cases:

• If $\tau ^* \in (\tau _0, \delta _0 h^{-1})$, then plugging this into the right hand side of (2) yields, for $\tau =\tau ^*$, that

  $$\begin{aligned} C(e^{c_1\tau } \Vert u\Vert _{L^2(B_{1/2})} + e^{-c_2\tau }\Vert u\Vert _{L^2(B_2)}) = 2 C\Vert u\Vert ^{\frac{c_2}{c_1 + c_2}}_{L^2(B_{1/2})}\Vert u\Vert _{L^2(B_2)}^{\frac{c_1}{c_1 + c_2}}. \end{aligned}$$

• If $\tau ^* \notin (\tau _0, \delta _0 h^{-1})$, we observe that $\tau< \delta _0 h^{-1}<\tau ^*$. We hence obtain that

  $$\begin{aligned} e^{c_1 \frac{\delta _0}{2} h^{-1}}\Vert u\Vert _{L^2(B_{1/2})}\le e^{c_1 \tau ^*}\Vert u\Vert _{L^2(B_{1/2})}= e^{-c_2 \tau ^*}\Vert u\Vert _{L^2(B_{2})} \le e^{- c_2 \frac{\delta _0}{2} h^{-1}}\Vert u\Vert _{L^2(B_2)}.\nonumber \\ \end{aligned}$$

  (29)

  Thus, since (2) holds for all $\tau \in (\tau _0, \delta _0 h^{-1})$ and by using (29), we have

  $$\begin{aligned} \Vert u\Vert _{L^2(B_1)} \le C(e^{c_1 \frac{\delta _0}{2} h^{-1}} \Vert u\Vert _{L^2(B_{1/2})} + e^{-c_2 \frac{\delta _0}{2} h^{-1} } \Vert u\Vert _{L^2(B_2)})\le 2\cdot e^{-c_2 \frac{\delta _0}{2} h^{-1} }\Vert u\Vert _{L^2(B_2)}. \end{aligned}$$

Combining both cases implies () with $\alpha = \frac{c_2}{ c_1 + c_2}$ and $c_0 = \frac{\delta _0}{2}c_2$. $\square$

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 $a_{jk}: (h\mathbb {Z})^d \rightarrow \mathbb {R}^{d \times d}$ be symmetric, bounded and uniformly elliptic with ellipticity constant $\lambda \in (0,1)$, i.e. assume that for all $\xi \in \mathbb {R}^{d}\setminus \{0\}$ we have

$$\begin{aligned} \lambda |\xi |^2 \le \sum _{i,j=1}^d \xi _i a_{ij} \xi _j \le \lambda ^{-1} |\xi |^2. \end{aligned}$$

Let $V: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ be uniformly bounded in h and $B:(h\mathbb {Z})^d \rightarrow \mathbb {R}^d$ be a uniformly bounded tensor field. Denote $B:=(B_j)_{j=1}^d$. Let $u: (h \mathbb {Z})^d \rightarrow \mathbb {R}$ be a weak solution of

$$\begin{aligned} \left( h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)D_{+,j}^ h D_{-,k}^h + h^{-1}\sum \limits _{j=1}^d B_j(n)D_{+,j}^h + V(n)\right) u(n) =0, \end{aligned}$$

in the sense that $u \in H^1_{{\text {loc}},h}((h\mathbb {Z})^d)$ and for all $v\in H^1((h\mathbb {Z})^d)$ with $\text {supp}(v)$ bounded, we have

$$\begin{aligned}&\sum \limits _{n \in (h \mathbb {Z})^d}\Big [ h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))(v(n + h e_k) - v(n)) \\&\quad - h^{-1}\sum \limits _{j=1}^d B_j(n)(u(n+h e_j)- u(n)) v(n) - V(n) u(n) v(n) \Big ] =0. \end{aligned}$$

Let $0<10h<r_1< r_1 + 100 h < r_2$. Then there exists a constant $C>1$ depending on $r_1, r_2, \Vert V\Vert _{L^{\infty }}, \Vert B\Vert _{L^{\infty }}$ such that

$$\begin{aligned} \sum \limits _{j=1}^d\Vert h^{-1}( u(\cdot +h e_j)- u(\cdot ))\Vert _{L^2(B_{r_1})}^2 \le C \Vert u\Vert _{L^2(B_{r_2})}^2 . \end{aligned}$$

Here $H^{1}_{{\text {loc}},h}((h \mathbb {Z})^d)$ and $H^{1}((h \mathbb {Z})^d)$ 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 $0 < h \le h_0$ for $h_0 \ll 1$ sufficiently small.

Let $\eta : (h \mathbb {Z})^d \rightarrow \mathbb {R}$ be a cut-off function which is equal to one on $B_1$ and vanishes outside of $B_2$. The function $(u \eta ^2)(n)$ is then an admissible test function in the Schrödinger equation for u. Inserting this, we obtain

$$\begin{aligned}&0 = \sum \limits _{n \in (h \mathbb {Z})^d}\Big [ h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))((u\eta ^2)(n + h e_k) - (u \eta ^2)(n))\\&\qquad - h^{-1}\sum \limits _{j=1}^d B_j(n)\big (u(n+h e_j)- u(n)\big ) (u \eta ^2)(n) - V(n) u(n) (u \eta ^2)(n) \Big ]. \end{aligned}$$

We first deal with the leading, second order contribution. Noting that

$$\begin{aligned} (u\eta ^2)(n + h e_k) - (u \eta ^2)(n) = \big (u(n + h e_k)- u(n)\big ) \eta ^2(n) + u(n + h e_k )\big (\eta ^2 (n + h e_k)-\eta ^2(n)\big ), \end{aligned}$$

we obtain that

$$\begin{aligned} \begin{aligned}&\sum \limits _{n \in (h \mathbb {Z})^d} h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))((u\eta ^2)(n + h e_k) - (u \eta ^2)(n))\\&\quad = \sum \limits _{n \in (h \mathbb {Z})^d} h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))(u(n + h e_k) - u(n))\eta ^2(n)\\&\qquad + \sum \limits _{n \in (h \mathbb {Z})^d} h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))u(n + h e_k )(\eta ^2 (n + h e_k)-\eta ^2(n)). \end{aligned} \end{aligned}$$

(30)

By virtue of the ellipticity of $a_{jk}$ we further infer that

$$\begin{aligned} \begin{aligned}&\sum \limits _{n \in (h \mathbb {Z})^d}h^{-2}\sum \limits _{j,k=1}^{d} a_{jk}(n)(u(n + h e_j)- u(n))(u(n + h e_k) - u(n))\eta ^2(n)\\&\quad \ge \lambda \sum \limits _{n \in (h \mathbb {Z})^d}h^{-2}\sum \limits _{j=1}^{d} (u(n + h e_j)- u(n))^2\eta ^2(n) \\&\quad = \lambda \sum \limits _{j=1}^{d} \Vert h^{-1}(u(\cdot + h e_j)- u(\cdot )) \eta \Vert ^2_{L^2((h \mathbb {Z})^d)}. \end{aligned} \end{aligned}$$

For the second contribution on the right hand side of (30), we rewrite $\eta ^2(n+he_k)-\eta ^2(n) = (\eta (n + he_k)-\eta (n))(\eta (n + he_k)+ \eta (n))$ and estimate from above:

$$\begin{aligned}&h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))u(n + h e_k )(\eta ^2 (n + h e_k)-\eta ^2(n))\nonumber \\&= h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))u(n + h e_k )(\eta (n + he_k)-\eta (n))(\eta (n + he_k)+ \eta (n))\nonumber \\&= h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))\eta (n) u(n + h e_k )(\eta (n + he_k)-\eta (n))\nonumber \\&\quad + h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))\eta (n + he_k) u(n + h e_k )(\eta (n + he_k)-\eta (n))\nonumber \\&= 2h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))\eta (n) u(n + h e_k )(\eta (n + he_k)-\eta (n))\nonumber \\&\quad + h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))(\eta (n+ h e_k)- \eta (n )) u(n + h e_k )(\eta (n + he_k)-\eta (n))\nonumber \\&\le h^{-2} a_{jk}(n)(u(n + h e_j)- u(n))^2 \eta ^2(n) + h^{-2} a_{jk}(n) u^2(n + h e_k )(\eta (n + he_k)-\eta (n))^2\nonumber \\&\quad + C h^{-2} a_{jk}(n)(u^2(n + h e_j) +u^2(n)+ u^2(n + h e_k )) (\eta (n + he_k)-\eta (n))^2 . \end{aligned}$$

(31)

Noting that $a_{ij} \le \frac{1}{2}\lambda ^{-1}$ (this follows from the ellipticity condition when choosing appropriate $\xi$) we obtain that

$$\begin{aligned} \begin{aligned}&\sum \limits _{n \in (h \mathbb {Z})^d} h^{-2} \sum _{j,k=1}^n a_{jk}(n)(u(n + h e_j)- u(n))u(n + h e_k )(\eta ^2 (n + h e_k)-\eta ^2(n))\\&\le \sum \limits _{n \in (h \mathbb {Z})^d} \dfrac{1}{2}\lambda ^{-1} \sum _{j=1}^d ( \eta ^{2}(n)h^{-2}(u(n + h e_j)-u(n))^2) \\&\qquad + C_{\lambda } \Vert u\Vert _{L^2(B_2)}^2 \sup \limits _{k}|h^{-1}(\eta (n+ h e_k)- \eta (n))|^2. \end{aligned} \end{aligned}$$

Combining this with the bounds for $B_j$ and V, we obtain

$$\begin{aligned} \begin{aligned}&\lambda \sum \limits _{j=1}^{d} \Vert h^{-1}(u(\cdot + h e_j)- u(\cdot )) \eta \Vert ^2_{L^2((h \mathbb {Z})^d)}\le \frac{\lambda ^{-1}}{2}\sum \limits _{j=1}^{d} \Vert h^{-1}(u(\cdot + h e_j)- u(\cdot )) \eta \Vert ^2_{L^2((h \mathbb {Z})^d)} \\&\quad + C_{\lambda }\sup \limits _{k}\Vert h^{-1}(\eta (\cdot + h e_k)- \eta (\cdot ))\Vert _{L^{\infty }((h\mathbb {Z})^d)}^2 \Vert u\Vert _{L^2(B_2)}^2 \\&\quad + \Vert V\Vert _{L^{\infty }(B_2)} \Vert u\Vert _{L^2(B_2)}^2 + \Vert B\Vert _{L^{\infty }(B_2)} \Vert h^{-1}(u(\cdot + h e_j) - u(\cdot )) \eta \Vert _{L^2((h \mathbb {Z})^d)}\Vert u\Vert _{L^2(B_2)}. \end{aligned} \end{aligned}$$

(32)

Here the first contribution in (32) originates from the first right hand side contribution in (31). We may absorb it from the right hand side of (32) into the left hand side of (32). Using Young’s inequality for the contribution

$$\begin{aligned} \begin{aligned}&\Vert B\Vert _{L^{\infty }(B_2)} \Vert h^{-1}(u(\cdot + h e_j) - u(\cdot )) \eta \Vert _{L^2((h \mathbb {Z})^d)} \Vert u\Vert _{L^2(B_2)} \\&\le \frac{\lambda }{4}\Vert h^{-1}(u(\cdot + h e_j) - u(\cdot )) \eta \Vert ^2_{L^2((h \mathbb {Z})^d)}+ C_{\lambda }\Vert B\Vert ^2_{L^{\infty }(B_2)} \Vert u\Vert _{L^2(B_2)}^2 \end{aligned} \end{aligned}$$

allows us to also absorb the gradient term in this contribution into the left hand side of (32). Due to the bounds on $\eta$, this concludes the proof of the Caccioppoli estimate. $\square$

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 $u: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ such that $P_h u(n) = 0$ for all $n \in B_4$. Fix $\varepsilon >0$ to be small enough and assume that $h_0>0$ is sufficiently small. We consider the function $w(n)=\theta (n)u(n)$, with $0\le \theta (x)\le 1$ a $C^{\infty }(\mathbb {R}^d)$ cut-off function defined as

$$\begin{aligned} \theta (x)= {\left\{ \begin{array}{ll} 0, \quad &{}x\in B_{\frac{1}{4}+\varepsilon +h}\cup B_{2-\varepsilon -h}^c\\ 1, \quad &{}x\in B_{\frac{3}{2}+h}\setminus B_{\frac{1}{2}-h}. \end{array}\right. } \end{aligned}$$

Using the equation for u, we then write

$$\begin{aligned}&h^{-2}\Delta _d w(n)=\sum _{j=1}^d\big (\theta (n+he_j)u (n+he_j)+\theta (n-he_j)u(n-he_j)-2\theta (n)u(n)\big )h^{-2}\\&=\theta (n)h^{-2}\Delta _du(n)+\sum _{j=1}^d\big ((\theta (n+he_j)-\theta (n))u(n+he_j)\\&\qquad +(\theta (n-he_j)-\theta (n))u(n-he_j)\big )h^{-2}\\&=\theta (n)h^{-2}\Delta _du(n) + \sum _{j=1}^d\Big ((\theta (n+he_j)-\theta (n))(u(n+he_j)-u(n-he_j))h^{-2}\\&\qquad +(\theta (n-he_j)-\theta (n))u(n-he_j)+(\theta (n+he_j)-\theta (n))u(n-he_j)\Big )h^{-2}\\&= \theta (n) V(n) u(n) + \theta (n) h^{-1}\sum \limits _{j=1}^{d} B_j(n) D^h_{+,j} u(n)\\&\quad + \sum _{j=1}^d\big ((\theta (n+he_j)-\theta (n))(u(n+he_j)-u(n-he_j))\big )h^{-2}\\&\qquad +h^{-2}\Delta _d\theta (n)\sum _{j=1}^d u(n-he_j)\\&= V(n) w(n) + h^{-1}\sum \limits _{j=1}^{d} B_j(n) D^h_{+,j} w(n)\\&\quad - h^{-1} \sum \limits _{j=1}^{d} B_j(n)(\theta (n+ h e_j) - \theta (n))u(n+he_j)) \\&\quad + \sum _{j=1}^d\big ((\theta (n+he_j)-\theta (n))(u(n+he_j)-u(n-he_j))\big )h^{-2}\\&\qquad +h^{-2}\Delta _d\theta (n)\sum _{j=1}^d u(n-he_j)\\&=: V(n) w(n) + h^{-1}\sum \limits _{j=1}^{d} B_j(n) D^h_{+,j} w(n) + T_{d,1} u(n) + T_{d,2}u(n) + T_{d,3} u(n). \end{aligned}$$

Applying the Carleman estimate (4) from Theorem 3, using Remark 1.1 and the triangle inequality, we obtain

$$\begin{aligned} \tau ^{\frac{3}{2}}\Vert e^{\phi } w\Vert + \tau ^{\frac{1}{2}} h^{-1} \Vert e^{\phi } D_+^hw\Vert \le C_{\text {Carl}}\Big ( \Vert e^{\phi } V w\Vert + h^{-1} \Vert e^{\phi }B D^h_+ w\Vert + \sum \limits _{\ell =1}^3\Vert e^{\phi } T_{d,\ell } u\Vert \Big ). \end{aligned}$$

(33)

Now choosing $\tau \ge 2C_{\text {Carl}}\max \{1,\Vert V\Vert _{L^{\infty }}^{\frac{2}{3}}, \Vert B\Vert _{L^{\infty }}^{2}\}$ allows us to absorb the first two contributions from the right hand side of (33) into the left hand side of (33). We thus obtain the bound

$$\begin{aligned} \tau ^{\frac{3}{2}}\Vert e^{\phi } w\Vert + \tau ^{\frac{1}{2}} h^{-1} \Vert e^{\phi } D_+^h w\Vert \le 2C_{\text {Carl}} \sum \limits _{\ell =1}^3\Vert e^{\phi } T_{d,\ell } u\Vert . \end{aligned}$$

(34)

We next deal with the errors on the right hand side of (34). On the one hand, since differences in $\theta$ are contained in a 2h-neighbourhood of the support of $\nabla \theta$, we have for $j\in \{1,3\}$

$$\begin{aligned} \Vert e^{\phi } T_{d,j} u\Vert ^2 \le C\big (\Vert e^{\phi }u\Vert ^2_{B_{\frac{1}{2}}\setminus B_{\frac{1}{4}+\varepsilon }}+\Vert e^{\phi }u\Vert ^2_{B_{2-\varepsilon }\setminus B_{\frac{3}{2}}}\big )\le C\big (e^{2\tau \varphi (1/4+\varepsilon )}\Vert u\Vert ^2_{B_{\frac{1}{2}}}+e^{2\tau \varphi (3/2)}\Vert u\Vert ^2_{B_{2}}\big ). \end{aligned}$$

On the other hand, for $T_{d,2}$

$$\begin{aligned}&\Vert e^{\phi } T_{d,j} u\Vert ^2 \le C \big (\Vert e^{\phi } D_su\Vert _{B_{2-\varepsilon }\setminus B_{\frac{3}{2}}}^2+\Vert e^{\phi }D_s u\Vert _{B_{\frac{1}{2}}\setminus B_{{\frac{1}{4}}+\varepsilon }}^2\big )\\&\qquad \le C \big (e^{2\tau \varphi (3/2)} \Vert u\Vert _{B_2}^2+e^{2\tau \varphi (1/4+\varepsilon )} \Vert u\Vert _{B_{\frac{1}{2}}}^2\big ), \end{aligned}$$

where we used the Caccioppoli estimate from Lemma 4.1.

Moreover, since $w\equiv u$ in $B_{3/2}\setminus B_{1/2}$, we have

$$\begin{aligned} \tau ^3 \Vert e^{\phi } w \Vert ^2\ge \tau ^3\Vert e^{\phi }u\Vert _{B_1\setminus B_{\frac{1}{2}}}^2\ge \tau ^3e^{2\tau \varphi (1)}\Vert u\Vert _{B_1\setminus B_{\frac{1}{2}}}^2. \end{aligned}$$

In view of the above, we get

$$\begin{aligned} \tau ^3e^{2\tau \varphi (1)}\Vert u\Vert _{B_1\setminus B_{\frac{1}{2}}}^2\le C \big (e^{2\tau \varphi (3/2)} \Vert u\Vert _{B_2}^2+e^{2\tau \varphi (1/4+\varepsilon )} \Vert u\Vert _{B_{\frac{1}{2}}}^2\big ), \end{aligned}$$

and since $\varphi$ is decreasing,

$$\begin{aligned} \Vert u\Vert _{B_1\setminus B_{\frac{1}{2}}}^2&\le C \big (\tau ^{-3}e^{2\tau \varphi (3/2)-2\tau \varphi (1)} \Vert u\Vert _{B_2}^2+\tau ^{-3}e^{2\tau \varphi (1/4)-2\tau \varphi (1)} \Vert u\Vert _{B_{\frac{1}{2}}}^2\big )\\&\le C\big (e^{-2c_2 \tau } \Vert u\Vert _{B_{2}}^2+e^{2c_1\tau } \Vert u\Vert _{B_{\frac{1}{2}}}^2\big ) \end{aligned}$$

for some constants $c_1, c_2>0$ with $c_1:=|\varphi (3/2)- \varphi (1)|$ and $c_2:=\varphi (1/4)-\varphi (1)>0$ (for which we choose the constant $c_{ps}>0$ in Theorem 3 and Lemma 3.1 sufficiently small). Since further trivially $\Vert u\Vert _{B_{\frac{1}{2}}}^2\le e^{2c_1\tau } \Vert u\Vert _{B_{\frac{1}{2}}}^2$, this concludes the proof. $\square$

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 (33) to (34)), we note that if V and B only satisfy the bounds

$$\begin{aligned} \Vert V\Vert _{L^{\infty }(B_{4})} \le \mu _0 h^{-\frac{3}{2}},\qquad \Vert B\Vert _{L^{\infty }(B_4)} \le \mu _0 h^{-\frac{1}{2}}, \end{aligned}$$

with $\mu _0 \le \frac{C_{Carl }}{10} \delta _0$, we can deduce that for some constants $\tilde{c}_1, \tilde{c}_2>0$ (independent of h)

$$\begin{aligned} \Vert u\Vert _{L(B_1)} \le C (e^{ \tilde{c}_1 h^{-1}} \Vert u\Vert _{L^2(B_{\frac{1}{2}})} + e^{- \tilde{c}_2 h^{-1}} \Vert u\Vert _{L^2(B_2)}). \end{aligned}$$

We also remark that while yielding quantitative propagation of smallness type estimates, as expected these estimates do not pass to the limit $h \rightarrow 0$. 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).

Remark 4.3

Let us further comment on possible extensions of our three balls inequalities: While we have formulated a three balls theorem with three concentric balls with ratio 1 : 2 : 4 (see also the remarks on scaling in the next section), the arguments used above are robust enough to deduce analogous results for balls of more general ratios $r_1:1:r_2$ for $0<r_1<1<r_2$ as long as $r_1, r_2$ are quantities “on continuum scales”, i.e. $r_1$, $r_2-r_1$ are not comparable to the lattice scales but are “substantially larger” than the lattice scales $h_0$. In this “continuum” case, all the arguments from above essentially persist with minor technical changes (e.g. different choices of supports for the functions f, $\psi _k, \theta$ etc) with constants depending on the choice of $r_1,r_2$ and $r_2-r_1$. Moreover, these estimates degenerate if the radii $r_1$, $r_2$ are such that either $r_1$ or $r_2-r_1$ approaches the order of the lattice spacing $h_0$.

5 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. We shall use the notation $\Delta _{d,h} =h^{-2}\Delta _d$.

Lemma 5.1

Let $u:B_4 \rightarrow \mathbb {R}$ be such that $\Delta _{d,h} u = 0$ in $B_R \subset (h\mathbb {Z})^d$. Then, for any $m \in \mathbb {N}$ such that $h m \le 2$, we also have $\Delta _{d,mh} u =0$ in $B_{R/m} \subset (mh \mathbb {Z})^d$ (i.e. with respect to the lattice $(mh \mathbb {Z})^d$).

Proof

We prove the statement inductively in m. For the case $m=2$ we have to show that

$$\begin{aligned} \sum \limits _{j=1}^d( u(x + h e_j) + u(x - he_j)-2u(x)) =0\quad \text{ for } \quad x \in (h \mathbb {Z})^d \end{aligned}$$

implies that

$$\begin{aligned} \sum \limits _{j=1}^d( u(x + 2h e_j) + u(x - 2he_j)-2u(x)) =0 \quad \text{ for } \quad x \in ( 2 h \mathbb {Z})^d. \end{aligned}$$

In order to observe this, we note that

$$\begin{aligned} u(x + 2h e_j) + u(x - 2he_j)-2u(x)&= (u(x + 2h e_j) + u(x) -2 u(x+ he_j)) \\&\quad + 2(u(x+he_j)+ u(x-he_j)-2 u(x))\\&\quad + (u(x-2h e_j) + u(x) - 2u(x-he_j)). \end{aligned}$$

Summing and noting that the corresponding contributions in the brackets yield the Laplacian on $(h\mathbb {Z})^d$ implies the claim for $m=2$.

Assuming the induction hypothesis for any m, i.e.,

$$\begin{aligned} \sum \limits _{j=1}^d( u(x + mh e_j) + u(x - mhe_j)-2u(x)) =0 \quad \text{ for } \quad x \in ( m h \mathbb {Z})^d, \end{aligned}$$

we prove the statement for $m+1$. We have

$$\begin{aligned}&u(x + (m+1)h e_j) + u(x - (m+1)he_j)-2u(x)\\&\,\, = (u(x + (m+1)h e_j) + u(x+(m-1)he_j)-2 u(x+ mhe_j))\\&\quad -\big (u(x+(m-1)he_j)+u(x-(m-1)he_j) -2u(x)\big ) \\&\qquad + 2\big (u(x+mhe_j)+ u(x-mhe_j)-2 u(x)\big )\\&\quad + (u(x-(m+1)h e_j) + u(x-(m-1)he_j) - 2u(x-mhe_j)). \end{aligned}$$

The conclusion follows from the cases $m=1$ (after translation) and the inductive steps for m and $m-1$. $\square$

Using the previous auxiliary result, we may infer rescaled versions of Theorem 2:

Corollary 5.2

Let $u: (h\mathbb {Z})^d \rightarrow \mathbb {R}$ be such that $\Delta _{d,h} u = 0$ in $B_R$. Assume that $u: (m^{-1} h \mathbb {Z}) \rightarrow \mathbb {R}$ is also such that $\Delta _{d,m^{-1}h} u = 0$. Then there exist $\alpha \in (0,1)$, $c_0>0$ $h_0 >0$ and $C>1$ (independent of u) such that for $h\in (0,h_0)$

$$\begin{aligned} \Vert u\Vert _{L^2(B_{m^{-1}})} \le C(\Vert u\Vert _{L^2(B_{m^{-1}/2})}^{\alpha }\Vert u\Vert _{L^2(B_{2m^{-1}})}^{1-\alpha } + 2^{ -c_0 h^{-1}}\Vert u\Vert _{L^2(B_{2m^{-1}})}). \end{aligned}$$

Proof

We consider the function $u_m(x):= u(m^{-1}x)$ with $x \in (h\mathbb {Z})^d$. By the considerations from Lemma 5.1 this is also harmonic on $(h\mathbb {Z})^d$. 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.