Uniform Strichartz estimates on the lattice

In this paper, we investigate Strichartz estimates for discrete linear Schr\"odinger and discrete linear Klein-Gordon equations on a lattice $h\mathbb{Z}^d$ with $h>0$, where $h$ is the distance between two adjacent lattice points. As for fixed $h>0$, Strichartz estimates for discrete Schr\"odinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis \cite{SK2005}. Our main result shows that such inequalities hold uniformly in $h\in(0,1]$ with additional fractional derivatives on the right hand side. As an application, we obtain local well-posedness of a discrete nonlinear Schr\"odinger equation with a priori bounds independent of $h$. The theorems and the harmonic analysis tools developed in this paper would be useful in the study of the continuum limit $h\to 0$ for discrete models, including our forthcoming work \cite{HY} where strong convergence for a discrete nonlinear Schr\"odinger equation is addressed.


Introduction
We consider a discrete linear Schrödinger equation We here define the discrete Laplacian by where {e j } d j=1 is the standard basis. In other words, we consider the harmonic oscillators interacting only with their nearest neighbors.
Discrete Schrödinger and discrete Klein-Gordon equations have been extensively studied in various aspects in the physics literature. Discrete Schrödinger equations describe periodic optical structures created by coupled identical single-mode linear waveguides [5,13,16]. They are also closely related to nonlinear dynamics of the Bose-Einstein condensates in optical lattices [1,2].
Meanwhile, discrete Klein-Gordon equations describe Fluxon dynamics in one dimension parallel array of Josephson Junctions [17], and also arises as a model for local denaturation of DNA [14]. In [12], the equations of motion of the model of DNA dynamics are reduced to the nonlocal discrete nonlinear Schrödinger equations, which shown rigorously to converges to fractional Schrödinger equations in continuum limit by [11]. For more informations on survey and general theory of discrete equations, see [9,10]. Our focus is particularly on developing analytic tools to explore the continuum limits h → 0 of the above equations. Precisely, we aim to establish inequalities quantitatively measuring decay properties of solutions, namely Strichartz estimates, but in the meantime, we also want them to hold uniformly in h ∈ (0, 1]. For 1 ≤ p < ∞ (respectively, p = ∞), the function space L p h consists of all complex-valued functions on Z d h satisfying If f ∈ L p h , then its L p h -norm is defined by Putting the constant h d p in the norm · L p h is natural in consideration of the continuum limit h → 0, since for f ∈ L p (R d ), By the above definitions, the previously-known dispersion and Strichartz estimates for the discrete Schrödinger equation (1.1) are written as follows.
Theorem 1.1 (Stefanov-Kevrekidis [15]). (i) (dispersion estimate) (ii) (Strichartz estimates) We say that (q, r) is discrete Schrödinger-admissible if Remark 1. (i) The |t| −d/3 -decay in the dispersion estimate (1.4) is weaker than that for the continuum equation due to the lattice resonance. Indeed, a solution to a discrete Schrödinger equation can be written as a certain oscillatory integral (see (1.12)), but its phase function may have degenerate Hessian.
Thus, it only allows a weaker dispersion estimate and Strichartz estimates with different admissibility conditions.
(ii) The inequalities (1.4), (1.6) and (1.7) cannot be directly applied to the continuum limit problems, because the constants blow up as h → 0 except the trivial case q =q = ∞.
The main observation of this article is that the h-dependence in (1.6) and (1.7) can be removed paying fractional derivatives on the right hand side, which compensates the lattice resonance. We also prove that putting such additional derivatives is necessary for uniform boundedness.
As for a fractional derivative, we here adopt the definition as the Fourier multiplier of symbol |ξ| s , and we use the homogeneous and the inhomogeneous Sobolev norms defined by Using the Sobolev norm (1.8), our main theorem is stated as follows. For any discrete Schrödinger-admissible pairs (q, r) and (q,r) (satisfying (1.5)), there exists C > 0, independent of h, such that Moreover, these inequalities are optimal in the sense that the range of (q, r) cannot be extended and for fixed (q, r) the required derivative loss is essential, as long as h uniform estimates are concerned. For a precise statements, see Proposition A.1. Remark 2. (i) On R d , combining the Sobolev inequality and the Strichartz estimates in Keel-Tao [8], the following Sobolev-Strichartz estimates are available, Here, (q * , r * ) lies on the painted trapezoid in Figure 1. The Strichartz estimate (1.9) corresponds to the red line in Figure 1 in the "formal" limit h → 0.
(ii) In the Strichartz estimates (1.9), the admissible conditions (1.5) must be satisfied due to the weaker dispersion (1.4) for each h > 0. Thus, the presence of the derivative |∇| 1/q cannot be avoided in the connection to its formal continuum limit (1.11).
We prove the Strichartz estimates (1.9) (as well as (1.10)) separating the bad high frequency part from the good low frequency part. Indeed, by the lattice Fourier transform, the solution e it∆ h u 0 can be written as the following oscillatory integral Here, the phase function has degenerate Hessian if and only if ξ j = ± π 2h for some j (in Figure 2, they correspond to the dashed line). For the high frequency part where the low frequencies, i.e., 2π h [− 1 8 , 1 8 ] d , are smoothly truncated out, we reduce to the problem on Z d from that on Z d h by a simple scaling argument, and make use of the previously-known result (1.6) to get the desired bound. For the remaining low frequency part where the Hessian of the phase function is non-degenerate, we decompose the It is not difficult to show that for fixed h > 0 the equation (1.13) is globally well-posed in L 2 h , and that its solutions conserve the mass (see Proposition 6.1). It follows from the mass conservation law and the inequality u L ∞ h ≤ C h u L 2 h that solutions to (1.13) are bounded in L r h for all r ≥ 2. Nevertheless, their upper bounds may depend on h > 0.
As an application of Theorem 1.3, we prove that the higher L r h -norms of solutions are uniformly bounded in a time average sense. Precisely, we prove that if initial data are bounded uniformly in h ∈ (0, 1], then their solutions are bounded in the Strichartz norm where ∞ − denotes a preselected arbitrarily large number 1 . (ii) (Global-in-time uniform bound) Let I max be the maximal interval of uniform boundedness, that is, the largest interval such that (1.17) holds on any compact interval I ⊂ I max . If λ = 1 Remark 3. In spite of the presence of the derivative on the right hand side in Strichartz estimates (1.9) and (1.10), we can still recover the optimal local theory in the discrete setting.
Finally, applying the aforementioned strategy to the discrete Klein-Gordon equation (1.2), we prove the following.

Preliminaries
In this section, we briefly introduce the preliminary L p theory, the Fourier transform and some elementary inequalities on a lattice (see also Section 17 in [3]).
2.1. L p h spaces and basic inequalities.
where for a set A ∈ Z d h , |A| denotes the standard normalized counting measure on a lattice, i.e., |A| = h d x∈A 1. On a lattice Z d h , we define the inner product by and the convolution by In the following propositions, we collect some basic inequalities and the interpolation theorems, whose proofs are omitted here because they follow from the standard arguments. .

Fourier transform. For a rapidly decreasing function
Indeed, since we here consider functions on the lattice Z d h , the Fourier transform is defined in the opposite way to what is done for periodic functions as Fourier series (see Figure 3). We also note that the Fourier and the inverse Fourier transforms formally converge to those on the whole space R d in the continuum limit h → 0: The Fourier transform (respectively, its inversion) can be extended to a larger class of functions, that is, the dual space of rapidly decreasing functions (respectively, that of smooth functions) via the duality relation Moreover, we have: Then, we have where with an abuse of notation, ψ 1 denotes the function ψ 1 restricted to the frequency domain T d h .
We now define the Littlewood-Paley projection operator P N = P N ;h as a Fourier multiplier such that Note that unlike the usual definition of Littlewood-Paley projections on R d , ψ N is not supported on an annulus on the Fourier side, because the entire Frequency space is a cube [− π h , π h ] d . As an analogue of the classical theory on the whole space R d , the Littlewood-Paley projections satisfy the following boundedness property.
where the implicit constant is independent of N and h.
Proof. It follows from Young's inequality that Then, a simple integration by parts using e 2πix·ξ = 1 2πixj ∂ ξj e 2πix·ξ deduces that Therefore, we conclude that
where the implicit constant is independent of h.
Proof. Replacing f by 1 Then, we write which is identity on the support of ψ N and supported near N h . Hence, by Bernstein's inequality (Lemma 2.3), we prove that As a consequence, we derive the Sobolev inequality except the sharp exponent.

Calderon-Zygmund theory on a lattice
We consider convolution operators on a lattice of the form Such operators are of course bounded on L p h by Young's inequality, because there is no singular kernel on a lattice. However, getting a uniform-in-h bound is not so obvious. In this section, we show boundedness of convolution operators satisfying the hypotheses similar to those for Calderon-Zygmund operators in the formal limit h → 0, extending the Calderon-Zygmund theory on the Euclidean space R d Theorem 3.1 (Calderón-Zygmund). Suppose that for all h ∈ (0, 1], Then, for 1 < p < ∞, there exists C p > 0, independent of h ∈ (0, 1], such that We prove Theorem 3.1 following the standard argument for instance in [4], involving the dyadic maximal function. We fix h ∈ (0, 1]. For each dyadic number N ∈ 2 Z with N ≥ 1, let Q N = Q N ;h be the family of cubes, open on the right, whose vertices are adjacent points of the lattice hN Z d .
Given f ∈ L 1 h , averaging over each cube in Q N , we introduce the average function Note that Q 1 ≈ Z d h and E 1 f (x) = f (x). Next, we define the dyadic maximal function by Using this maximal function, we decompose the domain of a function.

Theorem 3.2 (Calderón-Zygmund decomposition).
Given non-negative f ∈ L 1 h and λ > 0, there exists a collection {Q k } k of disjoint dyadic cubes such that Proof. In order to construct the desired dyadic cubes, we claim that Note that for x ∈ Ω N , N is the smallest dyadic numbers such that E N f (x) > λ.
For (1), For (2), we use the decomposition (3.4) to get If Ω N = kN Q kN , then E N f (x) has the same value on Q kN , which is 1 It remains to show (3). By the definition of the sets Ω N , the average of f over Q k is greater than λ. Let 2Q k be the dyadic cube containing Q k whose sides are twice as long. Then, the average of f over 2Q k is at most λ. Therefore, we prove that Now we are ready to show Theorem 3.1.
Proof of Theorem 3.1. By the Plancherel theorem with the bound (3.1), T K h is bounded on L 2 h . Thus, it suffices to show that for arbitrary non-negative f ∈ L 1 h and λ > 0, h . Consequently by interpolation, (3.3) holds for 1 < p ≤ 2, and then for 2 < p < ∞ by duality.
To show (3.5), applying the Calderón-Zygmund decomposition (Theorem 3.2) to given f ∈ L 1 h and λ > 0, we obtain the collection of disjoint dyadic cubes {Q k } k with the desired properties, and then we decompose f into the good function g and the bad functions b k 's: For the good function, we observe from Theorem 3.2 that g(x) ≤ 2 d λ. Hence, it follows from L 2 h boundedness that For the bad function, by a trivial estimate, we have where 2Q k is the cube with the same center as Q k and twice the length. For the first term, by On the other hand, for the second term, we write (3.6) We now recall that each b k is supported on Q k and that its average is zero, i.e., h d b k = 0.
Moreover, by the triangle inequality, Hence, we have where y k is the center of Q k . Here, the property h d k b k = 0 is used in the second identity, and the assumption (3.2) is used in the last inequality. Therefore, going back to (3.6) and summing (3.7) in k, we prove that Finally, collecting all, we conclude that

Hörmander-Mikhlin theorem and its applications
In this section, we present the Hörmander-Mikhlin multiplier theorem on a lattice and its applications.

4.1.
Hörmander-Mikhlin theorem. Given a symbol function m = m h on T d h , we consider the Fourier multiplier operator T m defined by We show that this multiplier operator is uniformly bounded if the symbol satisfies the assumption completely analogous to that in the multiplier theorem on R d . for all multi-index |α| ≤ d + 2. Then, for 1 < p < ∞, there exists C p > 0, independent of h, such that Proof. Since |m(ξ)| is bounded, it suffices to show that the integral kernelm satisfies (3.2). We can naturally extend the kernelm on Z d h to a function on R d . By the Littlewood-Paley projections, we decompose

It is obvious that
On the other hand, by integration by parts (d + 2) times with e ix·ξ = 1 ixj ∂ ∂ξj e ix·ξ , one can show that Summing in N , we get the bound, Using this, we finally check where the third one follows considering the Riemann summation and B > 0 is independent of h > 0.
Our proof is based on the following randomization technique. Lemma 4.3 (Khinchine's inequality for scalars [6]). Let z 1 , · · · , z N be complex numbers, and let ǫ 1 , · · · , ǫ N ∈ {−1, 1} be independent random signs, drawn from {−1, 1} with the uniform distributions. Then for any 0 < p < ∞ Lemma 4.4 (Khinchine's inequality for functions on Z h ). Let f 1 , · · · , f N ∈ L p h for some 1 < p < ∞, and let ǫ 1 , · · · , ǫ N ∈ {−1, 1} be independent of random signs, drawn from {−1, 1} with the uniform distributions. Then we have Proof. For x ∈ Z d h we apply (4.3) to the sequences f 1 (x), · · · , f N (x) and then take L p h norms Proof of Theorem 4.2. We first prove the second inequality in (4.2). By monotone convergence, it suffices to prove it assuming that the summation runs over finitely many N . That is, we suffices to show that for fixed M ≪ 1, A key observation is that for arbitrary ǫ N ∈ {−1, 1} the multiplier K≤N ≤1 ǫ N ψ N obeys the assumption (4.1) in the Hormander-Mikhlin Theorem. Thus we have Taking expectations on both sides and applying Khinchine's inequality (4.4) we get the desired result. Now we prove the first inequality in (4.2). Note that Plugging f + g into above identity we obtain Then we have by duality We introduce a partition of unity {χ j } d j=1 on the unit sphere S d−1 such that χ j (ξ) ≡ 1 on {|ξ j | ≥ 3|ξ|} but χ j (ξ) ≡ 0 on {|ξ j | ≤ 1 3 |ξ|}, and define the projection operator Γ j by Γ j f (ξ) = χ j (ξ)f (ξ). By direct calculations again, one can show that χ j (ξ)|ξ|( e ihξ j −1 h ) −1 satisfies (4.1). As By the same way, one can show norm equivalence for inhomogeneous Sobolev norms (see (1.8)).
Proposition 4.5. For any 1 < p < ∞, we have We can also prove the relation between homogeneous and inhomogeneous norm by the same argument.

Endpoint Sobolev inequality.
We close this section deriving the endpoint Sobolev inequality, which improves Proposition 2.5, by the Littlewood-Paley inequality.
If q > 2 > p, by above two cases we have f L q h f Ẇ s,p h . By interpolating this with trivial estimate f L p h f L p h , we get the desired result.

1.3)
In this section, we show Strichartz estimates for discrete Schrödinger equations. Now that harmonic analysis tools are at hand, their proof is reduced to the proof of the following frequency localized estimates.
Proposition 5.1 (Frequency localized dispersive estimate for discrete Schrödinger equations). Let h ∈ (0, 1]. Then, for any dyadic number N ∈ 2 Z with N ≤ 1, we have Proof of Theorem 1.3, assuming Proposition 5.1. By Proposition 5.1 and the trivial inequality  that the frequency localized Strichartz estimate holds for all admissible pairs (q, r) (see (1.5)). Letψ be a smooth function such thatψ ≡ 1 on supp ψ, and define the operatorP N as the Fourier multiplier of symbolψ( h N ·). Then, by the Littlewood-Paley inequality and the Minkowskii inequality with q, r ≥ 2, we prove that The second inequality in the theorem can be proved by the same way.
Proof of Proposition 5.1. By the Fourier transform, a solution to a discrete Schrödinger equation is represented as We observe that and so the Hessian Hϕ t is degenerate if and only if ξ j = ± π 2h for some j. Suppose that N = 1 or 1 2 or 1 4 . Then, by scaling hξ → ξ, we have for any x ∈ Z d , where F −1 1 is the inverse Fourier transform on Z d . Hence, it follows from the dispersion estimate on Z d (see (1.4)) that for all x ∈ Z d h , Therefore, going back to (5.2), we obtain (5.1).
If N ≤ 1 8 , then on the support of ψ( h N ·), the Hessian of the phase function is non-degenerate and moreover 2t cos hξ j |t| d .
Therefore, it follows from the standard oscillatory integral estimate that |I N,t (x)| |t| −d/2 and Finally, insertingP N defined in the proof of Theorem 1.3 and then employing Bernstein's inequality, we prove that

Uniform boundedness for
Moreover, it conserves the mass (1.14) and the energy (1.15).
Proof. We prove local well-posedness by a standard contraction mapping argument and the trivial embedding L 2 h ֒→ L ∞ h , that is, nothing but ℓ 2 ֒→ ℓ ∞ for sequences however whose implicit constant depends on h > 0.
Let I = [−T, T ] with small T > 0 to be chosen later. We define the nonlinear mapping Then, by unitarity of the linear propagator e it∆ h and the inequality u Similarly for the difference, using the fundamental theorem of calculus Then, taking small T > 0 depending on R and C h , we prove that Φ is contractive on a ball of radius R centered at zero in C(I; L 2 h ). Thus, the equation (1.13) has a unique strong solution, denoted by u h (t).
The conservation laws can be proved as usual by differentiating the mass and the energy, substituting ∂ t u h by the equation and then doing summation by parts. Note that unlike the Euclidean domain, the Laplacian ∆ h is bounded on L 2 h , and thus the energy is properly defined for L 2 hsolutions.
The mass conservation prevents a solution to blow up in L 2 h in finite time. Therefore, u h (t) exists globally in time.
Next, we will show the improved uniform boundedness (Theorem 1.4). To this end, we need the following nonlinear estimate. Lemma 6.2. Suppose p > 1. Then, Proof. By the norm equivalence (Proposition 1.2), we write Then, applying the fundamental theorem of calculus to Therefore, by the norm equivalence again, we obtain where the implicit constant is independent of h ∈ (0, 1].
where C > 0 is a uniform constant.
Next, we claim that there is α > 0 such that Indeed, if d = 2, 3, then by the assumption p < 1 + 4 d−2 , there exists small δ > 0 such that Hence, applying the Hölder inequality and the Sobolev inequality, and then using that ( By the same way but with the one-dimensional Sobolev inequality, one can prove the claim (6.5).
Inserting the bound (6.5) in (6.3), we get and going back to (6.4), we obtain that Therefore, we may increase τ up to It remains to show global-in-time bound (ii). If λ > 0 and 1 p > max{ d−2 d+2 , 0}, then by the energy conservation laws, any solution u h (t) satisfies On the other hand, if λ < 0 and p < 1 + 4 d , then it follows from the Gagliardo-Nirenberg inequality and the mass and the energy conservation laws that h is bounded uniformly in h ∈ (0, 1]. Therefore, in both cases, the a priori bound allows to iterate (6.7) with the uniform size of intervals so that Remark 5. In our analysis, high dimensions d ≥ 4 are not included due to lack of admissible pairs.
In (6.6), the admissible pair ( For discrete Klein-Gordon equation, it behaves like Schrödinger equation near the origin in the fourier side and we indeed get the same derivative loss as Schrödinger case. But in the large frequency region it is much like a wave equation. Thus higher regularity loss is required to compensate weak dispersion. As shown before, the Strichartz estimates follows from the dispersive estimates.
Proposition 7.1. Let h ∈ (0, 1]. Then for any dyadic number N ∈ 2 Z with N ≤ 1, we have Proof. By the Fourier transform, a solution to a discrete Klein-Gordon equation is represented as We observe that ϕ ′′ t (ξ) = From this we obtain (7.1) by applying Young's convolution inequality to (7.2).

Appendix A. Appendix
In this appendix we consider the optimality of Theorem 1.3. The next proposition says that (q, r) range and loss of derivative in Theorem 1.3 can not be improved. We prove sharpness by adapting the standard 'Knapp' example. We modify the example to make it applicable in our setting, i.e., to be defined on Z d h . And then we compute its norm and observe how it depends on h > 0 when h goes to zero. Our proof follows the mainstream of [15,Proposition 1].
Proposition A.1. Suppose for some s ∈ R and 2 ≤ p, q ≤ ∞ we have Then it should hold d 2 ≥ 3 q + d r and s ≥ 1 q .
Proof. By duality argument, (A.1) is equivalent to where (q ′ , r ′ ) is the Hölder conjugate of (q, r). Applying Plancherel's theorem we compute the left side e it∆ h |∇ h | −s f (t)dt .