An $L^p$-comparison, $p\in (1,\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box

It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\geq 2$, the tangential and normal component of the gradient $\nabla u$ on the sphere are comparable by means of the $L^p$-norms, $p\in(1,\infty)$, up to multiplicative constants that depend only on $d,p$. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the $d$-dimensional lattice with multiplicative constants that do not depend on the size of the box.

. Tangential and normal edges of a two dimensional box

Introduction
This paper formulates and proves a discrete analogue of a classical result in the continuum setting which states that the tangential and normal component of the gradient of a harmonic function on the boundary of a domain are comparable by means of L p -norms, p ∈ (1, ∞). For convenience we give a simplified version of this result in Theorem 1.1 below. For complete formulations and proofs we refer the reader to Maergoiz [17] (see, e.g., Theorems 1 and 2), Mikhlin [18] (see § 44 p. 208), and Bella, Fehrman, and Otto [1] (see Lemma 4). This result can be viewed as a stability estimate for harmonic extensions of given Dirichlet or Neumann boundary conditions. Thus, it plays an important role in the proof of a Liouville theorem for a class of elliptic equations with degenerate random coefficient fields (see formulas (40) and (41) in [1]) where the so-called idea perturbing around the homogenized coefficients is realized by harmonic extensions from given boundary conditions. The discrete analogue that we want to show here can be applied to prove a Liouville theorem for the random conductance model under degenerate conditions, which is the discrete analogue of [1] (see the paragraph below Lemma 4 in [1] and the PhD thesis of the author [20] (e.g., Section III.2.3 for an outline).
In order to formulate the discrete analogue of Theorem 1.1 let us introduce our notation. For the rest of this paper we always use the notation given in Setting 1.2 below. and let ∇u : (A × A) ∩ E d → Ã be the function which satisfies that for all x, y ∈ A with (x, y) ∈ E d it holds that ∇ (x,y) u = u(y) − u(x). For every d ∈ AE ∩ [2, ∞), p ∈ [1, ∞], every finite set A, and every function f : A → Ã let f L p (A) ∈ [0, ∞) be the real number given by ( 1.4) Note that in Setting 1.2 above the arguments of ∇u are edges. We will also introduce another notation for discrete derivatives which are functions of vertices (see Section 3). However, to formulate the main result let us temporarily use the notation in Setting 1.3 below. v(e) = 0 is thus a discrete Neumann condition. Here, the vanishing mean is a necessary condition for the Neumann problem to have a solution, which also holds in the continuum setting.
ii) it holds for all d, N ∈ [2, ∞) ∩ AE, p ∈ (1, ∞), f ∈ D d,N that  In addition, note that for fixed d, N ∈ [2, ∞) ∩ AE, g ∈ E ν d,N there exists no real numbers C ∈ (0, ∞) such that for every solution u to (1.12) it holds that ∇u L p (E τ d,N ) ≤ C g L p (E ν d,N ) . Indeed, e.g., in the case d = 2 we can freely change the value of u at the four corners of the rectangle in Fig. 1 to make ∇u L p (E τ d,N ) arbitrary large without damaging the fact that u is a solution to (1.12). Consequently, it is impossible to make any claims on the uniqueness of the family (Φ N d,N ) d,N ∈[2,∞)∩AE in the statement of Theorem 1.4.
Next, let us give a brief and rough explanation why Theorem 1.4 is useful for the idea of using harmonic extensions in the proof of the Liouville theorem in [20]. Let u be a function defined on the box in Fig. 1. We keep the Dirichlet condition of u at red points and replace the values of u at other points by an extension that is harmonic in the interior of the box. This will clearly erase the Neumann condition of u. However, Theorem 1.4 claims that the new Neumann condition can still be bounded by the remaining Dirichlet condition.
Discrete Laplacian and discrete harmonic functions are interesting topics that date back to 1920s (see, e.g., the fundamental works by Lewy, Friedrichs, and Courant [16], Heilbronn [11], Duffin [5]). Discrete boundary problems have been widely studied in numerical analysis, e.g., to approximate the continuum solutions (see, e.g., the classical work by Stummel [21] and for further references see, e.g., Gürlebeck and Hommel [12], [9], [10], who studied Dirichlet and Neumann boundary problems on general two-dimensional discretized domains using difference potentials, and the references therein).
Although discrete and continuum objects often have many similar properties, it is not always trivial to adapt things from the continuum case to the discrete case and vice verse. To the best of the author's knowledge, there exists no result in the discrete case which deals with the bounds (1.9) and (1.10), while L p -comparisons, p ∈ (1, ∞), between the tangential and non-tangential components of harmonic functions on Lipschitz and C 1 -domains and related topics have been studied by several papers, e.g., in chronological order: Mikhlin [18], Maergoiz [17], Calderon, Calderon, Fabes, Jodeit, and Rivièrie [2], Fabes, Jodeit, and Rivièrie [6], Jerison and Kenig [13], Verchota [22], Dahlberg and Kenig [3], Mitrea and Mitrea [19]. The main issue in the discrete case is to show that the functions C in (1.9) and (1.10) do not depend on the size N of the discrete box while in the continuum case this is not an issue due to a simple scaling argument. In fact, for (1.2) we only need to consider r = 1.
The proof of Theorem 1.4 that we represent here essentially mimics the proof of Lemma 4 in Bella, Fehrman, and Otto [1] who formulate and prove Theorem 1.1 with balls replaced by boxes in the continuum case. We separate the proof into several steps and organize the paper as follows. Section 2 formulates and proves a discrete counterpart of inequality (88) in [1], which was shown by using the continuum Poison kernels. In order to adapt this idea to the discrete case we use a result in Lawler and Limic [15] to approximate the discrete Poison kernels by the continuum Poison kernels. Estimates by means of the Marcinkiewicz multiplier theorem, e.g., inequalities (78), (79), (82), and (99) in [1] are adapted in Section 3 which focuses on discrete harmonic functions on haft spaces with periodic boundary conditions. In order to avoid many tedious calculations with higher derivatives of the multipliers we apply Cauchy's integral formula. In addition, with some elementary arguments, Section 3.4 provides a result of independent interest that the author has not found in the literature. Finally, Section 4 applies the results obtained in Sections 2 and 3 to prove the main result, Theorem 1.4. As Bella, Fehrman, and Otto [1] we call estimate (1.9) the Dirichlet case and estimate (1.10) the Neumann case and prove them separately. The main techniques here are basically to adapt two ideas learnt from [1] to the discrete case: i) returning to the case of periodic boundary conditions by using even and odd reflections and ii) reducing to the case of haft spaces. Concerning the idea of using reflections, Section IV.2.1 in the author's dissertation [20] may provide a simple illustration with figures in the two-dimensional case that may help to understand the general case. Another interesting application of even and odd reflections and the discrete Marcinkiewicz multiplier theorem is to prove L p -estimates for discrete Poisson equations (see Section 2.5.2 in Jovanović and Süli [14]).
For convenience, throughout this paper, the arguments here are often compared with that in the continuum case in [1]. However, since there are several differences between the discrete case and the continuum case, this paper is organized so that the reader can easily start from scratch.
Finally, the proof shows that the functions C in Theorem 1.4 may depend exponentially on the dimension: this result, as finite difference method in general, may not be quite useful for high-dimensional applications (the so-called curse of dimensionality).
Our notation will be defined clearly in the formulation of each result. In addition, remember that throughout this paper we always use the notation in Setting 1.2 above and the usual conventions in Setting 1.5 below.
Setting 1.5 (Conventions). Denote by i the imaginary unit. Denote by ℜ(z) and ℑ(z) the real and denote by x · y the standard scalar product of x and y, i.e., x · y = d i=1 x iȳi , and denote by |x| ∞ the maximum norm of x, i.e., |x| ∞ = max d i=1 |x i |. For every set A denote by |A| the cardinality of A. Partial derivatives will be denoted by ∂ i , ∂ ∂t i , ∂ ∂ξ i . When applying a result we often use a phrase like 'Lemma 3.8 with d ← d − 1' that should be read as 'Lemma 3.8 applied with d (in the notation of Lemma 3.8) replaced by d − 1 (in the current notation)' and we often omit a trivial replacement to lighten the notation, e.g., we rarely write, e.g., 'Lemma 3.28 with d ← d'.

Acknowledgement
This paper is based on a part of the author's dissertation [20] written under supervision of Jean-Dominique Deuschel at Technische Universität Berlin. The author thanks Benjamin Fehrman and Felix Otto for useful discussions and for sending him the manuscript of [1]. The author gratefully acknowledges financial support of the DFG Research Training Group (RTG 1845) "Stochastic Analysis with Applications in Biology, Finance and Physics" and the Berlin Mathematical School (BMS).
2 Potential-theoretic results for harmonic functions on haft spaces

Main result
In this section we essentially prove Corollary 2.2 below, which formulates a discrete analogue of inequality (88) in Bella, Fehrman, and Otto [1]. We basically follow the proof in [1]. However, to make the argument more illustrative we introduce a simple random walk in Setting 2.3. Lemmas 2.7 and 2.11 are discrete counterparts of inequality (92) and (93) in [1]. Combining Lemmas 2.7 and 2.11 with a Marcinkiewicz-type interpolation argument we obtain Corollary 2.12. Approximating the discrete Poisson kernels by the continuum counterparts we obtain Lemma 2.10. This and Corollary 2.12 imply Corollary 2.2.
The following proof relies on martingale theory. For an elementary proof see Appendix A.2.
Proof of Lemma 2.4. The assumption that ∀ (x, y) ∈ d−1 × AE: (△u)(x, y) = 0 and the assumption that u is bounded demonstrate for all x ∈ d−1 that (u(S n + (x, 0))) n∈AE 0 is a bounded martingale. The optional stopping theorem proves that This shows Item (i). Furthermore, (2.2), linearity, and periodicity imply for all y ∈ AE 0 that 3) The proof of Lemma 2.4 is thus completed.
Lemma 2.5. Assume Setting 2.1 and let d ∈ [2, ∞) ∩ AE, L ∈ AE, u ∈ À d,L,≥0 . Then it holds that Proof of Lemma 2.5. Throughout this proof we use the notation given in Setting 2.3. The fact that È-almost surely it holds that S T ∈ d−1 × {0} and the assumption on periodicity, i.e., ∀ (x, y) , y) imply that È-almost surely it holds that This and (2.5) establish that This completes the proof of Lemma 2.5.
Combining Lemma 2.5 with a Riesz-Thorin interpolation we obtain Corollary 2.6 below.
Proof of Lemma 2.7. Throughout this proof we use the notation given in Setting 2.3. First, observe that Lemma 2.4, Jensen's inequality, and linearity of show that Next, Jensen's inequality and Corollary 2.6 ensure that Combining this, (2.10), and the triangle inequality completes the proof of Lemma 2.7.

(2.13)
This completes the proof of Lemma 2.8.

Main result
In this section we continue considering harmonic functions on the discrete haft space with periodic boundary conditions, however, from the viewpoint of Fourier analysis. The main results are summarized in Corollary 3.1 below, whose main part is illustrated by Fig. 2. As Bella, Fehrman, and Otto [1] we call the first inequality in (3.1) the Dirichlet case and the second inequality in (3.1) the Neumann case. In order to show Corollary 3.1 we combine Corollary 3.14 and, in particular, Corollary 3.27 (the Neumann case) Corollary 3.29 (the Dirichlet case). As in [1] our proof is based on Marcinkiewicz-type multiplier theorems and the observation that the tangential derivatives and the normal derivatives of harmonic functions on haft space are related by mean of Fourier multipliers. After having finished his dissertation [20], the author realized that for the argument with telescope sequences (see the paragraph below inequality (88) in [1]) it suffices to consider haft spaces instead of strips. The calculations here are therefore much simpler than that in [20]. However, we still have to overcome some tedious calculations with the discreteness when estimating the higher derivatives of the multipliers. Another issue is to adapt carefully the paragraph between (83) and (84) in [1] into the discrete case for which we have to work with the dyadic sets, see Section 3.4.

Notation and settings
Instead of d , d ∈ AE, we will work with h d , h ∈ π/AE, d ∈ AE. In fact, our notation in Setting 3.2 is inspired by Jovanović and Süli [14, Section 2.5] so that we can easily use the Marcinkiewicz multiplier therein. To make the notation consistent we introduce Setting 3.3.
be the set of all 2π-periodic functions defined on h d , i.e., which is distinguished from · L p (A) in (1.4) by a normalized factor h d . Denote by F the so-called discrete Fourier transform, i.e., F : is the operator which satisfies for every Laplacian with mesh h, i.e., the function which satisfies for all (x, y) In order to obtain Corollary 3.14 using a Riesz-Thorin interpolation argument we choose Ã = in Setting 3.3 above. For other results we only need Ã = Ê.

Some simple calculations
The main results of this subsection, Corollaries 3.11 and 3.12, prove that the discrete normal and tangential derivatives are related by means of Fourier multipliers. We start with Setting 3.4 below that defines the functions which are used to represent the Fourier transform of harmonic functions and their discrete derivatives. It is useful to consider Q and f in (3.10) as functions of a complex variable. The names Q and λ are inspired by Guadie [8] who considers harmonic functions on infinite strips with L 2 ( d−1 ) boundary conditions.
and let c ∈ (0, ∞) be the real number (cf.
Lemma 3.7 below is a classical result and is included for convenience of the reader.
This completes the proof of Lemma 3.7.
Lemma 3.8 is straightforward and its proof is therefore omitted.
Proof of Lemma 3.9. Throughout the proof let v : defined through its Fourier transform, and let v, u : d−1 × AE 0 → be the function given by hy) and u(x, y) = u(hx, hy).
Proof of Lemma 3.10. Observe that (3.9), Lemma 3.8 (with d ← d − 1, f ← u(·, 0)), and the fact that and The proof of Lemma 3.10 is thus completed.
Proof of Corollary 3.11. Lemma 3.10 and (3.11) prove that in the case k = 0 it holds that and in the case k = 0 it holds that Q(λ(hk)) = 1 and This completes the proof of Corollary 3.11.
In Corollary 3.12 below we see that in the Dirichlet case there are (d − 1) multipliers, which are the quotients ∩ , and therefore not everywhere defined. Fortunately, we can still show that for each dyadic rectangle there is a multiplier well-defined on it. In Section 3.4 we will develop a Marcinkiewicz-type multiplier theorem to deal with this situation.
Proof of Corollary 3.12. First, note that Lemma 3.10 implies Item (i). Next, observe that (3.27), the assumption that k i = 0, and the assumption that Lemma 3.10 and (3.11) therefore show that This completes the proof of Corollary 3.12.
Proof of Lemma 3.13. Observe that (3.11) and Lemma 3.5 show for all t ∈ [−π, π] d−1 that This, Bernoulli's inequality, and the assumption that N h ≥ r show for all k ∈ Á d−1 This proves Item (i). Observe that (3.7) and the assumption that x∈ω d−1 h u(x, 0) = 0 imply that [F(u(·, 0))](0) = 0. The Plancherel identity (for details see Lemma 3.7), Lemma 3.9, and (3.30) hence demonstrate that Combining Lemma 3.13, Lemma 2.5, a scaling argument, and a Riesz-Thorin-type interpolation argument we obtain the following result, Corollary 3.14. For later use we only need the fact that the multiplicative constants do not depend on N .

A Marcinkiewicz-type theorem for more than one multipliers
This subsection slightly extends the classical Marcinkiewicz multiplier theorem to the case of more than one multipliers (see Corollary 3.22). It will be used to bound the normal component by (d − 1) tangential components. In this case there are (d − 1) multipliers, however, each multiplier is not everywhere well-defined as seen in Corollary 3.12. In the continuum setting this issue is overcome by considering a partition of unity (see the paragraph between (83) and (84) in [1]). The argument here also relies on local properties of the multipliers. Roughly speaking, the function locvar in Setting 3.15 below, called the local variation, measures the variation of a function on each dyadic rectangle. Corollary 3.22 proves that we still obtain L p -estimates, p ∈ (1, ∞), if for each dyadic rectangle there is a nice multiplier defined on it. Moreover, in order to conveniently verify an assumption in Corollary 3.22 we use Lemma 3.18.
Setting 3.15. Let Setting 3.2 be given. Let D(ℓ) ⊆ Ê, ℓ ∈ , be the intervals given by → Ê be the so-called total variation, i.e., the function which satisfies for all d, L ∈ AE, a ∈ P 2L d , that For every β ∈ {0, 1} and every finite set A ⊆ we write In Lemma 3.16 below we explain the purpose of introducing (3.36) and (3.37).
Proof of Lemma 3.16. Let us shows that for all α ∈ {0, 1} d it holds that For the proof of Lemma 3.18 below we use the mean value theorem. This is a routine idea (cf. the proof of Item (b) in Theorem 2.49 in [14]). The proof is included only for convenience of the reader. , assume that J = ∅, let A ∈ C(J, ), a ∈ P 2L d , satisfy for all ν ∈ J ∩ d that a(k) = A(k), and assume for all Then it holds that locvar(a, k) ≤ M .

Combining (3.74), (3.76), and the triangle inequality then shows that for all
This (with i ← J(k)) and (3.73) prove for all This, (3.12), and the chain rule prove for all    [20] for a simple illustration in the two-dimensional case). Section 4.3 adapts inequality (79) in [1] into Corollary 4.24. Here, we also construct a telescope series of harmonic functions on haft spaces, however, now by means of Neumann conditions. The reader will see that there are quite a lot of similarities between the Dirichlet and the Neumann case. However, the two cases are not identical and it is necessary to adapt rigorously every step of the proof due to the discreteness. In Section 4.4 we prove carefully the main theorem in the Neumann case, see Theorem 4.27, although the argument is quite straightforward in the continuum case, as said in the last sentence in the proof of Lemma 4 in [1]. The idea of even reflections is explained in Lemma 4.26 where some minor arguments are used to deal with the discreteness (see Section IV.2.1 in [20] for an illustration in the two-dimensional case). Throughout this section we always use the notation given by Setting 4.1 below.
. For every finite set A and every function

Construction of Dirichlet extensions
let À d,L,≥0 be the set of all bounded functions u : d−1 × AE 0 → Ê with the properties that Setting 4.3 and Lemma 4.4 below prepare two important inequalities, which follow from the results in the last sections. We will bound the telescope series by a geometric series using the fact that  1 (d, p) is the smallest real extended number such that for all N, L ∈ AE, u ∈ À d,L,≥0 with 1/4 ≤ N/L and u Á d−1 it holds that . This (with r ← 4 and u ← D ± i u for i ∈ [1, d] ∩ , L, N ∈ AE, u ∈ À d,L,≥0 with N/L ≤ 4) implies that for all L, N ∈ AE with N/L ≤ 4, u ∈ À d,L,≥0 it holds that Next, Corollary 2.6 (with u ← D + d u and N ← N − 1) shows for all L ∈ AE, u ∈ À d,L,≥0 that Hence, Corollary 3.1 shows that there exists c 2 ∈ (0, ∞) such that for all L ∈ AE, u ∈ À d,L,≥0 it holds Combining (4.4) and (4.6) then yields that there exists c 3 ∈ (0, ∞) such that for all N, L ∈ AE, i ∈ [1, d − 1] ∩ , u ∈ À d,L,≥0 with N/L ≤ 4 it holds that This shows that C 2 (d, p) < ∞. The proof of Lemma 4.4 is thus completed.
Existence and uniqueness of the solutions to the Dirichlet problems on haft spaces (shown, e.g., by means of Fourier transforms in Section 3) ensure that the sequences (u k ) k∈AE in Setting 4.5 below are well-defined by (4.10)-(4.12). be the sequences given by Proof of Lemma 4.6. Observe that (4.10) and and (4.14) This and an induction argument prove for all n ∈ AE that
Lemma 4.8 shows the existence of solutions to Dirichlet problems. Combining this with the uniqueness, which easily follows, e.g., from the maximum principle, we obtain Corollary 4.9 below.
The existence and uniqueness, stated in Corollary 4.9, and Lemma 4.8 imply Corollary 4.10 below.
it holds that Then it holds that . This and (4.33) yield that ∀ x ∈ d−1 × {0, N } : w(x) = w(x). Corollary 4.9 hence shows for all . This and Lemma 4.11 (with , applied to the j-th coordinate) complete the proof of Lemma 4.12.
The sets E # d,N , E τ d,N , V τ d,N in Setting 4.13 below are illustrated in Fig. 1: E # d,N consists of all red and blue edges; E τ d,N consists of all red edges; V τ d,N consists of all red points. Furthermore, this setting provides two ingredients that we need for the next step: Corollary 4.10 (see (4.39)) and the Poincaré inequality (see (4.40)).
Proof of Corollary 4.15. Lemma 4.14 (with u ← u − a for a ∈ Ê) and (4.40) show that

Construction of the Neumann extensions
In the Neumann case we also use a telescope sequence. First of all, instead of Setting 4.3 we start with Setting 4.16 below with Neumann conditions on the right hand sides of (4.64) and (4.65).
and (see (4.67)). Next, using Corollary 2.6 (with the function replaced by the derivative) and Item (ii) in Corollary 3.1 we bound the tangential differences on the top and bottom by the normal differences on the bottom (see (4.70)). Furthermore, using Corollary 2.2 we bound the differences with respect to all edges with one endpoint on the face {x 1 = 0} by the tangential differences on the bottom (see (4.68)) and hence again by the normal differences on the bottom. A permutation of the coordinates then shows that we can bound the differences with respect to all edges with one endpoints on the boundary by the normal differences on the bottom.
Setting 4.18 below introduces a telescope sequence which is similar to that in the Dirichlet case (cf. Setting 4.5). In (4.74) the means on each layer are set to be zero, since otherwise the Neumann problems on the haft spaces do not determine unique solutions.  be the sequences which satisfy that Proof of Lemma 4.6. The assumption that (u 2k+1 ) k∈AE 0 ⊆ À d,L,≥0 in (4.73) and Corollary 2.6 (with Similarly, the assumption that (u 2k+2 ) k∈AE 0 ⊆ À d,L,≤N in (4.73) and Corollary 2.6 (together with a simple change of coordinates) show for all k ∈ AE 0 that Next, (4.73), (4.64), and possibly a simple change of coordinates show for all k ∈ AE 0 that and Combining (4.76)-(4.79), an induction argument, and (4.76) (with k ← 0) proves that for all n ∈ AE it holds that This (with u ← u n for n ∈ AE and combined with (4.73)), and Lemma 4.19 imply for all x ∈ Á d−1 (4.82) The fact that C 1 (d, p) < 1 and the fact that ∀ x ∈ (0, 1) : (4.83) Then it holds that and This completes (4.84). Next, observe that (4.83), the triangle inequality, (4.65) (with u ← u k for k ∈ AE), Lemma 4.20, the fact that ∀ x ∈ (0, 1) : ∞ k=1 x k−1 = (1 − x) −1 , and (4.74) ensure that and Proof of Lemma 4.22. Let v 1 , v 2 , v 3 ∈ d,L,N be the functions which satisfy that This construction and the fact that v Á d−1 (4.93) Next, the fact that and such that for all i ∈ {1, 2} it holds that    and let E # d,L,N ⊆ E d be the set of edges given by

Proof of the main result in the Neumann case
Due to the discreteness we choose N − 1 as the period in Lemma 4.26 below. and Then it holds that (4.106) Furthermore, (4.105), (4.102), the fact that j = d, and the periodicity in (4.101) and (4.105) imply that let V ν d,N ⊆ d be the set of vertices given by  (i.e., v is the Neumann condition of u) and such that ∇u  To this end we distinguish two cases: ℓ + 1 = d and ℓ + 1 < 1. First, when ℓ + 1 < d, then the odd reflection in (4.122) implies (4.123). Next, we consider the case ℓ + 1 = d. Note that in this case [ℓ + 2, d] ∩ = ∅ and we therefore cannot use (4.122). In this step, to shorten the notation, for every i ∈ [1, d] ∩ , j ∈ {0, N } let F j i be the set given by The fact that v ∈ N d,N implies that v V ν d,N = 0 and hence that This and (4.125) prove that Furthermore, (4.112) and (4.113), following from the induction hypothesis, imply that  137) The rest of the proof is now clear. We only give a sketch. We write C(d, p) to denote possibly different real numbers that only depend on d, p and write for i ∈ [1, d] ∩ to lighten the notation Then Corollary 4.24 shows that The proof of Theorem 4.27 is thus completed.

A.2 The simple random walk representation without martingale theory
For convenience of the reader we include an elementary proof without using martingales.