Abstract
It is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇u on the sphere are comparable by means of the Lp-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.
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Acknowledgments
This paper is based on a part of the author’s dissertation [4] 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 [3]. 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).
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Appendix: A
Appendix: A
For convenience we include here some simple results.
1.1 A.1 Some Basic Results
Lemma A.1 (Complex square root)
There exists a unique function \(R\in C\left (\mathbb {C}\setminus (-\infty ,0), \mathbb {C}\right )\) such that \(R\upharpoonright _{\mathbb {C}\setminus (-\infty ,0]}\) is holomorphic and \(\forall z\in \mathbb {C}\setminus (-\infty ,0)\colon R(z)^{2}={z}\).
Proof Proof of Lemma A.1
Let \(\log \colon \mathbb {C}\setminus \{0\}\to \mathbb {C}\) be the principle branch of the logarithm, i.e., it holds for all \(z\in \mathbb {C}\setminus \{0\}\) that \(\exp (\log z)=z \) and \(-\pi \leq \Im (\log z) \leq \pi \), where I denotes the imaginary part (see, e.g., Theorem I.2.11 in Freitag and Busam [21]), and let \(R\colon \mathbb {C}\setminus (-\infty ,0)\to \mathbb {C}\) be given by
An elementary property of the function \(\exp \) then shows that
Furthermore, the fact that \(\exp \) and \(\log \upharpoonright _{\mathbb {C}\setminus (-\infty ,0]}\) are holomorphic (cf. Theorem I.5.8 in [21]) and the chain rule then show that \(R\upharpoonright _{\mathbb {C}\setminus (-\infty ,0]}\) is holomorphic. Finally, we prove by contradiction that R is continuous at 0. Suppose there exist \((z_{n})_{n\in \mathbb {N}}\subseteq \mathbb {C}\setminus (-\infty ,0)\), 𝜖 ∈ (0, 1) such that \(\lim _{n\to \infty } z_{n}=0\) and \(\forall n\in \mathbb {N}\colon |R(z_{n})|>\epsilon \) and without lost of generality assume for all \(n\in \mathbb {N}\) that |zn| < 1. Then Eq. A.2 implies for all \( n\in \mathbb {N}\) that |R(zn)|2 = |zn|≤ 1. The Bolzano theorem hence proves that there exists a sequence \((n_{k})_{k\in \mathbb {N}}\subseteq \mathbb {N}\) such that \((R(z_{n_{k}}))_{k\in \mathbb {N}}\) converges. This, Eq. A.2, and the fact that \(\lim _{n\to \infty } z_{n}=0\) then demonstrate that \(\lim _{k\to \infty }|R(z_{n_{k}})|^{2}= \lim _{k\to \infty }|z_{n_{k}}|=0\). This contradicts the assumption that \(\forall n\in \mathbb {N}\colon |R(z_{n})|>\epsilon \). Thus, R is continuous at 0. The proof of Lemma A.1 is thus completed. □
Lemma A.2
-
i)
It holds that \( 0<\inf _{s\in [-\pi ,\pi ]\setminus \{0\}}\frac {1-{\cos \limits } (s)}{s^{2}}< \sup _{s\in [-\pi ,\pi ]\setminus \{0\}}\frac {1-{\cos \limits } (s)}{s^{2}}<\infty \),
-
ii)
it holds that \(0< \inf _{s\in [-\pi ,\pi ]\setminus \{0\}}\left |\frac {e^{-\mathbf {i} s}-1}{s}\right |\leq \sup _{s\in [-\pi ,\pi ]\setminus \{0\}}\left |\frac {e^{-\mathbf {i} s}-1}{s}\right |<\infty \), and
-
iii)
it holds that \(\sup _{s\in [-\pi ,\pi ]\setminus \{0\}}\left |s^{2}\frac {d}{ds}\left (\frac {1}{e^{-\mathbf {i} s}-1}\right )\right |<\infty \).
Proof Proof of Lemma A.2
Throughout the proof let \(g,h\colon [-\pi ,\pi ]\to \mathbb {R}\) be the functions which satisfy for all s ∈ [−π,π] ∖{0} that
The fact that \(\lim _{s\to 0}\frac {1-{\cos \limits } (s)}{s^{2}}= \frac {1}{2}\) and the fact that \(\left |\frac {e^{-\mathbf {i} s}-1}{s}\right |=1\) show that \(g,h\in C([-\pi , \pi ],\mathbb {R})\). The extreme value theorem and the fact that \(\forall s\in [-\pi ,\pi ]\colon \min \limits \{g(s),h (s)\}> 0\) then imply that
and
This proves Items (i) and (ii). Finally, Item (iii) follows from Item(ii). The proof of Lemma A.2 is thus completed. □
1.2 A.2 The Simple Random Walk Representation Without Martingale Theory
For convenience of the reader we include an elementary proof without using martingales.
Proof Proof of Item (i) in Lemma 2.4 without martingale theory
First, it holds for all \(n\in \mathbb {N}\) that {Sn− 1 = x,T > n − 1} depends only on X1,…,Xn− 1 and is therefore independent of Xn. The fact that \(\forall n\in \mathbb {N}\colon S_{n}=S_{n-1}+X_{n}\), the assumption on the distribution of Xn, \(n\in \mathbb {N}\), and the assumption that \(\forall x\in \mathbb {Z}^{d-1}\times \mathbb {N}\colon \bigtriangleup u(x)=0\) imply for all \(x\in \mathbb {Z}^{d-1}\times \mathbb {N}\), \(n\in \mathbb {N}\) that
This and the fact that \(\forall x\in \mathbb {Z}^{d-1}\times \{0\}, n\in \mathbb {N}\colon \mathbb {P}(S_{n-1} =x,T >n-1)=0\) prove for all \(x\in \mathbb {Z}^{d-1}\times \mathbb {N}_{0}\), \(n\in \mathbb {N}\) that and
This and the assumption that u is bounded yield that
An induction argument shows for all \(n\in \mathbb {N}\), \((x,y)\in \mathbb {Z}^{d-1}\times \mathbb {N}_{0}\) that \(\mathbb {E} \left [u(S_{n\wedge T})\right ] =\mathbb {E} [u(S_{0})]\) and \(\mathbb {E} \left [u(S_{n\wedge T})|S_{0}=(x,y)\right ]=\mathbb {E} [u(S_{0})|S_{0}=(x,y)]=u(x,y)\). The bounded convergence theorem then ensures with n tending to infinity that for all \(x\in \mathbb {Z}^{d-1}\), \(y\in \mathbb {N}_{0}\) it holds that \(\mathbb {E}[u(S_{T})|S_{0}=(x,y)]= u(x,y)\). This (with \(u\leftarrow u(\cdot +(x,0))\) and \((x,y)\leftarrow (0,y) \) for \(x\in \mathbb {Z}^{d-1}\), \(y\in \mathbb {N}_{0}\)) establishes that for all \((x,y)\in \mathbb {Z}^{d-1}\times \mathbb {N}_{0}\) it holds that . The proof is thus completed. □
1.3 A.3 An Interpolation Argument
Lemma A.3 below gives a version of the Marcinkiewicz interpolation theorem in the discrete case. Its formulation is unfortunately not found in the literature, although its proof is quite routine. We follow the proof of Theorem 9.1 in the book by DiBenedetto [22].
Lemma A.3 (\(L_{w}^ p\)-\(L^{\infty }\)-interpolation)
Let \(N\in \mathbb {N}\), let \(E\subseteq \mathbb {Z}^{N}\) be a finite set, let \(p,r\in [1,\infty )\), \(N_{p},N_{\infty }\in (0,\infty )\), assume that \(1\leq p<r<\infty \), let \(T\colon E^{\mathbb {R}}\to E^{\mathbb {R}}\) be linear and satisfy for all \(f\colon E\to \mathbb {R}\) that
Then it holds for all \(f\colon E\to \mathbb {R}\) that
Proof Proof of Lemma A.3
Throughout this proof let \(f\colon E\to \mathbb {R}\) and let \(f_{i}= (f_{i}^{t,\lambda }(x))_{t,\lambda \in (0,\infty ),x\in E}\), i ∈{1, 2}, be the functions which satisfy that
First, Markov’s inequality and Eq. A.9 show for all λ,t > 0 that
Observe that Eq. A.9 shows for all \(\lambda \in [0,(2N_{\infty })^{-1}]\), \(t\in (0,\infty )\) that
Hence, it holds for all \(\lambda \in [(2N_{\infty })^{-1},\infty )\), \(t\in (0,\infty )\) that
The fact that T is linear, the fact that f1 + f2 = f (see Eq. A.11), the triangle inequality, and Eq. A.12 therefore show for all \(\lambda \in [(2N_{\infty })^{-1},\infty )\), \(t\in (0,\infty )\) that
The fact that Tonelli’s theorem, and a direct calculation hence yield for all \(\lambda \in [(2N_{\infty })^{-1},\infty )\), \(t\in (0,\infty )\) that
This (with \(\lambda \leftarrow (2N_{\infty })^{-1}\)) implies that \(\|f\|_{L^{r}(E)}^{r}\leq \frac {r}{r-p}(2N_{p})^{p}(2N_{\infty })^{r-p}\). This and the fact that f was arbitrary complete the proof of Lemma A.3. □
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Nguyen, T.A. An Lp-Comparison, \(p\in (1,\infty )\), on the Finite Differences of a Discrete Harmonic Function at the Boundary of a Discrete Box. Potential Anal 56, 351–407 (2022). https://doi.org/10.1007/s11118-020-09888-8
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DOI: https://doi.org/10.1007/s11118-020-09888-8
Keywords
- Discrete harmonic function
- Discrete boundary problems
- Discrete Fourier multiplier theorem
- Discrete Poisson kernel