2.1 Solutions of the Laplace equation

Seeking solutions of $\Delta _{} U=0$ , we fix k and take the horizontal Fourier transform in n:

$$\begin{align*}\widehat U(\theta,k)=\sum_{n\in\mathbb Z}U(n,k)\,\mathrm e^{-\mathrm i n\theta}. \end{align*}$$

We repeatedly use the shift rule:

(2.1) $$ \begin{align} \mathcal F\, \big[U(n\pm1,k)\big] =\mathrm e^{\pm\mathrm i\theta}\,\widehat U(\theta,k). \end{align} $$

Using (2.1) and linearity,

$$ \begin{align*} 0=\mathcal F[\Delta_{}{U}](\theta,k) =\big(2\cos\theta-4\big)\,\widehat U(\theta,k) +\widehat U(\theta,k+1)+\widehat U(\theta,k-1). \end{align*} $$

We look for separated solutions of the form

$$\begin{align*}\widehat U(\theta,k)=C(\theta)\,\rho(\theta)^{\,k}, \end{align*}$$

with $C(\theta )$ independent of k. Substituting gives a quadratic equation with two real reciprocal roots:

(2.2) $$ \begin{align} \begin{aligned} \rho+\rho^{-1}+2\cos\theta-4=0. \end{aligned} \end{align} $$

We select the root in $(0,1]$ :

(2.3) $$ \begin{align} \rho(\theta)=2-\cos\theta-\sqrt{(2-\cos\theta)^2-1}. \end{align} $$

This choice yields normalized, positive kernels $P_{k}\in \ell ^{1}(\mathbb {Z})$ with $\sum _{n}P_{k}(n)=1$ and produces the unique $\ell ^{p}$ -bounded solution of the Dirichlet problem on $\mathbb {Z}\times \mathbb {N}$ (Lemma 2.1).

2.2 The Poisson extension

Let the boundary sequence be $U(n,0)=a_n$ , so the Fourier transform at $k=0$ is $ \widehat U(\theta ,0)=\widehat a(\theta ). $ Evaluating $ \widehat U(\theta ,k)=C(\theta )\,\rho (\theta )^{\,k}$ at $k=0$ fixes $C(\theta )=\widehat a(\theta )$ ; hence,

$$ \begin{align*} \widehat U(\theta,k)=\rho(\theta)^{\,k}\,\widehat a(\theta),\qquad k\ge0. \end{align*} $$

By taking the inverse transform in n,

(2.4) $$ \begin{align} U(n,k)=\frac1{2\pi}\int_{-\pi}^{\pi}\rho(\theta)^{\,k}\,\widehat a(\theta)\,\mathrm e^{\mathrm i n\theta}\,d\theta. \end{align} $$

Definition 2.1 (Discrete Poisson kernel)

For $k\in \mathbb {N}$ and $n\in \mathbb {Z}$ , let

(2.5) $$ \begin{align} P_k(n)=\frac1{2\pi}\int_{-\pi}^{\pi} \rho(\theta)^{\,k}\, \mathrm e^{\mathrm i n\theta}\,d\theta. \end{align} $$

For $a \in \ell ^p(\mathbb {Z})$ with $p \ge 1$ , we have

$$\begin{align*}U(\cdot,k)=P_k*a,\qquad U(n,k)=\sum_{m\in\mathbb Z}P_k(n-m)\,a_m. \end{align*}$$

Indeed, by Lemma 2.1 below, the convolution is well-defined and in $\ell ^p(\mathbb {Z})$ for every $a \in \ell ^p(\mathbb {Z})$ with $p \ge 1$ .

2.3 The conjugate-Poisson kernel

Definition 2.2 On the Fourier side, we define the conjugate-Poisson kernel as follows:

(2.7) $$ \begin{align} \widehat{V}(\theta,k) =\bigl(-\mathrm i\,\mathrm{sgn}\ \theta\bigr)\, \rho(\theta)^{\,k+\frac12}\, \mathrm e^{\mathrm i\theta/2}\, \widehat a(\theta), \qquad k\ge 0. \end{align} $$

The factors $\mathrm e^{\mathrm i\theta /2}$ and $\rho \left ( \theta \right )^{1/2}$ shift V half a unit horizontally and vertically, respectively, locating V at $(n+\tfrac 12,k+\tfrac 12)$ . Let $ V(n+\tfrac 12,k+\tfrac 12)=\mathcal F^{-1}[\widehat V(\,\cdot \,,k)](n) $ , and let us define

$$ \begin{align*}Q_k(n):=\frac{1}{2\pi}\int_{-\pi}^{\pi} \bigl(-\mathrm i\,\mathrm{sgn}\ \theta\bigr)\,\rho(\theta)^{k+\frac12}\,e^{\mathrm i(n+\frac12)\theta}\,d\theta. \end{align*} $$

We note that $Q_k(n)$ is real-valued, which follows from $\rho (-\theta )=\rho (\theta )$ . As a convolution formula, for $a \in \ell ^p(\mathbb {Z})$ with $p>1$ , we have

$$ \begin{align*}V\!\left(n+\tfrac12,k+\tfrac12 \right)=\sum_{m\in\mathbb{Z}}Q_k(n-m)\,a_m. \end{align*} $$

For notational convenience, we may relabel this as $V(n,k)$ , so that both U and V are indexed on $\mathbb {Z}\times \mathbb {N}$ . As we will see, this definition is imposed by the Laplacian and our requirement that U and V satisfy a Cauchy–Riemann system.

Lemma 2.2 (Harmonicity)

For each fixed sequence $a\in \ell ^p(\mathbb {Z})$ with $p>1$ , the extensions

$$\begin{align*}U(n,k)=(P_k\ast a)_n,\qquad V(n,k)=(Q_k\ast a)_n \end{align*}$$

satisfy $\Delta _{} U=\Delta _{} V=0$ on $\mathbb {Z}\times \mathbb {N}$ .

4.1 A discrete Hilbert transform

The above constructions lead us to the following definition of a discrete Hilbert transform, $H_{\mathrm {d}}$ , for a sequence $(a_n)_{n \in \mathbb {Z}}$ (Definition 1.1):

$$\begin{align*}(H_{\mathrm{d}}a)_n =\mathcal F^{-1}\!\bigl[ (-\mathrm i\,\mathrm{sgn}\ \theta)\, \rho(\theta)^{1/2}\mathrm e^{\mathrm i\theta/2} \widehat{a}(\theta)\bigr](n). \end{align*}$$

We have thus proven all assertions of Theorem 1.1, and in this section, we will consider several properties of $H_{\mathrm d}$ .

The discrete Hilbert transform we are considering has the Fourier multiplier:

$$\begin{align*}m_{d}(\theta)=-\mathrm i\ \mathrm{sgn}\ \theta\,\rho(\theta)^{1/2}\,\mathrm e^{\mathrm i\theta/2}. \end{align*}$$

This multiplier was forced by the Cauchy–Riemann system we considered. Nevertheless, since the Fourier multiplier is bounded, we immediately get the boundedness of the discrete Hilbert transform on $\ell ^2(\mathbb Z)$ . We prove stronger boundedness properties below.

4.2 Properties of the discrete Hilbert transform

The kernel of our discrete Hilbert transform model is given by

(4.1) $$ \begin{align} \begin{aligned} K_{\mathrm{d}}(n) = &\frac{1}{2\pi}\!\int_{-\pi}^{\pi}\!(-\mathrm i\,\mathrm{sgn}\ \theta)\,\rho(\theta)^{1/2}\,\mathrm e^{\mathrm i(n+\frac12)\theta}\,d\theta \\ \;=&\; \frac{1}{\pi}\!\int_{0}^{\pi}\!\rho(\theta)^{1/2}\,\sin\!\bigl((n+\tfrac12)\theta\bigr)\,d\theta, \end{aligned} \end{align} $$

so $H_{\mathrm {d}}a=K_{\mathrm {d}}*a$ .

By using the Taylor expansion of $\rho (\theta )$ , it is possible to integrate and express $K_{\mathrm {d}}$ explicitly as an infinite series in n. However, the above integral representation of $K_{\mathrm {d}}$ is more useful when exploring properties and weighted bounds of $H_{\mathrm {d}}$ , and we will not give the more complicated formula for it in terms of a series representation.

Proposition 4.1 (Kernel antisymmetry and cancellation)

The kernel $K_{\mathrm {d}}$ is real-valued. Moreover, it is antisymmetric:

$$\begin{align*}K_{\mathrm{d}}(-n-1)=-K_{\mathrm{d}}(n), \quad n\in\mathbb{Z}, \end{align*}$$

and in particular, $\sum _{n\in \mathbb {Z}}K_{\mathrm {d}}(n)=0$ (understood as pairwise cancellation between pairs n and $-n-1$ ).

Proof Formula (4.1) shows that the kernel is real-valued. Let $m_d(\theta )=(-\mathrm i\,\mathrm {sgn}\ \theta )\rho ^{1/2}\mathrm e^{\mathrm i\theta /2}$ be the Fourier multiplier. Then,

$$ \begin{align*} m_d(-\theta)=& \, (-\mathrm i\,\mathrm{sgn}(-\theta))\rho^{1/2}\mathrm e^{-\mathrm i\theta/2} \\= & \, \mathrm i\,\mathrm{sgn}\ \theta\,\rho^{1/2}\mathrm e^{-\mathrm i\theta/2} \\ = &-\,\mathrm e^{-\mathrm i\theta}\,m_d(\theta). \end{align*} $$

Thus, we get that

$$ \begin{align*} K_{\mathrm{d}}(-n-1) &=\frac{1}{2\pi}\!\int_{-\pi}^{\pi} m_d(\theta)\,\mathrm e^{-\mathrm i(n+1)\theta}\,d\theta \\ &=-\frac{1}{2\pi}\!\int_{-\pi}^{\pi} m_d(-\theta)\, e^{\mathrm i \theta} \mathrm e^{-\mathrm i n\theta}\,\mathrm e^{-\mathrm i\theta}\,d\theta \\ &=-\frac{1}{2\pi}\!\int_{-\pi}^{\pi} m_d(\theta)\,\mathrm e^{\mathrm i n\theta}\,d\theta \\ &=-K_{\mathrm{d}}(n). \end{align*} $$

The pairwise cancellation of the series $\sum _{n}K_{\mathrm {d}}(n)$ follows.

4.3 Asymptotics of $\boldsymbol{K_{\mathrm {d}}(n)}$ and $\boldsymbol{\ell ^p(w)}$ bounds

We next compare $H_{\mathrm d}$ to the kernel of the Riesz–Titchmarsh transform

$$ \begin{align*} K_{H^+}(n)& = \frac{1}{2\pi}\!\int_{-\pi}^{\pi}\!(-\mathrm i\,\mathrm{sgn}\ \theta)\,\,\mathrm e^{\mathrm i(n+\frac12)\theta}\,d\theta \\ & =\frac{1}{\pi}\!\int_{0}^{\pi}\!\sin \bigl((n+\tfrac12)\theta\bigr)\,d\theta \\ &=\frac{1}{\pi}\frac{1}{n+\tfrac12}. \end{align*} $$

We define the difference kernel $D(n):=K_{\mathrm {d}}(n)-K_{H^{+}}(n)$ and prove that $D\in \ell ^1(\mathbb {Z})$ , while giving an explicit bound.

Weighted $\ell ^p(w)$ bounds are an immediate corollary. A weight is a function $w:\mathbb {Z}\to (0,\infty )$ . For integers $M\le N$ , write $I=[M,N]\cap \mathbb {Z}$ and $|I|=N-M+1$ . For $1<p<\infty $ , the Muckenhoupt class $A_{p}$ [Reference Hunt, Muckenhoupt and Wheeden10] consists of all w for which there exists $C<\infty $ such that, for every pair $M\le N$ ,

(4.2) $$ \begin{align} \Big(\sum_{k=M}^{N} w(k)\Big)\, \Big(\sum_{k=M}^{N} w(k)^{-1/(p-1)}\Big)^{p-1} \;\le\; C\,|I|^{p}. \end{align} $$

For $A_{1}$ , one requires that, for all $M\le N$ ,

(4.3) $$ \begin{align} \Big(\sum_{k=M}^{N} w(k)\Big)\, \max_{k\in I} \frac{1}{w(k)} \;\le\; C\,|I|. \end{align} $$

Now, if we set

$$\begin{align*}f(\theta):=\rho(\theta)^{1/2}-1,\qquad \theta\in[0,\pi], \end{align*}$$

the difference kernel is given by

(4.4) $$ \begin{align} D(n)=K_{\mathrm{d}}(n)-K_{H^+}(n) \;=\;\frac{1}{\pi}\int_{0}^{\pi} f(\theta)\,\sin \bigl((n+\tfrac12)\theta\bigr)\,d\theta. \end{align} $$

We prove that $D(n)=O(n^{-2})$ ; hence, $K_{\mathrm {d}}(n)=\frac {1}{\pi (n+\frac 12)}+O(n^{-2})$ as $|n|\to \infty $ .

Starting from Equation (3.2): $ (\rho -1)^{2}=4\sin ^{2}(\tfrac {\theta }{2})\,\rho , $ for $\theta \in [0,\pi ]$ , we get that

$$ \begin{align*} \rho(\theta)^{1/2} = \sqrt{1+\sin^{2}\!\tfrac{\theta}{2}}\;-\;\sin\tfrac{\theta}{2}. \end{align*} $$

Expanding around $\theta =0$ , we get that as $\theta \to 0$ :

$$ \begin{align*} \rho(\theta)^{1/2} &= \Big(1+\tfrac12\sin^{2}\tfrac{\theta}{2}+O(\theta^{4})\Big) -\sin\tfrac{\theta}{2} \notag\\ &= 1 - \frac{\theta}{2} + \frac{\theta^{2}}{8} + O(\theta^{3}), \end{align*} $$

so that

(4.5) $$ \begin{align} f(\theta)=\rho(\theta)^{1/2}-1 = -\frac{\theta}{2} + \frac{\theta^{2}}{8} + O(\theta^{3}). \end{align} $$

In particular, $f(0)=0$ , $f'(0)=-\tfrac 12$ , and $f"(0)=\frac 14$ . Using the explicit formula for $f(\theta )$ , one also finds that $f'(\pi )=0$ and that $f"\in C[0,\pi ]$ .

Lemma 4.2 The difference kernel $D=K_{\mathrm {d}}-K_{H^+}$ belongs to $\ell ^1(\mathbb {Z})$ , the asymptotic formula

$$ \begin{align*}K_{\mathrm{d}}(n)=\dfrac{1}{\pi(n+\frac12)}+O(n^{-2})\end{align*} $$

is valid as $|n|\to \infty $ , and for every $1\le p\le \infty $ ,

$$\begin{align*}\|H_{\mathrm{d}}-H^+\|_{\ell^p\to\ell^p}\;\le\;\|D\|_{\ell^1}\;<\infty. \end{align*}$$

Proof Integrating by parts twice in (4.4), we obtain

(4.6) $$ \begin{align} \begin{aligned} |D(n)| \;=& \;\frac{1}{\pi}\Big|\int_{0}^{\pi} f(\theta)\,\sin\!\bigl((n+\tfrac12)\theta\bigr)\,d\theta\Big| \\ \; \le & \; \frac{1}{\pi\left(n+\frac12\right)^{2}}\int_{0}^{\pi} |f"(\theta)|\,d\theta\\[2pt] \; = &\; \frac{C_0}{\left(n+\frac12\right)^{2}} \sim \frac{1}{n^2}. \end{aligned} \end{align} $$

Here, we used the fact that $f \in C^2[0,\pi ]$ , so that $\int _{0}^{\pi } |f"(\theta )|\,d\theta < \infty $ . This proves that $D(n)=O(n^{-2})$ , which implies that $K_{\mathrm {d}}(n)=\frac {1}{\pi (n+\frac 12)}+O(n^{-2})$ as $|n|\to \infty $ . Moreover, by using $\displaystyle \sum _{n \in \mathbb {Z}} \frac {1}{\left ( n+\frac 12\right )^2}=\pi ^2$ , we find that $\|D\|_{\ell ^1} < \infty $ .

The difference

$$\begin{align*}(H_{\mathrm{d}}-H^+)a = D*a \end{align*}$$

is convolution with the $\ell ^1$ kernel D, and the final claimed bound follows from the discrete Young’s inequality.

Using the fact that $H^+$ is a Calderón–Zygmund operator, and hence bounded on $\ell ^p(\mathbb {Z})$ and $\ell ^p(\mathbb {Z}; w)$ when $w \in A_p$ , we obtain the following results.

Proposition 4.3 (Weak-type $(1,1)$ and $\ell ^p$ bounds)

For $p>1$ , we have

$$\begin{align*}\|H_{\mathrm{d}}a\|_{\ell^p}\;\le\;\|H^{+}a\|_{\ell^p}+\|D\|_{\ell^1}\,\|a\|_{\ell^p}. \end{align*}$$

In addition, since $H^{+}$ is of weak type $(1,1)$ ,

$$\begin{align*}\#\big\{n:\ |H_{\mathrm{d}}a(n)|>\lambda\big\} \;\le\;\frac{C\,\|a\|_{\ell^1}}{\lambda}\;+\;\frac{\|D\|_{\ell^1}\,\|a\|_{\ell^1}}{\lambda} \qquad(\lambda>0), \end{align*}$$

so $H_{\mathrm {d}}$ is weak type $(1,1)$ and bounded on $\ell ^p$ for $1<p<\infty $ .

By the proof of Lemma 4.2, the function $K(n):=C_0 \,(|n|+\frac 12)^{-2} \in \ell ^1(\mathbb {Z})$ is a radially decreasing majorant of D. Then, $(|D|*|a|)(n)\ \le \ C\,M a(n)$ , where M is the discrete Hardy–Littlewood maximal operator. The weighted bounds for M give [Reference Stein14]

$$ \begin{align*} \|D*a\|_{\ell^{p}(w)} & \lesssim\ [w]_{A_{p}}^{1/(p-1)}\ \|a\|_{\ell^{p}(w)}, \qquad 1<p<\infty,\\[2pt] w\big(\{|D*a|>\lambda\}\big) & \lesssim\ \frac{[w]_{A_{1}}}{\lambda}\,\|a\|_{\ell^{1}(w)}, \qquad \lambda>0. \end{align*} $$

Thus, $H_{\mathrm {d}}$ is bounded on $\ell ^p(w)$ , if $w \in A_p$ , strongly for $1<p<\infty $ and weakly for $p=1$ .

We will compare $H_{\mathrm {d}}$ and $H^{+}$ at the level of their Fourier multipliers:

$$\begin{align*}m_{\mathrm{d}}(\theta) =(-\mathrm i\,\mathrm{sgn}\ \theta)\,\rho(\theta)^{1/2}\,\mathrm e^{\mathrm i\theta/2},\qquad m_{H^+}(\theta)=(-\mathrm i\,\mathrm{sgn}\ \theta)\,\mathrm e^{\mathrm i\theta/2}. \end{align*}$$

The expansion (4.5) gives

$$\begin{align*}m_{\mathrm{d}}(\theta)-m_{+}(\theta) =(-\mathrm i\,\mathrm{sgn}\ \theta)\bigl(\rho^{1/2}(\theta)-1\bigr)\mathrm e^{\mathrm i\theta/2} = \frac{\mathrm i}{2}\,|\theta| + O(\theta^{2})\qquad(\theta\to0). \end{align*}$$

Thus, the difference multiplier vanishes to first order at the origin. On the circle, a $C^{1}$ multiplier with a simple zero at $0$ has Fourier coefficients decaying as $O(|n|^{-2})$ . This is precisely (4.6) proved by two integrations by parts, with the vanishing of $f(0)$ removing the $O(1/n)$ boundary term from the first integration by parts.

5.1 The $\boldsymbol{(2s{+}3)}$ -point Laplacian and the Poisson extension

The fully discrete $(2s{+}3)$ -point Laplacian on $\mathbb {Z}^{s}\times \mathbb {N}$ is

$$ \begin{align*} (\Delta_{s}{F})(x,k) = \sum_{j=1}^{s}\big(F(x+e_{j},k)+F(x-e_{j},k)\big) + F(x,k+1)+F(x,k-1) -2(s{+}1)F(x,k), \end{align*} $$

so $\Delta _{s}=\sum _{j=1}^{s}\delta ^{-}_{x_{j}}\delta ^{+}_{x_{j}} + \partial _{k}^{-}\partial _{k}^{+}$ . Given a boundary sequence $a:\mathbb {Z}^{s}\to \mathbb {C}$ at height $k=0$ , we seek $U:\mathbb {Z}^{s}\times \mathbb {N}\to \mathbb {C}$ with $U(\cdot ,0)=a$ and $\Delta _{s}{U}=0$ .

Let $\mathbb {T}^{s}=(-\pi ,\pi ]^{s}$ and set

$$\begin{align*}\widehat{a}(\theta)=\sum_{x\in\mathbb{Z}^{s}}a(x)\,e^{-\mathrm i x\cdot\theta}, \qquad \mathcal F^{-1}_{s}[g](x)=\frac{1}{(2\pi)^{s}}\int_{\mathbb{T}^{s}}g(\theta)\,e^{\mathrm i x\cdot\theta}\,d\theta, \; \theta=(\theta_{1},\dots,\theta_{s}). \end{align*}$$

Taking the horizontal Fourier transform of $\Delta _{s}{U}=0$ yields, for each fixed $\theta \in \mathbb {T}^{s}$ ,

$$\begin{align*}\widehat U(\theta,k+1)+\widehat U(\theta,k-1) +\Big(2\sum_{j=1}^{s}\cos\theta_{j}-2(s{+}1)\Big)\widehat U(\theta,k)=0. \end{align*}$$

Looking for separated solutions $\widehat U(\theta ,k)=C(\theta )\,\rho (\theta )^{k}$ gives

$$\begin{align*}\rho+\rho^{-1}=2\Big((s{+}1)-\sum_{j=1}^{s}\cos\theta_{j}\Big). \end{align*}$$

We keep the root in $(0,1]$ to ensure boundedness as $k\to \infty $ :

(5.1) $$ \begin{align} \rho(\theta)\;=\;\alpha(\theta)-\sqrt{\alpha(\theta)^{2}-1}, \qquad \alpha(\theta):=(s{+}1)-\sum_{j=1}^{s}\cos\theta_{j}\in [1,2{s}+1]. \end{align} $$

Imposing $\widehat U(\theta ,0)=\widehat a(\theta )$ yields

$$\begin{align*}\widehat U(\theta,k)=\rho(\theta)^{k}\,\widehat a(\theta), \qquad U(x,k)=\mathcal F^{-1}_{s}\!\big[\rho(\cdot)^{k}\widehat a\big](x) =\sum_{y\in\mathbb{Z}^{s}}P_{k}^{(s)}(x-y)\,a(y), \end{align*}$$

where the s-dimensional discrete Poisson kernel is

$$\begin{align*}P_{k}^{(s)}(x)=\frac{1}{(2\pi)^{s}}\int_{\mathbb{T}^{s}}\rho(\theta)^{k}\,e^{\mathrm i x\cdot\theta}\,d\theta. \end{align*}$$

Similarly as in Lemma 2.1, $P_{k}^{(s)}\ge 0$ , $P_{k}^{(s)}\in \ell ^{1}(\mathbb {Z}^{s})$ , and $\sum _{x\in \mathbb {Z}^{s}}P_{k}^{(s)}(x)=1$ for every $k\in \mathbb {N}$ .

5.2 A vector conjugate function and the discrete Cauchy–Riemann system

For the conjugate part, we construct s components $V=(V_{1},\dots ,V_{s})$ .

We introduce

$$ \begin{align*} \omega_j(\theta)= \begin{cases} \dfrac{\mathrm{sgn}\ \theta_j\,\sin(\theta_j/2)}{\Big(\sum_{j=1}^{s}\sin^{2}\!\frac{\theta_{j}}{2}\Big)^{1/2}},& \theta\neq(0,0,\ldots,0),\\[6pt] 0,& \theta=0, \end{cases} \qquad j=1,\ldots,s. \end{align*} $$

The key algebraic identity now reads

(5.2) $$ \begin{align} \rho(\theta)-1 \;=\; -\,\rho(\theta)^{1/2}\sum_{j=1}^{s} \omega_{j}(\theta)\,\bigl(e^{\mathrm i\theta_{j}}-1\bigr)\, \bigl(-\mathrm i\,\mathrm{sgn}\ \theta_{j}\bigr)\,e^{-\mathrm i\theta_{j}/2}, \end{align} $$

and it is derived using the identity:

$$ \begin{align*} \frac{(\rho(\theta)-1)^{2}}{\rho(\theta)} \;=\; 2\sum_{j=1}^{s}\bigl(1-\cos\theta_{j}\bigr) \;=\; 4\sum_{j=1}^{s}\sin^{2}\!\frac{\theta_{j}}{2}, \qquad \theta\in\mathbb{T}^{s}. \end{align*} $$

We define $\widehat V_{j}$ on the Fourier side by

(5.3) $$ \begin{align} \widehat V_j(\theta,k) =\bigl(-\mathrm i\,\mathrm{sgn}\ \theta_j\bigr)\,\omega_j(\theta)\, \rho(\theta)^{\,k+\frac12}\,e^{\mathrm i\theta_j/2}\,\widehat a(\theta), \qquad \theta\in\mathbb{T}^{s},\ k\in\mathbb{N}. \end{align} $$

If we let $V_{j}(x,k)=\mathcal F^{-1}_{s}[\widehat V_{j}(\cdot ,k)](x)$ , then each $V_{j}$ is harmonic with respect to $\Delta _{s}$ .

We obtain the following equations, analogous to the generalized Cauchy–Riemann system in higher dimensions [Reference Stein14, p. 120].

Proposition 5.1 (Discrete Cauchy–Riemann system in $\mathbb {Z}^{s}\times \mathbb {N}$ )

With U and $V=(V_{1},\dots ,V_{s})$ defined above, for every $x\in \mathbb {Z}^{s}$ and $k\ge 1$ , we have

(5.4) $$ \begin{align} \partial_{k}^{+}U(x,k) &= -\,\sum_{j=1}^{s}\delta^{-}_{x_{j}}V_{j}(x,k), \end{align} $$

(5.5) $$ \begin{align} \delta^{+}_{x_j}U(x,k) &=\,\partial_{k}^{-}V_{j}(x,k)\qquad (j=1,\dots,s), \end{align} $$

(5.6) $$ \begin{align} \delta^{+}_{x_j}V_{l}(x,k) &=\,\delta^{+}_{x_l}V_{j}(x,k)\qquad (j,l=1,\dots,s). \end{align} $$

Proof Using the shift rules and the definitions,

$$\begin{align*}\mathcal F_{s}\!\big[\partial^{+}_{k}U\big] =(\rho-1)\,\rho^{k}\widehat a,\qquad \mathcal F_{s}\!\big[\delta^{-}_{x_j}V_j\big] =(1-\mathrm e^{-\mathrm i\theta_j})\,(-\mathrm i\,\mathrm{sgn}\ \theta_j)\,\omega_j\, \rho^{k+\frac12}\,\mathrm e^{\mathrm i\theta_j/2}\widehat a. \end{align*}$$

Therefore, (5.4) reduces to (5.2). The proof of (5.5) is similar.

To prove (5.6), we use that, for all j and l, the following identity holds:

$$ \begin{align*} \big(\mathrm e^{\mathrm i\theta_j}-1\big)\,\bigl(-\mathrm i\,\mathrm{sgn}\ \theta_l\bigr) \omega_l(\theta)\,\mathrm e^{\mathrm i\theta_l/2} =\big(\mathrm e^{\mathrm i\theta_l}-1\big)\,\bigl(-\mathrm i\,\mathrm{sgn}\ \theta_j\bigr) \omega_j(\theta)\,\mathrm e^{\mathrm i\theta_j/2}. \\[-32pt]\end{align*} $$

Thus, the complex vector field $F=(U,V_{1},\dots ,V_{s})$ obeys a lattice Cauchy–Riemann system whose elimination produces the $(2s{+}3)$ -point Laplace equation.

5.3 Boundary transforms and properties

At height $k=0$ , (5.3) gives the boundary operators $T_{j}$ with Fourier multipliers

$$ \begin{align*} m_j(\theta)&=\bigl(-\mathrm i\,\mathrm{sgn}\ \theta_j\bigr)\,\omega_j(\theta)\,\rho(\theta)^{1/2}\,e^{\mathrm i\theta_j/2} \;\\&=\; -\,\mathrm i\,\frac{\sin(\theta_j/2)}{\Big(\sum_{j=1}^{s}\sin^{2}\!\frac{\theta_{j}}{2}\Big)^{1/2}}\,\rho(\theta)^{1/2}\,e^{\mathrm i\theta_j/2}. \end{align*} $$

Therefore, for $x=(x_1,\dots ,x_s)\in \mathbb {Z}^{s}$ , we have

$$\begin{align*}(T_j a)(x)=\sum_{y\in\mathbb{Z}^{s}}K_{T_j}(x-y)\,a(y), \quad\text{where}\quad K_{T_j}(x) =\frac{1}{(2\pi)^{s}}\int_{\mathbb{T}^{s}} m_j(\theta)\,\mathrm e^{\mathrm i x\cdot\theta}\,d\theta. \end{align*}$$

We get that the kernel $K_{T_j}$ is real-valued. Moreover, for every $x\in \mathbb {Z}^{s}$ ,

$$\begin{align*}K_{T_j}(-x-e_j)=-\,K_{T_j}(x), \end{align*}$$

and we have a similar cancellation property as in Proposition 4.1: $\sum _{x\in \mathbb {Z}^{s}}K_{T_j}(x)=0$ .

Consider the following Riesz transforms $R_{j}^{+}$ on $\mathbb {Z}^s$ , defined by their Fourier multipliers

$$ \begin{align*} \widehat{R_{j}^{+}a}(\theta) = m_{R_{j}}(\theta)\,\widehat a(\theta),\qquad m_{R_{j}}(\theta):=\bigl(-\mathrm i\,\mathrm{sgn}\ \theta_{j}\bigr)\,\omega_{j}(\theta)\,e^{\mathrm i\theta_{j}/2}. \end{align*} $$

These operators are Calderón–Zygmund operators on the discrete space $\mathbb {Z}^{s}$ . Indeed, the size and smoothness estimates for their corresponding kernels follow from the expansions of the multipliers near $\theta =0$ : $m_{R_j}(\theta ) = -\mathrm i\,\frac {\theta _j}{|\theta |}+O(|\theta |).$

As before, we obtain the mapping properties of $T_{j}$ by comparing them to the transforms $R_j^+$ . For $1<p<\infty $ and weights $w\in A_{p}(\mathbb {Z}^{s})$ , the operators are bounded on $\ell ^p(w; \mathbb {Z}^s)$ . Moreover, when $w\in A_{1}(\mathbb {Z}^{s})$ , each $T_{j}$ is of weak type $(1,1)$ ; that is, $T_j:\ell ^1(w; \mathbb {Z}^s) \to \ell ^{1,\infty }(w; \mathbb {Z}^s)$ .