Problem 1: Find all functions $f:\mathbf{Z}^2 \to \mathbf{R}$ which are harmonic in the sense that $f(x,y) = \frac{f(x+1,y) + f(x-1,y) + f(x,y+1) + f(x,y-1)}{4}$for all $(x,y)\in\mathbf{Z}^2$, and which are also Lipschitz in the sense that the gradients $f(x+1,y)-f(x,y)\\f(x-1,y)-f(x,y)\\f(x,y+1)-f(x,y)\\f(x,y-1)-f(x,y)$are all globally bounded.
Obvious examples: $f = 1$, $f=x$, $f=y$, and linear combinations of these. Is this all?
Problem 2: What further examples do we get if we weaken the Lipschitz condition, say by allowing the gradients to grow at most linearly (with respect to distance from $(0,0)$)?
Problem 3: How much do the character of our examples change if we replace the generating set $S = \lbrace (1,0),(-1,0),(0,1),(0,-1)\rbrace$ by another (symmetric) set $S$ which generates $\mathbf{Z}^2$? For example, what if we require that $f(x) = \frac{1}{|S|}\sum_{s\in S}f(x+s),$for, say, $S = \lbrace s: \|s\|_2\leq 100\rbrace$? Does the dimension of the space of Lipschitz harmonic functions change?
[Background: I'm trying to understand Kleiner's theorem, which states that if a finitely generated group $G$ has polynomial growth then the space of Lipschitz harmonic functions on $G$ has finite dimension. The simplest example $G=\mathbf{Z}$ is pretty simple, but the second simplest example $G=\mathbf{Z}^2$ already seems nonobvious to me.]