Questions about harmonic functions on Cayley graphs

Question

I am reading A proof of Gromov’s theorem by Terence Tao, where I encountered harmonic (and Lipschitz) functions on Cayley graphs, here is the definition:

Let $G$ be an infinite group generate by a finite symmetric set $S$, a function $f: G \rightarrow \mathbb{R}$ is harmonic if: $$ f(x) = \frac{1}{|S|}\sum_{s \in S}f(xs)$$ for all $x \in G$, and $f(x)$ is Lipschitz if: $$ |f(x) - f(xs)| \leq C $$ for all $x\in G$ and $s \in S$, and some $C < \infty $.

And later, he states that: Consider some Lipschitz harmonic functions $u_1,\ldots,u_D$, which we normalise to all vanish at the identity. Let $V$ be the space spanned by $u_1,\ldots,u_D$. For each $R$, the $L^2(B_S(R))$ inner product gives a quadratic form $Q_R$ on $V$. Using this quadratic form, we can build a Gram matrix determinant:

$$ \det( Q_R(u_i,u_j) )_{1 \leq i,j \leq D}.$$

From the Lipschitz nature of the harmonic functions, we have a bound of the form $$\det( Q_R(u_i,u_j) )_{1 \leq i,j \leq D} \ll R^{D}.$$ as $R \rightarrow \infty$.

And I do not know how to get the last inequality: I think that the condition of Lipschitz controls the growth of $f(x)$, but I do not know how to use the harmonic condition here. Is there any information that I missed in the context or any misunderstanding I get?

Edit: I forgot to say that Tao added the condition for group $G$ to be bounded doubling, which says that $|B_s(2R)| \leq C|B_{s}(R)|$ for some fixed constant $C$ and all $R>0$, I think this is also needed to prove the inequality above.

Moreover, since I am quite unfamiliar with this topic, I want to know are there any relevant materials that I can learn from, thanks!

Answer 1

The blog post you are reading is a kind of announcement or sketch accompanying an arxiv preprint https://arxiv.org/abs/0910.4148. The expected level of accuracy for such a medium is similar to that of a talk. Tao probably wrote the whole thing in 15 minutes. To get all the details exactly right, you want to look at the actual paper. In this case see equation (24) on page 19 and the argument on page 21.

In detail, the quadratic form $Q_R$ is defined in the paper by $$Q_R(u,v) = \sum_{x \in B(R)} (u(x) - u(1))(v(x)-v(1)).$$ Assuming $u$ is $1$-Lipschitz, $|u(x) - u(1)| \le R$ for $x \in B(R)$, and similar for $v$, so $|Q_R(u,v)| \le R^2 |B(R)|$. Let $e_1, \dots, e_D$ be a $Q_R$-orthonormal basis for the span of $u_1, \dots, u_D$ and write $u_i = \sum_j u_{ij} e_j$ for each $i$. Then $\sum_j u_{ij}^2 = Q_R(u_i,u_i) \le R^2 |B(R)|$. Let $U$ be the matrix $(u_{ij})$. By the Hadamard bound it follows that $\det U \le R^D |B(R)|^{D/2}$. On the other hand $$Q_R(u_i,u_j) = \sum_k u_{ik} u_{jk} = (UU^T)_{ij},$$ so $$\det(Q_R(u_i,u_j)) = \det(UU^T) = \det(U)^2 \le R^{2D} |B(R)|.$$

In summary there are two inaccuracies in Tao's blog post:

1. In the blog post (unlike the paper) Tao seems to have in mind the normalized variant of $Q_R$, in which the sum over $B(R)$ is replaced by an average. Then $|Q_R(u,v)| \le R^2$.
2. Where Tao wrote $\det(Q_R(u_i,u_j))$ he really meant $\det(Q_R(u_i,u_j))^{1/2}$. In the paper this quantity is referred to as $\operatorname{Vol}_R(u_1, \dots, u_D)$.