Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.

If $g:[0,\infty)\to\mathbb [0,\infty)$ is a function having the following two properties:

1. $\sum_{z\in \mathbb Z^d}g(\|z\|)$ converge;

2. there is a constant $K\in \mathbb R$ (which depends only on $g$) such that for any $x,y\in\mathbb Z^d$, we have
   $$ \sum_{z\in\mathbb Z^d}g(\|x-z\|_1)g(\|z-y\|_1)\leq K g(\|x-y\|_1), $$

Can we determine lower bounds for the ratio decay of $g(\|z\|_1)$ when $\|z\|_1$ goes to infinity ?

Ps1: For any $\varepsilon>0$ $$ g(\|z\|_1)=\frac{1}{1+\|z\|_1^{d+\varepsilon}} $$ `, has the properties 1 and 2.

But, if $g$ decays fast as $$ g(\|z\|_1)=e^{-r\|z\|_1}, $$ where $r>0$, the property 2 is not satisfied.

Ps2: I tried to perform a spectral analysis of the related Toeplitz operators. I thought $g(x-y)$ as matrix elements of a Toeplitz operator $A$ from $L^1(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$ to itself. So the question becomes, what are the Toeplitz operators satisfying $(A^2)_{xy}\leq K A_{xy}$ where $(A^2)_{xy}$ is the $xy$ element of the matrix $A^2$.