Let be $d>0$ an integer number. Consider the Cartesian product $\mathbb Z^d$ and the Lebesgue space 
 $L^1(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$, where $\sharp$ denotes the counting measure. 
If $g$ is a function having the following two properties:
 
1) $g\in L^1(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$, i.e, $\sum_{z\in \mathbb Z^d}g(z)$ 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  
$$
\sum_{z\in\mathbb Z^d}g(\|x-z\|_1)g(\|z-y\|_1)\leq K g(\|x-y\|_1),
$$
where $\|z\|_1=\sum_{j=0}^d|z_j|$.


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

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

But, if $g$ decays fast as
$$
g(z)=e^{-r\|z\|_1},
$$
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 $A^2$.

I used the measure theoretical notation, to make simple to ask the natural generalizations of this question.