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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 $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.

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

1. $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ is convergent;

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

Question: Can we determine lower bounds for the ratio decay of $g(\|x-y\|_1)$ when $\|x-y\|_1$ goes to infinity ?

Examples:

Ex1: For any $\varepsilon>0$ $$ g(\|z\|_1)=\frac{1}{1+\|z\|_1^{d+\varepsilon}} $$ has the properties 1 and 2.

For the other hand, $$ g(\|z\|_1)=e^{-r\|z\|_1}, $$ where $r>0$, breaks the property 2.

Edit:

I added the Toeplitz operator tag, because of the asymptotic behavior for $g$ (in terms of the lower bounds) could be obtained thinking $g(\|x-y\|_1)$ as matrix elements of a Toeplitz operator $A:L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)\to L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$. In fact, in this point of view, we ask for lower bounds for the entries of a Toeplitz operator satisfying $(A^2)_{xy}\leq K A_{xy}$, where $(A^2)_{xy}$ is the $xy$ element of the matrix $A^2$.

Remark: The issues pointed out by the Thomas Kragh in the comments were fixed by not considering $g$ as a function of space $\mathbb Z^d$ and requiring it to be positive.

Any reference or help, even for partial answer is very welcome.

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 $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.

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

1. $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ is convergent;

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

Question: Can we determine lower bounds for the ratio decay of $g(\|x-y\|_1)$ when $\|x-y\|_1$ goes to infinity ?

Examples:

Ex1: For any $\varepsilon>0$ $$ g(\|z\|_1)=\frac{1}{1+\|z\|_1^{d+\varepsilon}} $$ has the properties 1 and 2.

For the other hand, $$ g(\|z\|_1)=e^{-r\|z\|_1}, $$ where $r>0$, breaks the property 2.

Edit:

I added the Toeplitz operator tag, because of the asymptotic behavior for $g$ (in terms of the lower bounds) could be obtained thinking $g(\|x-y\|_1)$ as matrix elements of a Toeplitz operator $A:L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)\to L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$. In fact, in this point of view, we ask for lower bounds for the entries of a Toeplitz operator satisfying $(A^2)_{xy}\leq K A_{xy}$, where $(A^2)_{xy}$ is the $xy$ element of the matrix $A^2$.

Remark: The issues pointed out by Thomas Kragh in the comments were fixed by not considering $g$ as a function of space $\mathbb Z^d$ and requiring it to be positive.

Any reference or help, even for partial answer is very welcome.

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. $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ 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.

Things I tried ... to perform a spectral analysis of the related Toeplitz operators. I thought $g(\|x-y\|_1)$ 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 $A^2$.

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 $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.

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

1. $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ is convergent;

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

Question: Can we determine lower bounds for the ratio decay of $g(\|x-y\|_1)$ when $\|x-y\|_1$ goes to infinity ?

Examples:

Ex1: For any $\varepsilon>0$ $$ g(\|z\|_1)=\frac{1}{1+\|z\|_1^{d+\varepsilon}} $$ has the properties 1 and 2.

For the other hand, $$ g(\|z\|_1)=e^{-r\|z\|_1}, $$ where $r>0$, breaks the property 2.

Edit:

I added the Toeplitz operator tag, because of the asymptotic behavior for $g$ (in terms of the lower bounds) could be obtained thinking $g(\|x-y\|_1)$ as matrix elements of a Toeplitz operator $A:L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)\to L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$. In fact, in this point of view, we ask for lower bounds for the entries of a Toeplitz operator satisfying $(A^2)_{xy}\leq K A_{xy}$, where $(A^2)_{xy}$ is the $xy$ element of the matrix $A^2$.

Remark: The issues pointed out by the Thomas Kragh in the comments were fixed by not considering $g$ as a function of space $\mathbb Z^d$ and requiring it to be positive.

Any reference or help, even for partial answer is very welcome.

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 $A^2$.

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. $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ 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.

Things I tried ... to perform a spectral analysis of the related Toeplitz operators. I thought $g(\|x-y\|_1)$ 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 $A^2$.