Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??  

$f(x)=(Kg)(x)=\int\_{\Omega}K(x,y)g(y)dy$   

$K:L^2(\Omega)\rightarrow L^2(\Omega) , f , g \in L^2(\Omega) $   

1. $ K(x,y) : \Omega\times\Omega\rightarrow R $ where $\Omega$ is a compact set of $R^2$ and   

2. It has the property that $\forall y \in \Omega $   
 $\partial\_{x\_{1}}K(x,y)$= $\partial\_{x\_{2}}K(x,y)$ where $x=(x\_1,x\_2)$ like a general map $K(x,y)=h(ax\_1+ax\_2)h(ay\_1+ay\_2), $ with $h$ a differentiable map $R \rightarrow R $ and $a$ is a scalar  




A special Integral Kernel 
The function
$$
k(x,y)=\frac{H(x)H(y)}{\pi(x+y)},\quad \text{with}\quad H=\mathbf 1\_{\mathbb R\_+}
$$
is the kernel of the Hardy operator which is bounded on $L^2(\mathbb R)$ with operator norm 1.


