Does there exist either one / general class of nonnegative 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



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. 

