Does there exist either one / general class of positive definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\Omega)\rightarrow L^2(\Omega)$

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 $x=(x_1,x_2)$
   

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