Does there exist   either one /  general class of non-negative definite , symmetric  Integral  Kernel map  satisfying the following properties ??<br>
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$ <br>
$K:L^2(\Omega)\rightarrow L^2(\Omega) , f , g \in L^2(\Omega) $ <br>
1. $ K(x,y) : \Omega\times\Omega\rightarrow R $ where $\Omega$ is a compact  set of $R^2$  and  <br>
2. It has the property that  $\forall y \in \Omega $ <br>  $\partial_{x_{1}}K(x,y)$= $\partial_{x_{2}}K(x,y)$  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<br>