Let $(E,\mathcal E,\mu)$ be a probability space, $p:E^2\to[0,\infty)$ be $\mathcal E^{\otimes2}$-measurable with $$p(y,x)=p(x,y)\;\;\;\text{for all }x,y\in E\tag1$$ and $g\in\mathcal L^2(\mu)$ be nonnegative. Are we able to show that $$\int\mu({\rm d}x)\left|\int\mu({\rm d}y)p(x,y)(g(y)-g(x))\right|^2\le c\int\mu({\rm d}x)|g(x)|^2\tag2$$ for some $c\ge0$?
The "usual" inequalities are not sharp enough. Are we able to deduce the desired claim from $(1)$? Or do we need to impose further assumtpions?