Vtyjkyuk

Suppose $Omega subset mathbbR^2$ is open and bounded. Let $g in C^infty_c(mathbbR^2)$ and $f in L^2(mathbbR^2)$. Let
beginequation*
h(x,x',y,y') := frac1^2)^r int_mathbbR^2 g(t)(( f(x'+t)-f(y'+t))^2-(f(x+t)-f(y+t))^2) dt.
endequation*
I wonder if $exists r > 0$ such that $h in L^infty(Omega^4)$.

I've been able to bound some of the terms, but when I expand the products, the mixed terms are problematic for me. Here is my approach for the pure terms:

beginequation*
beginsplit
&int g(t) [f^2(x'+t)+f^2(y'+t)-f^2(x+t)-f^2(y+t)] dt \
&= int f^2(t) [g(t-x')+g(t-y')-g(t-x)-g(t-y)] dt \
&leq C int f^2(t) (|x'-x|^2+|y'-y|^2)^frac12 \
&= C | f |_L^2(mathbbR^2) (|x'-x|^2+|y'-y|^2)^frac12
endsplit
endequation*
where the second to last line follows by the Holder continuity of $g$. For the mixed terms I've been playing around with adding and subtracting terms, but I'm not having a lot of luck. Any help would be greatly appreciated.