Originally Posted by

**vincisonfire** Hi, I just gonna give some thoughts.

The region $\displaystyle \Omega$ is a wedge below $\displaystyle y=x$ in the first quadrant.

The function $\displaystyle Y(t,s):=\int_{s}^{t}X(t,u)\mathrm{d}u$ integrates the function $\displaystyle X$ along the line $\displaystyle (t,s)\rightarrow (t,t)$.

It seems to me that the integrand $\displaystyle X$ could take negative values as long as it overshoots fast enough close to the region $\displaystyle y=x$ since you always have to integrate that part.

I think you can "build" such a function slice by slice ($\displaystyle t$ by $\displaystyle t$). You could take some function that starts at zero, assume negative values until $\displaystyle y=x/2$ where it goes back to zero. Then it assumes large positive values from $\displaystyle y=x/2$ to $\displaystyle y=x$. Find some way to make that $\displaystyle t$-dependent, and you should have a counter-example. You could take for example a $\displaystyle \sin$ which period (and amplitude) varies with $\displaystyle t$ for the lower part and a sharp $\displaystyle \tanh$ for the upper part.