Let$(X,Y)$ be an RV of the continous type with PDF $f(x,y)$.Let $Z=X+Y$,then the Convolution of probability distributions told us the PDF of $Z$ is $f_{Z}(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$.
If we suppose that
$$f(x,y)= \left\{\begin{matrix} p(x,y)& (y\ge0) \wedge (x\le y)\\ 0& \text{otherwise}\\ \end{matrix}\right.$$
From above formula for the distribution of the sum $Z=X+Y$,
I get the PDF of $Z$ as follows: $$f_{Z}(z)=\left\{\begin{matrix} \int_{-\infty}^{z}p(x,z-x)dx &z\le0\\ \int_{-\infty}^{z/2}p(x,z-x)dx &z> 0 \\ \end{matrix}\right.$$ I'm not sure if this result is correct, please correct me if necessary.