I'm trying to understand the reason behind the integrability of the products $p\phi$ and $p\frac{\partial\phi}{\partial t}$ as shown below: $$ \int_{-\infty}^{V_F}p(v,T)\phi(v,T)\,dv $$ $$ \int_0^T\int_{-\infty}^{V_F}p(v,t)\frac{\partial}{\partial t}\phi(v,t)\,dv\,dt. $$ Everything I know about these functions is $p\in L^{\infty}(\mathbb{R}^+;L_+^1(-\infty,V_F))$ and $\phi \in C^{\infty}((-\infty,V_F]\times [0,T])$, $v \frac{\partial \phi}{\partial v}, \frac{\partial^2 \phi}{\partial v^2} \in L^{\infty}((-\infty,V_F)\times (0,T))$. Some of this conditions might not be of use, they're just part of another issue which I believe has nothing to do with what I ask.
Any help would be greatly appreciated.