What can I say about the function $\varphi$, if $\varphi\in L^p\left((a,b)\times (c,d)\right)$ and $$ \int_c^d \varphi(x,y) dy=0\mbox{ a.e. }x\in (a,b) ? $$ Can I say that $\varphi$ is independent of $x$? If I can not, when will I be able to conclude that $\varphi$ independs on $x$?
Of course, what other properties this $\varphi$ can have?
If $\phi (x,y)=f(x)g(y)$ with $\int_c^{d} g(y)dy=0$ then the hypothesis is satisfied but $\phi $ is not independent of $x$, even if $f$ and $g$, (hence $\phi $) are smooth. I have no idea why you thin k that $\phi $ should be independent of $x$.