Suppose $u: A\times B\to (0,\infty)$ be a function such that $$ \int_{x\in A}\int_{y\in B}u^2(x,t)\,dx dt\leq \int_{x\in A}\int_{y\in B}f(u(x,t))\,dx dt. $$ Now let us fix $\hat{t}\in B$, and define $v(x,\hat{t})=u(x,\hat{t})$, then is it possible to get $$ \int_{x\in A}\int_{B}v^2(x,\hat{t})\,dx dt\leq \int_{x\in A}\int_{B}f(v(x,\hat{t}))\,dx dt. $$
Note that is this holds then we get $$ \int_{x\in A}u^2(x,\hat{t})\,dx\leq\int_{x\in A}f(u(x,\hat{t}))\,dx. $$
You can take any function $f$. Basically, my question is if one can get the second one from the fist when one fixes the variable $t$ and get the same estimate for the one variable function $v(x,\hat{t})$?
Can somebody kindly help me?
Thanking you.