I have one question about the Hölder's inequality.
Let $f:\Omega \times [0,\infty) \rightarrow \mathbb{R}$ be in $L^{\infty}(\Omega\times[0,\infty))$ and $g: \Omega \rightarrow \mathbb{R}$ be in $L^1(\Omega)$. By using the Hölder's inequality, can we conclude that $\int_{\Omega}f(x,t)g(x)dx \leq ||f||_{L^{\infty}(\Omega\times[0,\infty))}||g||_{L^1(\Omega)}$ ?.
I wasn't sure because domains of $f$ and $g$ are different, $\Omega \times [0,\infty)$ and $\Omega$. How can we directly use Hölder's inequality?