Suppose $$\frac{1}{r} = \frac{1}{p} + \frac{1}{q}.$$ The usual Holder's inequality for scalar valued functions says that if $f \in L^p$ and $g \in f^q$, then $fg \in L^r$ and $$\|fg\|_{L^r} \leq \|f\|_{L^p}\|g\|_{L^q}.$$
I am looking if any analogous statement holds for vector-valued $L^p$ spaces. For example, suppose $f \in L^p(0, T; L^u)$ (takes values in $L^u$) and $g \in L^q(0, T; L^v)$. Does Holder's inequality still apply? More specifically, is $fg \in L^r(0, T; L^w)$? If so, what is the requirement on $w$? Is it that $$\frac{1}{w} = \frac{1}{u} + \frac{1}{v}?$$ I could not find anything in my textbooks or online.
EDIT: Based on Nick's comment, I was able to write out $$\|fg\|_{L^r} = \Big(\int_0^T\|fg\|_{L^w}^r\Big)^{1/r} \leq \Big(\int_0^T\|f\|_{L^u}^r\|g\|_{L^v}^r\Big)^{1/r}.$$ I am now working on bounding the above by the product of each integral and hence finish the proof.