When studying commutator estimates, I have encountered the following problem.
Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$.
My question is what would $\|\nabla f(x)\|_p$ indicate?
I am not sure if the question has been clearly stated, please leave comments if you feel the question is not well formed.
When you have a function $v \colon X \to E$ on a measure space $X$ with values in a Banach space $E$, saying that $v\in L^p(X,\,E)$ usually means that $v$ is a measurable function such that the real-valued function $n \colon x \mapsto \lVert v(x)\rVert$ is in $L^p(X,\,\mathbb{R})$, and $\lVert v\rVert_p = \left(\int \lVert v(x)\rVert_E^p\right)^{1/p}$.
In the case that $E = \mathbb{R}^d$, firstly, that is equivalent to each component function itself being in $L^p$, and secondly, since all norms on $\mathbb{R}^d$ are equivalent, it doesn't matter which norm we choose on $\mathbb{R}^d$, they all produce equivalent norms on $L^p(X,\,\mathbb{R}^d)$. You then get the most convenient computations if you choose the corresponding $\ell^p$-norm on $\mathbb{R}^d$, which becomes
$$\lVert v\rVert_p = \left(\int_X \sum_{i=1}^d \lvert v_i(x)\rvert^p \,d\mu\right)^{1/p},$$
the $p$-th power of the norm of $v$ is the sum of the $p$-th powers of the norms of the components.