I was looking up the Gagliardo–Nirenberg–Sobolev inequality. In its formulation
\begin{equation} \|u\|_{L^{p^*}(\mathbf{R}^n)}\leq C \|Du\|_{L^{p}(\mathbf{R}^n)}. \end{equation}
I didn't really understand what $Du$ was. Is it the gradient of $u$? Just a partial derivative? Something else? And what is the $L^p$ norm $\lVert Du\rVert$ for this vector valued function?
$Du$ denotes the transposed gradient. The norm is usually given by $$ \lVert Du \rVert_{L^p(\mathbb{R}^n)} = \left(\sum_{j = 1}^n \int_{\mathbb{R}^n} \lvert \partial_{x_j} u(x) \rvert^p ~\mathrm{d}x \right)^\frac{1}{p} $$ or $$ \lVert Du \rVert_{L^p(\mathbb{R}^n)} = \sum_{j = 1}^n \left(\int_{\mathbb{R}^n} \lvert \partial_{x_j} u(x) \rvert^p ~\mathrm{d}x \right)^\frac{1}{p}. $$ Both are equivalent norms. They are known as sobolev norms.
This notation does, admittedly, take some time getting used to but it is very common in functional analysis and PDE theory.
