I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below.
Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, Q:=M\times[0,T]$.
For a sufficiently differentiable function $u:Q\to\mathbb{R}^q$, we want to give a quantity $|u|^{(2+\alpha, 1+\frac{\alpha}{2})}_Q$ as (4.16) below. But I cannot understand what is \begin{eqnarray} \langle D^2_x u\rangle^{(\alpha)}_x \end{eqnarray} . Please tell me how to define the quantity above in usual. Thanks a lot.
In the image below, the covariant derivative $\nabla$ denotes the Levi-Civita connection w.r.t. the Riemannian metric $g$.
