Let $T(x,y)$ be a tensor locally defined in $\mathcal{U}$ which is $C^{\infty}$ on $\mathcal{U}\setminus U$, then for a non-negative integer $k$ we say as $r\rightarrow 0$

1. (1)

   $T(x,y)=O^{\prime}(r^{k})$ if for each $\epsilon>0$

   (3.14)

   $\displaystyle\Big|\partial_{x}^{I}\partial_{y}^{J}T(x,y)\Big|=\begin{cases}O(r^{-\epsilon}),&|J|\leq k,\\ O(r^{k-|J|-\epsilon}),&|J|>k,\end{cases}$

   for all multi-indices $I$ and $J$. In particular, if $T\in C^{\infty}(\mathcal{U})$, then $T=O^{\prime}(r^{k})$ for all $k\in\mathbb{Z}$.

2. (2)

   $T(x,y)=r^{k}O^{\prime}(1)$ if $r^{-k}T(x,y)=O^{\prime}(1)$.

3. (3)

   $T(x,y)=\widetilde{O}(r^{k})$ if $T(x,y)\in C^{\infty}(\mathcal{U})$ and

   (3.15)

   $$T(x,y)=r^{k}O^{\prime}(1).$$

   In other words, $T(x,y)$ is smooth in $\mathcal{U}$ and has vanishing normal derivatives along $U$ up to order $k-1$.