Denote by $H_{s}(\mathbb{R}^{n})$ the $s$-th Sobolev space on $\mathbb{R}^{n}$. Fix a real $s$ and a non-negative integer $k$ and let $A: H_{s}(\mathbb{R}^{n}) \rightarrow H_{s-k}(\mathbb{R}^{n})$ be a continuous linear operator. Then by the Schwartz kernel theorem $A$ has a distributional kernel, which is a distribution on $\mathbb{R}^{2n}$. What can be said about the order of this distribution?
Any reference would be greatly appreciated.