Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, and let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$. Let $R \in \Psi^{-\infty}(U)$ be a smoothing pseudodifferential operator with Schwartz kernel $K_R \in C^\infty(U \times U)$. For every $s,t \in \mathbb{R}$ and every $\chi,\psi \in C_c^\infty(U)$, define the localized kernel $k_{\chi,\psi}: U \times U \to \mathbb{C}$ by $k_{\chi,\psi}(x,y)=\chi(x)K_R(x,y)\psi(y)$. Let $\rho_U: H^s(\mathbb{R}^n) \to H^s(U)$ denote the restriction map and let $\mathcal{D}_s(U):=\rho_U(C_c^\infty(\mathbb{R}^n)) \subset H^s(U)$ be the dense subspace of restrictions of global test functions. Define the localized smoothing operator $S_{\chi,\psi}: \mathcal{D}_s(U) \to C^\infty(U)$ as follows: for $u=\rho_U F$ with $F \in C_c^\infty(\mathbb{R}^n)$,

\begin{align*} (S_{\chi,\psi}u)(x)=\int_U k_{\chi,\psi}(x,y)u(y)\,d\mathcal{L}^n(y). \end{align*}

This definition is independent of the chosen representative $F$. Then $S_{\chi,\psi}$ extends uniquely to a bounded [linear map](/page/Linear%20Map)

\begin{align*} S_{\chi,\psi}: H^s(U) \to H^t(U), \end{align*}

where $H^s(U)$ and $H^t(U)$ are the Sobolev spaces obtained by restriction from $\mathbb{R}^n$ and equipped with the corresponding quotient norms.