The following, stated as e.g. Theorem 8.2.13 in Hörmander’s *The Analysis of Linear Partial Differential Operators I*, is standard:

**Theorem 0.** If $ X \subseteq \mathbb{R}^n$ , $ Y \subseteq \mathbb{R}^m$ , $ Z \subseteq \mathbb{R}^p$ are open, $ K_1 \in \mathscr{D}'(X \times Y)$ , $ K_2 \in \mathscr{D}'(Y \times Z)$ , the projection $ \mathrm{supp} K_2 \ni (y,z) \mapsto z$ is proper, and (using Hörmander’s notation) $ $ (*) \quad WF'(K_1)_Y \cap WF(K_2)_Y = \emptyset,$ $ then the composition $ \mathcal{K} = \mathcal{K}_1 \circ \mathcal{K}_2$ of the operators $ \mathcal{K}_2 : C_\mathrm{c}^\infty(Z) \to \mathscr{D}'(Y)$ , $ \mathcal{K}_1 : C_\mathrm{c}^\infty(Y) \to \mathscr{D}'(X)$ is defined and continuous from $ C_\mathrm{c}^\infty(Z)$ to $ \mathscr{D}'(X)$ , and for its kernel $ K \in \mathscr{D}'(X \times Z)$ it holds that $ $ (**) \quad WF'(K) \subseteq [WF'(K_1) \circ WF'(K_2)] \cup [WF(K_1)_X \times 0_{T^*Z}] \cup [0_{T^*X} \times WF'(K_2)_Z].$ $

# Doing without $ (*)$ ?

Suppose we had, instead, the following setup: $ X \subseteq \mathbb{R}^n$ , $ Y \subseteq \mathbb{R}^m$ , $ Z \subseteq \mathbb{R}^p$ are open, $ K_1 \in \mathscr{D}'(X \times Y)$ , $ K_2 \in \mathscr{D}'(Y \times Z)$ , and the associated continuous operators $ \mathcal{K}_2 : C_\mathrm{c}^\infty(Z) \to \mathscr{D}'(Y)$ , $ \mathcal{K}_1 : C_\mathrm{c}^\infty(Y) \to \mathscr{D}'(X)$ have the following properties:

- for some $ s \in \mathbb{R}$ , $ \mathcal{K}_2$ happens to map $ C_\mathrm{c}^\infty(Z)$ continuously to the space $ H^s_\mathrm{c}(Y)$ of compactly supported, order-$ s$ Sobolev distributions on $ Y$ ;
- $ \mathcal{K}_1$ admits a continuous extension to a map from $ H^s_\mathrm{c}(Y)$ to $ \mathscr{D}'(X)$ .

Clearly, in this scenario the composite map $ \mathcal{K} = \mathcal{K}_1 \circ \mathcal{K}_2$ is automatically well-defined and continuous from $ C_\mathrm{c}^\infty(Z)$ to $ \mathscr{D}'(X)$ .

**Is this information alone enough to extract a bound for $ WF(K)$ similar to $ (**)$ ? Is $ (**)$ itself still valid in this scenario?**

The reason I am using as “intermediate” space a Sobolev-type space $ H^s_\mathrm{c}$ , and am not instead allowing for any arbitrary space $ \mathscr{X}$ of distributions with compact support on $ Y$ such that $ C_\mathrm{c}^\infty(Y) \hookrightarrow \mathscr{X} \hookrightarrow \mathscr{D}'(Y)$ , is that I am *guessing* that the answer to my second question is negative, while I am *hoping* that a useful bound may be described if one is ready to use $ H^s$ wave front sets (and perhaps to add further assumptions). Which leads me to the next part of this question…

# An $ H^s$ analog of Theorem 0?

Returning to the Theorem 0, I wonder if condition $ (*)$ there may be replaced with a weaker one along the following lines: $ WF’^s(K_1)_Y \cap WF^t(K_2)_Y$ must be empty for some suitable $ s, t \in \mathbb{R}$ . If this is indeed the case, what is the resulting estimate for the ($ H^s$ ) wave front set of the composite map?