Would someone please help me to think of examples of functions belonging to the symbol class $S^{1}_{\Lambda}$ introduced at the bottom of page 12 - top of page 13 of http://arxiv. …

I've been thinking about the following propagation of singularities result: Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose …

Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wav …

In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ i …

If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described …

I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is the setup: All …

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle. For a morphism of smooth varieties $f: X \to Y$ write $f_{\pi}: T^*Y …

Assume $u\in L^2(\mathbb{R}^n)$ and let $(x_0, \xi _0) \in T^\ast \mathbb{R}^n = \mathbb{R}^n_x \times \mathbb{R}^n_\xi $. Assume I can find $a\in C^\infty (T^\ast \mathbb{R}^n)$ w …

Suppose that $a\in C^{\infty}_{c}(\mathbb{R}^n)$ and $\phi\in C^{\infty}(\mathbb{R}^n)$ are real-valued. How would I show that $WF_{h}(ae^{\frac{i}{h}\phi})=\{(x,\partial_{x}\phi …

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave fro …

Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $ C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topo …