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I want first to change your notations, sticking to the usual variables $x,\xi$ in the phase space. As a general statement about pseudo-differential operator with a symbol $a(x,\xi)$, I wish to write $ …

Hörmander's definition of a pseudo-differential operator on an open subset $\Omega$ of $\mathbb R^n$ in the class $\text{Op}S^m$ ($m\in \mathbb R$) is the following: take a symbol $a$, that is a smoot …

Let me start by altering a bit your notations: we consider a differential operator $P$ defined by $$ P=\sum_{1\le j\le n}p_j(x) D^j, \quad D=-i\frac{d}{dx}. $$ The formal adjoint is (there is a typo i …

You need some ellipticity condition to start off. Let us assume as you do that $$ c=c(x,\xi, h), \quad \vert\partial_x^\alpha\partial_{\xi}^\beta c\vert\le C_{\alpha\beta} h^{\vert\beta\vert}, \quad\ …

Analytic regularity of a $C^\infty$ function can be characterized by using an infinite number of derivatives. A function $f\in C^\infty(\Omega)$ where $\Omega$ is an open subset of $\mathbb R^n$ is r …

The answer to the first question is negative on the Euclidean sphere $\mathbb S^2$. It is possible to prove that the Laplace operator on the sphere $\mathbb S^2$ is NOT the sum of two squares of smoot …

A preliminary remark. The operator $(d/dx)^\gamma$ is never continuous on the Schwartz space (and thus on tempered distributions) except if $\gamma$ is a non-negative integer, since you introduce a s …

First a classical result about elliptic operators: in dimension $\ge 3$ the order of elliptic operators is even. In dimension 2, say in $\mathbb R^2$ you have elliptic vector fields such as $$ \bar \p …

Too long for a comment. I want to use a version of the Lojaciewicz theorem of division of distributions by an analytic function (in fact Hörmander's result of division by a polynomial). We may assume …

You will find a study of self-adjoint operators of type $$ \sum_{j=1}^r(X_j^*-iY_j^*)(X_j+iY_j), $$ where $X_j, Y_j$ are real-valued vector fields for instance in the Helffer-Nier Lecture Note (Lectur …

Let us consider the classical Harmonic Oscillator (a selfadjoint operator) $$ \mathcal H=\frac12\left(-\frac{d^2}{dx^2}+x^2\right),\quad\text{with spectrum $\frac12+\mathbb N$.} $$ This one-dimensiona …

For $\epsilon >0$, $u\in H^{\frac{n}{2}+2\epsilon}$, $N_0=-\Delta+1$, $$ \Vert N_0^{\frac{n}{4}+\epsilon} u\Vert_{L^2}=\Vert u\Vert_{H^{\frac{n}{2}+2\epsilon}}\ge c_{n,\epsilon} \Vert u\Vert_{L^\inf …

When $A_K$ is a (pseudo)differential operator with analytic coefficients, a good answer to your question is given by Theorem 9.5.1 on page 353 of the first volume of Hörmander's ALPDO (Springer Grund. …

Let us take a look at the case where $Y\equiv x_n=0$ in $\mathbb R^n$. Let $a(\underbrace{\overbrace{x_1,\dots,x_{n-1}}^{x'},x_n}_x; \xi_n)$ be a symbol in $S^m(\mathbb R^n\times \mathbb R)$ and let …

An operator is trace class whenever it is the product of Hilbert-Schmidt operators. There is a simple characterization of Hilbert-Schmidt operators pseudodifferential operators: a pseudodifferential o …