### ca.classical analysis and odes - Elliptic pseudodifferential operator estimate

If $P$ is an elliptic pseudodifferential operator of order 1 in the sense that its principal symbol is invertible, then we have the a priori estimate $\|u\|_{H^1(U)} \le C (\|Pu\|_{L^2(W)} + \|u\|_{L^2(W)}), \quad (C>0 \text{ a constant})$for any $\overline{U}\subset W\subset \mathbb{R}^n$ with $W$ bounded. Say I'm only interested in an upper bound of $\|u\|_{L^2(U)}$. Then the term $\|u\|_{L^2(W)}$ on the right hand side above feels a bit redundant. Can I obtain $\|u\|_{L^2(U)} \le C\|Pu\|_{L^2(W)} \quad \text{?}$...Read more

### Applications of pseudodifferential operators to PDE

I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole picture. So my question is, after setting up the stage and covering the basic stuff on pseudodifferential operators, including the algebra and mapping properties, what are my choices on the "main part" of the course? One application per answer, and please include references if possible.To get started, I have the following examples (please expand thes...Read more

### universal algebra - What are the invariant Pseudo-differential operators on a Lie group?

It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group. On the other hand $\Psi$Do are generalization of the differential operators on general smooth manifold.My question is that:Does any algebraic description for the $G$-invariant $\Psi$DOs on the Lie groups (or more general on homogeneous spaces $G/H$) exist ? Thanks...Read more

### pseudo differential operators - Exponential decay for the gradient of a solution

Dear all,I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$:$$\sqrt{-\Delta+m^2} u +u= f(u),$$where I can assume that $m \neq 0$, that $f$ is smooth and that the solution $u \in H^{1/2}(\mathbb{R}^N)$ is at least Hölder-continuous. I also know that $u(x) = O(e^{-C|x|})$ at infinity. If the equation were local (like $-\Delta u + u = f(u)$), the usual approach would consist in using interior Schauder estimates for $\nabla u$ (or some Harnack-like estimate); I did not find any p...Read more