# Tagged Questions

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### Extension of pseudodifferential operators

I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( ...
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### Closure of pseudodifferential operators of order 0 on compact manifolds

Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...
Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. HÃ¶rmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that $$L_1 ... 1answer 86 views ### Support-preserving pseudodifferential operators Let A = F^{-1}\sigma F be a pseudodifferential operator acting on functions on \mathbb R^n, where F, F^{-1} are the direct and inverse Fourier transforms respectively and \sigma is the ... 0answers 27 views ### Vanishing of non commutative ( Wodzicki) residue on pseudo differential projections Its a known fact that the non-commutative (Wodzicki) residue of a pseudo-differential projection is always zero. My question is: Is it possible to get this result by looking at structure of the ... 1answer 288 views ### Pseudo-differential operators with compactly supported symbols If the symbol p(x,\xi) of a pseudodifferential operator P has compact x-support, then for any Schwartz function f, Pf has compact x-support. Is the reverse true? Namely that if some PDO ... 1answer 210 views ### On the generalization of the Mittag-Leffler function and fractional derivative The Mittag-Leffler function E_{\alpha}(x) has an important property:$$ \frac{\partial^{\alpha}}{\partial x^{\alpha}} E_{\alpha}(x^{\alpha}) = E_{\alpha}(x^{\alpha}). $$I tried to find an ... 1answer 368 views ### Is this kernel space of finite dimension ? Assume that P \in \Psi^{m}(X) (X is a C^{\infty} manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ... 2answers 527 views ### Why take 'complex powers' of pseudo-differential operators? Given a pseudo-differential operator P of order zero, Seeley showed that the holomorphic family of operators \lbrace P^{z} : z\in \mathbb{C} \rbrace of all complex powers is contained in the ... 2answers 660 views ### when a pseudo-differential operators to be compact? In the theory of Pseudo-differential operators,when a symbol a(x,\xi)\in S^{0},then the operator a(x,D) defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$is L^2 bounded.  My ... 2answers 748 views ### (sharp)Garding's inequality and inequality with lower bounds The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let$$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$with principal part ... 6answers 601 views ### Fractional Leibniz formula Let T=(-\Delta)^{1/2}. Can we have estimates, similar to the one below$$ \| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p,  hold in $L^p$, where ...
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...