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micro local analysis - Associating a pseudo-differential operator to the symbol in the SG setting

We all know that given a symbol $a(x,\xi) \in S^{\mu,\rho}(\mathbb{R}^n,\mathbb{R}^n)$, a pseudo-differential operator can be defined as\begin{equation}Op(a)u(x)=(2\pi)^{-n}\int \int e^{i(x-x')\cdot \xi}a(x,\xi)u(x')dx'd\xi \end{equation}for $u$ in Schwartz class of test functions, $\mathcal{S}(\mathbb{R}^n)$. $Op(a)$ is Fourier inverse operator.Now, for symbols in the class $S^{\mu,\mu',\rho, \rho'}(\mathbb{R}^{2n} \times \mathbb{R}^{2n})$, which are defined as fuctions $a(x,x',\xi, \xi')\in C^{\infty}(\mathbb{R}^{2n}_{x,x'}\times \mathbb{R}^{...Read more

sp.spectral theory - Functions of Pseudodifferential Operators

Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can consider the operator $f(A)$ defined by functional calculous.Is $f(A)$ again a pseudo-differential operator and if yes, how are the symbols related?In what way does the type of operator or the type of function matter?...Read more

Schwartz kernel of a pseudodifferential operator with singular symbols

If we have a smooth symbol $r(x,\xi)$ of order $d$ the corresponding pseudodifferential operator $P$ is integral operator with kernel given by $$K(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi} r(x,\xi)d\xi$$and $K$ is $C^k$ in $(x,y)$ if $d+m<-n$. ( see for instance Gilkey’s book, Invariance theory...1984, lemma1.2.2 and lemma 1.2.5).For example if the symbol is smooth and of order $-\infty$ then $K(x,y)$ is smooth.My question is: Is there any similar statement in the case of pseudo differential operators whose symbol is singular. Some result...Read more

micro local analysis - Pseudo-differential operators which are independent of lower order perturbations

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 $P$ is zeroth order real principal type operator with principal symbol $q(x,\xi),$ then for a lower order perturbation $R$ we can always find a zeroth order elliptic operator $E$ such that $E^{-1}(P+R)E=q(x,D).$ The reason for this independence of lower terms in elliptic type operator is because the principal term can be inverted and in real principal type operator is ther...Read more

fa.functional analysis - 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 question is when add what condition (as least as possible) to the symbol $a(x,\xi)$ can assure the operator $a(x,D)$ to be compact on $L^2$ ?Or is there a equivalent condition of this ?I guess some decayed assumption on $a(x,\xi)$ (about $\xi$) is necessary.but I'm not sure.some references about this are also appreciatedAdded:There is a equivalent condi...Read more

sobolev spaces - about smoothing pseudodifferential operators

Hi,I have a question which involves pdo.Let us consider a pseudodifferential operator $A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) $ whose symbol $a(x,\xi)$ lives in the $S_{0,0}^0$ class : $$ \forall \alpha,\beta \in \mathbb{N}^d \qquad |\partial_x^{\alpha} \partial_\xi^{\beta} a(x,\xi) |\leq C(\alpha,\beta) $$Let be $f$ and $g$ smooth real functions on $\mathbb{R}^d$ such that $$ \forall \alpha \in \mathbb{N}^d\qquad |\partial_x^\alpha f(x)|+ |\partial_x^\alpha g(x)|\leq C(\alpha) $$If the supports of $f$ and $g$ are disjoint (one may assu...Read more

dg.differential geometry - Criteria for Positivity of Pseudoddifferential Operators on Manifolds

Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L_0^2$ be the subspace which is orthogonal to the constant functions. When is a pseudodifferential operator on $M$ a positive operator on $L^2_0$?For second order operators the Laplacian $\Delta$ is the main example.For order zero, the obvious examples are multiplication by $f$ where $f \in C^\infty(M)$ is a smooth function and $f > 0$. Conversely if $f < 0$ anywhere then it is clear that the multiplication operator ...Read more

pseudo differential operators - Understanding the analytic index map of the Atiyah-Singer index theorem

Hi,I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn.I do not understand why the analytic index map $\operatorname{ind}\colon K_{cpt}(T^\ast X) \to Z$, as defined in chapter III.$13 in equation (13.8), agrees with the Fredholm index of an elliptic pseudo-differential operator.Recall how the analytic index map is constructed (this is chapter III.§13 in the book):Given an element $u \in K_{cpt}(T^\ast X) \cong K(DX, \partial DX)$ we can represent it b...Read more

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

pseudo differential operators - Paraproduct and Fourier series

I know that it's possible to define the paraproduct $T_a u$ when $a=a(x)\in L^{\infty}$ and $u\in H^s$ and in this case $T_a u\in H^s$.Remark: $T_a u$ is the pseudo differential operator with symbol $\sigma^{\psi}_a(x,\xi)=(\mathcal{F}^{-1}_{\eta}\psi(\eta,\xi)\star a)(x)$, where $\psi$ is an admissible cut off function.Considering the Bony's choice for the admissible cut off function, we can write $T_a u$ as a part of the product $a u$ with $a$ and $u$ written with the Paley-Littlewood decomposition.I want to prove the following estimate$$\lV...Read more

microlocal analysis - Differentiation of an operator raised to a variable power

I am studying first order systems of the form\begin{equation} L=\partial_t+K(t,x,D_x)\text{ where }D_x=-i\partial_x\end{equation}There is a change of variable and operator of our concern becomes \begin{equation} L_1=\big<D_x\big>^{-t}L\big<D_x\big>^{t}\end{equation}where $\big<\xi\big>=(1+|\xi|^2)^{1/2}$. $L_1$ takes the form\begin{equation} L_1=\partial_t+K(t,x,D_x)+\log{\big<D_x\big>}+A\end{equation}where operator $A$ corresponds to lower order terms. My question is about the form of $L_1$.How is it that $\bi...Read more

pseudo differential operators - 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 analogue of such function that satisfies$$ \frac{\partial^{\alpha}}{\partial t^{\alpha}} G_{\alpha}((a+bt)^{\alpha}) = CG((a+bt)^{\alpha}).$$I looked through some papers on fractional differentiating but didn't find such analogue. Then I tried to construct it myself. I found fractional derivative of order $\alpha$ of $(a+bt)^{n \alpha}$. If it was somethi...Read more