6-9 November 2017
Saint Flour
Europe/Paris timezone

Bounded $H^\infty$-calculus for Cone Differential Operators

7 Nov 2017, 09:00
Hôtel des Planchettes (Saint Flour)

Hôtel des Planchettes

Saint Flour

7 Rue des Planchettes, 15100 Saint-Flour


Elmar Schrohe


We model a conic manifold by a manifold $\mathbb B$ with boundary $\partial \mathbb B=:X$. In a collar neighborhood we introduce coordinates $(t,x)$, where $t$ is the distance to $\partial \mathbb B$ and $x$ the variable in $X$. A cone differential operator of order $\mu$ is an operator $A: C^\infty_c(\mathbb B^\circ)\to C^\infty_c(\mathbb B$ that can be written near $\partial \mathbb B$ in the form $$A=t^{-\mu}\sum_{k=0}^\mu a_k(t)(-t\partial_t)^k\quad \text{with }a_k\in C^\infty([0,1)\mathrm{Diff}^{\mu-k}(X)). $$ We consider an extension of $A$ in a weighted cone Sobolev space $\mathcal H^{s,\gamma}_p(\mathbb B)$ with domain $\mathcal D(A) = \mathcal H^{s+\mu,\gamma+\mu}_p(\mathbb B)\oplus \mathcal E$, where $\mathcal E$ is a space of asymptotics functions. Given a sector $$\Lambda_\theta= \{re^{i\phi}: r\ge0, \theta\le\phi\le2\pi-\theta\}, \quad 0<\theta<\pi,$$ we show that any extension of $A$ which is parameter-elliptic with respect to $\Lambda_\theta$ has a bounded $H^\infty$-calculus on $\mathbb C\setminus \Lambda_\theta.$ Parameter-ellipticity with respect to $\Lambda_\theta$ here requires the following \begin{enumerate} \item Denote by $\sigma_\psi^\mu(A)$ the principal symbol of $A$. Then $\sigma_\psi^\mu(A)-\lambda$ is invertible for $\lambda \in \Lambda_\theta$, even up to the boundary, if one takes into account the degeneracy. \item The principal conormal symbol $\sigma_M^\mu(A)(z)$ is invertible for all $z\in \mathbb C$ with ${\rm Re}\, z= \frac{n+1}2-\gamma -\mu $ or ${\rm Re}\, z= \frac{n+1}2-\gamma$. \item $\Lambda_\theta$ is a sector of minimal growth for the model cone operator $$\widehat A= t^{-\mu}\sum_{k=0}^\mu a_k(0)(-t\partial_t)^k$$ acting on cone Sobolev spaces over he infinite cone. \end{enumerate} Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities. (Joint work with J\"org Seiler (Torino))

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