$L^p$-bounds in Safarov pseudo-differential calculus on manifolds with bounded geometry

Abstract

Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ constructed by Safarov. Our result recovers classical Fefferman's theorem, and extends it to the following two situations: $ρ>1/3$ and $\nabla$ symmetric; and $\nabla$ flat with any values of $ρ$ and $δ$. Moreover, as a consequence of our main result, we obtain boundedness on Sobolev and Besov spaces and some $L^p-L^q$ boundedness. Different examples and applications are presented.

Figures (7)

• Figure 1: In red the possible values of $p$ when $\rho>1/2$, in green the possible values of $p$ when $\rho>1/3$ and $\nabla$ is symmetric, in blue the possible values of $p$ when $\rho>0$ and $\nabla$ is flat.
• Figure 2: The coloured red area (triangle) represents the values of $p$ and $q$ one can insert in inequality (\ref{['pq']}) when $0<-\theta<n$ and $1<p\leq2\leq q<\infty$.
• Figure 3: The values of $p$ and $q$ one can insert in inequality (\ref{['pq']}) when $0<-\theta<\frac{n}{2}$, $1<p\leq q \leq 2$ and, $\rho=1$ in black, $\rho=\frac{1}{2}$ in red, $\rho=\frac{1}{3}$ in green, $\rho=0$ in blue.
• Figure :
• Figure :

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2: Fefferman Fefferman
• Remark 1.3
• Definition 1.4
• Lemma 1.5
• Definition 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10