(-tP/h)$ (microlocally) for $0\le t\ll 1$ and to show that in this case [$$\label{in.15}
\| \exp -\frac{tP}{h}\| \le C\exp (-\frac{t^{k+1}}{Ch}),$$]{} for some constant $C>0$. Noting that that implies that $\| \exp -\frac{tP}{h}\| ={\cal O}(h^\infty )$ for $t\ge h^\delta $ when $\delta (k+1)<1$, and using the formula [$$\label{in.16}
(z-P)^{-1}=-\frac{1}{h}\int_0^\infty \exp (\frac{t(z-P)}{h}) dt,$$]{} leads to (\[in.14\]). (This has some relation to the works of A. Cialdea and Maz’ya [@CiMa05; @CiMa06] where the $L^p$ dissipativity of second order operators is characterized.)

The most direct way of studying $\exp (-tP/h)$, or rather a microlocal version of that operator, is to view it as a Fourier integral operator with complex phase ([@Mas; @Ku; @MeSj76; @Ma]) of the form [$$\label{in.17}
U(t)u(x)=\frac{1}{(2\pi h)^n}{\int\hskip -2mm\int}e^{\frac{i}{h}(\phi (t,x,\eta )-y\cdot \eta )}a(t,x,\eta ;h) u(y)dy d\eta ,$$]{} where the phase $\phi $ should have a non-negative imaginary part and satisfy the Hamilton-Jacobi equation: [$$\label{in.18}
i\partial _t\phi +p(x,\partial _x\phi )={\cal O}((\Im \phi )^\infty ), \hbox{ locally uniformly,}$$]{} with the initial condition [$$\label{in.19}
\phi (0,x,\eta )=x\cdot \eta .$$]{} The amplitude $a$ will be bounded with all its derivatives and has an asymptotic expansion where the terms are determined by transport equations. This can indeed be carried out in a classical manner for instance by 