First Band of Ruelle Resonances for Contact Anosov Flows in Dimension 3

Abstract

We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3-dimensional manifold $\mathcal {M}$ has only finitely many Ruelle resonances in the vertical strips $\{ s\in \mathbb {C}\ |\ \mathrm{Re}(s)\in [-\nu _{\min }+\varepsilon ,-\frac{1}{2}\nu _{\max }-\varepsilon ]\cup [-\frac{1}{2}\nu _{\min }+\varepsilon ,0]\}$ for all $\varepsilon >0$, where $0<\nu _{\min }\le \nu _{\max }$ are the minimal and maximal expansion rates of the flow (the first strip only makes sense if $\nu _{\min }>\nu _{\max }/2$). We also show polynomial bounds in s for the resolvent $(-X-s)^{-1}$ as $|\mathrm{Im}(s)|\rightarrow \infty$ in Sobolev spaces, and obtain similar results for cases with a potential. This is a short proof of a particular case of the results by Faure–Tsujii, using that $\dim E_u=\dim E_s=1$.

Appendix A: An Exotic Symbol Class

Let $\mathcal {M}$ be a closed n-manifold equipped with a Riemannian metric g, let $k\ge 0$ and $0<\rho <1$. We use the usual notation $\langle \xi \rangle =(1+|\xi |^2)^{1/2}$, $h>0$ will be a small semiclassical parameter, we let $\overline{T}^*\mathcal {M}$ be the fiber radial compactification of $T^*\mathcal {M}$ as defined in [DyZw, Section E.1.3] and $H_p$ will denote the Hamiltonian vector field of $p\in C^\infty (T^*\mathcal {M})$. Given an operator P, we write ${{\,\mathrm{Im}\,}}P = \frac{P - P^*}{2i}$ for the imaginary part of P and ${{\,\mathrm{Re}\,}}P = \frac{P + P^*}{2}$ for the real part; then $P = {{\,\mathrm{Re}\,}}P + i {{\,\mathrm{Im}\,}}P$. Recall that for $x\in \mathbb {R}$ we write $x_+ = \max (x, 0)$.

Most of the results we gather in this appendix are simple extensions of classical results in semiclassical analysis that can be found in the books [Zw] and [DyZw, Appendix E]. We shall only point out the main differences with our setting.

For each $m\in \mathbb {R}$, we define the exotic pseudo-differential calculus $\Psi _{h,\rho , k}^m(\mathcal {M})$ by saying that $A\in \Psi _{h,\rho , k}^m(\mathcal {M})$ if its Schwartz kernel $K_A$ is in $\mathcal {O}_{C^N(\mathcal {M}\times \mathcal {M})}(h^N)$ for all $N>0$ outside a neighborhood of the diagonal, and near the diagonal can be written in local coordinates as

$$\begin{aligned} K_A(x,y)=(2\pi h)^{-n}\int _{\mathbb {R}^n} e^{\frac{i}{h}(x-y)\xi }a(x,\xi )d\xi , \end{aligned}$$

where the local symbols are in the class $S^{m}_{h,\rho ,k}(\mathbb {R}^{2n})$ defined by the property: $a\in S^{m}_{h,\rho ,k}(\mathbb {R}^{2n})$ if $a\in C^\infty (\mathbb {R}^{2n})$ is an h-dependent function and satisfies in local coordinates (for some $C_{\alpha ,\beta }$ uniform in h)

$$\begin{aligned} |\partial _x^{\alpha }\partial _\xi ^{\beta }a(x,\xi )|\le C_{\alpha ,\beta }\langle \xi \rangle ^{m-|\beta |}h^{-\rho (|\alpha |-k)_+}. \end{aligned}$$

(A.1)

We notice that it is important here, for the calculus, that the loss of $h^{-\rho }$ happens only in the x-derivatives and not in the $\xi$ derivatives. First, observe the basic properties for $0\le k'\le k$

$$\begin{aligned}&a\in S^{m}_{h,\rho ,k}(\mathbb {R}^{2n}),\, b\in S^{m'}_{h,\rho ,k'}(\mathbb {R}^{2n}) \Longrightarrow ab\in S^{m+m'}_{h,\rho ,k'}(\mathbb {R}^{2n}),\nonumber \\&a\in S^{m}_{h,\rho ,k}(\mathbb {R}^{2n}) \Longrightarrow \partial ^\alpha _xa\in h^{-\rho (|\alpha |-k)_+}S^{m}_{h,\rho ,(k-|\alpha |)_+}(\mathbb {R}^{2n}) , \,\, \partial ^\alpha _\xi a\in S^{m-|\alpha |}_{h,\rho ,k}(\mathbb {R}^{2n}),\nonumber \\&\quad \quad \quad \forall j\in [0,k], \quad h^{j\rho }S^{m}_{h,\rho ,k-j}(\mathbb {R}^{2n})\subset S^{m}_{h,\rho ,k}(\mathbb {R}^{2n}). \end{aligned}$$

(A.2)

We define $S^m_{h,\rho ,k}(T^*\mathcal {M})$ to be $C^\infty (T^*\mathcal {M})$ functions that, using local coordinates on $\mathcal {M}$, are in $S^m_{h,\rho ,k}(\mathbb {R}^{2n})$. First, one directly sees from the formula of symbols under a change of coordinates [Zw, Theorem 9.9] that the symbol in local coordinates being in $S^{m}_{h,\rho ,k}(\mathbb {R}^{2n})$ is invariant by change of coordinates, and moreover there is a principal symbol map

$$\begin{aligned} \sigma : \Psi _{h,\rho , k}^m(\mathcal {M})\rightarrow S^{m}_{h,\rho ,k}(T^*\mathcal {M})/hS^{m-1}_{h,\rho ,k}(T^*\mathcal {M}). \end{aligned}$$

Using local charts and a partition of unity, we fix a semi-classical quantization $\mathrm{Op}_h:S^{m}_{h,\rho ,k}(T^*\mathcal {M})\rightarrow \Psi _{h,\rho ,k}^m(\mathcal {M})$, which satisfies

$$\begin{aligned} \sigma (\mathrm{Op}_h(a))=a \, \, \mathrm{mod}\, \, hS^{m-1}_{h,\rho ,k}(T^*\mathcal {M}). \end{aligned}$$

We first check that symbols in this class are closed under composition. Recall from [Zw, Theorem 4.14] that if $A\in \Psi ^{m}_{h,\rho ,k}(\mathbb {R}^n)$ and $B\in \Psi ^{m'}_{h,\rho ,k}(\mathbb {R}^n)$ have full symbol a, b then AB has full symbol (as an oscillatory integral)

$$\begin{aligned} a \# b(x,\xi ) =(2\pi h)^{-n}\int _{\mathbb {R}^{2n}} e^{-\frac{i}{h}(x'.\xi ')}a(x,\xi +\xi ')b(x+x',\xi )dx'd\xi ' \end{aligned}$$

(A.3)

which has expansion

$$\begin{aligned} a \# b =\sum _{|\alpha |\le N} \frac{(-ih)^{|\alpha |}}{\alpha !} \partial _\xi ^\alpha a \partial _x^\alpha b +\mathcal {O}_{S_{h,\rho ,0}^{m+m'-N-1}(\mathbb {R}^{2n})}(h^{(N+1)(1-\rho )}). \end{aligned}$$

(A.4)

Here $|\partial _\xi ^\alpha a \partial _x^\alpha b|\le C_{\alpha }h^{-|\alpha |\rho }\langle \xi \rangle ^{m+m'-|\alpha |}$ so that higher order terms in the expansion are higher powers of h and of $\langle \xi \rangle ^{-1}$.

Lemma A .1

Let $a \in S^{m_1}_{h, \rho , k}(\mathbb {R}^{2n})$ and $b \in S^{m_2}_{h, \rho , k}(\mathbb {R}^{2n})$. Then $a \# b \in S^{m_1 + m_2}_{h, \rho , k}(\mathbb {R}^{2n})$.

Proof

This follows from (A.4) and (A.2). $\quad \square$

For $A\in \Psi _{h,\rho ,k}^{m}(\mathcal {M})$, we say that $(x_0,\xi _0)\in \overline{T}^*\mathcal {M}$ is not in $\mathrm{WF}_h(A)$ if there is a small neighborhood U of $(x_0,\xi _0)$ in $\overline{T}^*\mathcal {M}$ so that the full local symbol of A restricted to U is in $h^{N}S^{-N}_{h,\rho ,0}(U)$ for all $N>0$. We also define the elliptic set $\mathrm{ell}_h(A)$ of $A\in \Psi _{h,\rho ,k}^m(\mathcal {M})$ to be the set of points $(x_0,\xi _0)\in \overline{T}^*\mathcal {M}$ so that for a neighborhood U of $(x_0,\xi _0)$ there is $c_0>0$ so that $\langle \xi \rangle ^{-m}|\sigma (A)(x,\xi )|\ge c_0$, for $(x, \xi ) \in U$. We finally list some properties of the $\Psi _{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$ calculus.

Proposition A.1

The following properties hold:

• 1. Let $A \in \Psi _{h, \rho , k}^m(\mathcal {M})$. If $B, B' \in \Psi _h^{{{\,\mathrm{comp}\,}}}(\mathcal {M})$ with $\mathrm{WF}_h(B) \cap \mathrm{WF}_h(B') = \emptyset$, then $BAB' \in h^\infty \Psi _h^{{{\,\mathrm{comp}\,}}}(\mathcal {M})$.

• 2. The principal symbol is well-defined as a map

  $$\begin{aligned} \sigma : \Psi ^m_{h, \rho , k}(T^*\mathcal {M}) \rightarrow S^m_{h, \rho , k}(T^*\mathcal {M}) / hS^{m-1}_{h, \rho , k}(T^*\mathcal {M}) \end{aligned}$$

  with kernel $h\Psi _{h,\rho ,k}^{m-1}(\mathcal {M})$, and it satisfies for any $A\in \Psi ^m_{h, \rho , k}(\mathcal {M})$ and $B\in \Psi ^{m'}_{h, \rho , k}(\mathcal {M})$

  $$\begin{aligned} \begin{aligned} \sigma (AB) = \sigma (A) \sigma (B) \,\,\,\mathrm {mod}\,\,\, h^{1-\rho }S^{m+m'-1}_{h,\rho ,k}(T^*\mathcal {M}), \\ \sigma (AB) = \sigma (A) \sigma (B) \,\,\,\mathrm {mod}\,\,\, hS^{m+m'-1}_{h,\rho ,k-1}(T^*\mathcal {M}) \text { if }k\ge 1. \end{aligned} \end{aligned}$$

• 3. If $A \in \Psi ^{m_1}_{h, \rho , k}(\mathcal {M})$ and $B \in \Psi ^{m_2}_{h, \rho , k}(\mathcal {M})$ for $k\ge 1$, then

  $$\begin{aligned} \begin{aligned} \, [A, B]\in h^{1 - \rho } \Psi ^{m_1 + m_2 - 1}_{h, \rho , k}(\mathcal {M}), \\ h^{-1}[A, B] \in \Psi _{h, \rho , k - 1}^{m_1 + m_2 - 1}(\mathcal {M}) \text { if }k\ge 1 , \end{aligned} \end{aligned}$$

  and $\sigma (h^{-1}[A, B]) = -i\{\sigma (A), \sigma (B)\}$.

• 4. If $P = -ihX$ for X a vector field on ${{\,\mathrm{\mathcal {M}}\,}}$, $\Theta \in \Psi ^m_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$ and $H_p \sigma (\Theta ) \in S^m_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$, where $p =\sigma (P)= \xi (X)$, then $[P, \Theta ] \in h\Psi ^m_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$.

• 5. If $A \in \Psi _{h, \rho , k}^m(\mathcal {M})$, then $A^* \in \Psi _{h, \rho , k}^m(\mathcal {M})$ and

  $$\begin{aligned}\begin{aligned} \sigma (A^*)=\overline{\sigma (A)} \,\,\,\mathrm {mod}\,\,\, h^{1-\rho }S^{m+m'-1}_{h,\rho ,k}(T^*\mathcal {M}),\\ \sigma (A^*)=\overline{\sigma (A)} \,\,\,\mathrm {mod}\,\,\, hS^{m-1}_{h,\rho ,k-1}(T^*\mathcal {M}) \text { if }k\ge 1.\end{aligned}\end{aligned}$$

• 6. Each $A \in \Psi ^0_{h, \rho , 0}(\mathcal {M})$ is bounded and for each $\varepsilon > 0$

  $$\begin{aligned}\Vert A\Vert _{L^2 \rightarrow L^2} \le (1 + \varepsilon ) \sup _{h, x, \xi } |\sigma _h(A)(x, \xi )| + \mathcal {O}_{\varepsilon }(h^{\infty }).\end{aligned}$$

  Moreover, for any $A \in \Psi _{h, \rho , k}^m(\mathcal {M})$ and any $s \in \mathbb {R}$ we have

  $$\begin{aligned}\Vert A\Vert _{H_h^s \rightarrow H_h^{s-m}} \le (1 + \varepsilon ) \sup _{h, x, \xi }|\langle {\xi }\rangle ^{-m} \sigma (A)| + \mathcal {O}_{\varepsilon }(h^\infty ).\end{aligned}$$

• 7. Let $P \in \Psi _{h, \rho , k}^p({{\,\mathrm{\mathcal {M}}\,}})$, $A \in \Psi ^m_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$ and $B_1 \in \Psi ^l_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$. Assume $\mathrm{WF}_h(A) \subset {{\,\mathrm{ell}\,}}_h(P) \cap {{\,\mathrm{ell}\,}}_h(B_1)$. Then for all $s\in \mathbb {R}$, $N > 0$, and u with $B_1Pu \in H^{s-p-l}_h({{\,\mathrm{\mathcal {M}}\,}})$

  $$\begin{aligned}\Vert Au\Vert _{H^{s-m}_h} \le C\Vert B_1Pu\Vert _{H^{s - p - l}_h} + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}.\end{aligned}$$

• 8. Assume $k {\ge } 1$ and let $P \in \Psi _{h, \rho , k}^1({{\,\mathrm{\mathcal {M}}\,}})$ with ${{\,\mathrm{Re}\,}}P \in \Psi ^1_h({{\,\mathrm{\mathcal {M}}\,}})$ and ${{\,\mathrm{Im}\,}}P \in h\Psi ^0_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$. Denote $p := \sigma (P)$ and assume that for each $(x,\xi )\in \mathrm{WF}_h(A)\subset \overline{T}^*\mathcal {M}$, there is $T>0$ such that $e^{-TH_p}(x,\xi )\in \mathrm{ell}_h(B)$ and $e^{-tH_p}(x,\xi )\in \mathrm{ell}_h(B_1)$ for $t \in [0, T]$. Then for each $u\in L^2$ with $Pu \in L^2({{\,\mathrm{\mathcal {M}}\,}})$, and every $N > 0$ there is a $C > 0$ such that

  $$\begin{aligned}\Vert Au\Vert _{L^2} \le C\Vert Bu\Vert _{L^2} + Ch^{-1} \Vert B_1 P u\Vert _{L^2} + Ch^N \Vert u\Vert _{H_h^{-N}}.\end{aligned}$$

Proof

• 1. This follows from the composition formula (A.4).

• 2. This was discussed above.

• 3. From the composition formula (A.4), locally we have

  $$\begin{aligned} a\# b-b\# a \sim h(D_\xi a \partial _x b {-} D_\xi b \partial _x a) + \frac{1}{2}h^2(D_\xi ^2 a \partial _x^2 b - D_\xi ^2 b \partial _x^2 a) + \frac{1}{6}h^2(D_\xi ^3 a \partial _x^3 b - D_\xi ^3 b \partial _x^3 a) {+} \cdots \end{aligned}$$

  where, after taking the expansion to a high enough order, the remainder is in $h^NS^{m-N}_{h,\rho ,0} (\mathbb {R}^{2n})$ for some large $N>0$. By (A.2), all these terms are in $h^{1-\rho }S^{m_1+m_2-1}_{h,\rho ,k}(\mathbb {R}^{2n})\cap hS^{m_1+m_2-1}_{h,\rho ,k-1}(\mathbb {R}^{2n})$ if $k\ge 1$, and the principal symbol of $h^{-1}[A,B]$ is $-i(\partial _\xi a \partial _x b-\partial _x a \partial _\xi b)$.

• 4. This follows from the composition formula (A.4), the fact that $\partial _\xi ^{\alpha } \sigma (P) = 0$ for $|\alpha | \ge 2$ and Item 3 above.

• 5. This follows from the fact that, if A has full local symbol a in local coordinates, the full symbol $a^*$ of $A^*$ is

  $$\begin{aligned} a^*(x, \xi ) \sim \sum _\alpha \frac{h^{|\alpha |}}{\alpha !} \partial _\xi ^\alpha D_x^\alpha \overline{a(x, \xi )}. \end{aligned}$$

• 6. The argument is standard (see [GrSj, Theorem 4.5]) and we just make a brief summary. Let $a := \sigma (A) \in S^0_{h, \rho , k}(\mathcal {M})$. For $M=\Vert a\Vert _\infty$, we can construct $B_0 \in \Psi ^0_{h, \rho , 0}(\mathcal {M})$ with principal symbol $b_0:=\sqrt{(1 + \varepsilon )^2M^2-|a|^2}\ge \varepsilon M>0$, so that

  $$\begin{aligned} C:=(1 + \varepsilon )^2 M^2 - A^*A = B_0^*B_0 + h^{1-\rho }R_0, \end{aligned}$$

  for some $R_0=R_0^* \in \Psi _{h, \rho , 0}^{-1}(\mathcal {M})$ by part 2.⁵ Next we can choose $B_1=B_0 + \tfrac{h^{1-\rho }}{2}\mathrm {Op}_h(b_0^{-1})^*R_0$ so that $B_1^*B_1=C - h^{2(1-\rho )}R_1$ with $R_1\in \Psi ^{-2}_{h, \rho , 0}(\mathcal {M})$. We iterate this procedure to find $B_N \in \Psi _{h, \rho , 0}^{0}(\mathcal {M})$ such that $(1 + \varepsilon )^2 M^2 - A^*A=B_N^*B_N+h^{(N+1) (1-\rho )}R_N$ with $R_N\in \Psi _{h,\rho ,0}^{-N-1}(\mathcal {M})$. It is a routine argument to show that $\Vert R_N\Vert _{L^2\rightarrow L^2}=\mathcal {O}(h^{N(1-\rho ) - n})$ thus for any $u \in L^2$

  $$\begin{aligned} \Vert Au\Vert _{L^2}^2 = (1 + \varepsilon )^2 M^2\Vert u\Vert _{L^2}^2 - \Vert B_Nu\Vert _{L^2}^2 + \mathcal {O}(h^{2(N + 1)(1-\rho )-n}) \Vert u\Vert ^2_{L^2} \end{aligned}$$

  which shows the desired estimate by choosing N large. The Sobolev bound is an easy consequence of this.

• 7. This follows from the parametrix construction in the elliptic set, the main thing to notice is that if $B \in \Psi _{h, \rho , k}^m(\mathcal {M})$, $A \in \Psi _{h, \rho , k}^{l}(\mathcal {M})$ and $\mathrm{WF}_h(A) \subset {{\,\mathrm{ell}\,}}_h(B)$, then

  $$\begin{aligned} \frac{\sigma (A)}{\sigma (B)} \in S_{h, \rho , k}^{l-m}(\mathcal {M}). \end{aligned}$$

• 8. We follow the proof of [DyZw, Theorem E.47] and divide the proof into steps.

  Step 0: an escape function. Fix $\beta \ge 0$. There is a $g \in C^\infty ({\overline{T}}^*\mathcal {M})$ with ${{\,\mathrm{supp}\,}}g \subset {{\,\mathrm{ell}\,}}_h(B_1)$, such that

  $$\begin{aligned} g \ge 0, \quad g > 0 \,\,\mathrm {on}\,\, \mathrm{WF}_h(A), \quad H_p g \le - \beta g, \end{aligned}$$

  where the last condition holds outside ${{\,\mathrm{ell}\,}}_h(B)$. Step 1. Note that $g \in S^0(T^*{{\,\mathrm{\mathcal {M}}\,}})$ and define

  $$\begin{aligned} G:= {{\,\mathrm{Op}\,}}_h(\langle {\xi }\rangle ^s g) \in \Psi _h^s({{\,\mathrm{\mathcal {M}}\,}}), \quad \mathrm{WF}_h(G) \subset {{\,\mathrm{ell}\,}}_h(B_1). \end{aligned}$$

  We can that $u \in C^\infty ({{\,\mathrm{\mathcal {M}}\,}})$. If we write $f = Pu$,

  $$\begin{aligned} {{\,\mathrm{Im}\,}}\langle {f, G^*Gu}\rangle = \underbrace{{{\,\mathrm{Im}\,}}\langle {({{\,\mathrm{Re}\,}}P) u, G^*Gu}\rangle }_{\mathrm {term\,\, T_1}} + \underbrace{{{\,\mathrm{Re}\,}}\langle {({{\,\mathrm{Im}\,}}P) u, G^*Gu}\rangle }_{\mathrm {term\,\, T_2}}. \end{aligned}$$

  We will bound the two terms on the right separately.

  Step 2: term $T_1$. Since $\mathrm{Re}(P)\in \Psi _h^1(\mathcal {M})$ is non exotic, this step is exactly the same as in the proof of [DyZw, Theorem E.47] and we get

  $$\begin{aligned} T_1 \le (C_1 - \beta )h \Vert Gu\Vert _{L^2}^2 + Ch\Vert Bu\Vert ^2_{H_h^s} + Ch^2 \Vert B_1 u\Vert ^2_{H^{s-\frac{1}{2}}_h} + \mathcal {O}(h^\infty )\Vert u\Vert _{H_h^{-N}}^2. \end{aligned}$$

  Step 3: term $T_2$. Write

  $$\begin{aligned} T_2 = {{\,\mathrm{Re}\,}}\langle {({{\,\mathrm{Im}\,}}P) u, G^*Gu}\rangle = \langle {({{\,\mathrm{Im}\,}}P)Gu, Gu}\rangle + {{\,\mathrm{Re}\,}}\langle {[G, {{\,\mathrm{Im}\,}}P]u, Gu}\rangle . \end{aligned}$$

  We estimate the two terms on the right hand side separately. Firstly

  $$\begin{aligned} |\langle {({{\,\mathrm{Im}\,}}P)Gu, Gu}\rangle | = h |\langle {(h^{-1}{{\,\mathrm{Im}\,}}P)}Gu, Gu\rangle | \le C_2 h \Vert Gu\Vert _{L^2}^2 + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}^2, \end{aligned}$$

  where we used the boundedness property (item 6) for the exotic operator $h^{-1} {{\,\mathrm{Im}\,}}P$. For the second term, we need to deal with a commutator. Note first that

  $$\begin{aligned} {{\,\mathrm{Re}\,}}G^*[G, {{\,\mathrm{Im}\,}}P] = h {{\,\mathrm{Re}\,}}G^* [G, h^{-1} {{\,\mathrm{Im}\,}}P] \in h^2 \Psi ^{2s-1}_{h, \rho , k-1}({{\,\mathrm{\mathcal {M}}\,}}), \end{aligned}$$

  by item 3. above. As $\mathrm{WF}_h(G^*[G, {{\,\mathrm{Im}\,}}P]) \subset {{\,\mathrm{ell}\,}}_h(B_1)$, by the elliptic estimate

  $$\begin{aligned} |\langle {{{\,\mathrm{Re}\,}}(G^*[G, {{\,\mathrm{Im}\,}}P]) u, u}\rangle |&= h^2|\langle {h^{-2}{{\,\mathrm{Re}\,}}(G^*[G, {{\,\mathrm{Im}\,}}P]) u, Yu}\rangle | + \mathcal {O}(h^\infty )\Vert u\Vert _{H_h^{-N}}^2\\&\le Ch^2 \Vert B_1 u\Vert ^2_{H_h^{s - \frac{1}{2}}} + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}^2, \end{aligned}$$

  where $Y \in \Psi _h^0({{\,\mathrm{\mathcal {M}}\,}})$ is such that $Y = 1 + \mathcal {O}(h^\infty )$ microlocally on $\mathrm{WF}_h({{\,\mathrm{Re}\,}}(G^*[G, {{\,\mathrm{Im}\,}}P]))$ and $\mathrm{WF}_h(Y) \subset {{\,\mathrm{ell}\,}}_h(B_1)$. Here we used the boundedness in the exotic class item 6., item 1., the elliptic estimate in the exotic class (item 7.). Adding the two estimates we finally obtain

  $$\begin{aligned} |{{\,\mathrm{Re}\,}}\langle {({{\,\mathrm{Im}\,}}P)u, G^*Gu}\rangle | \le C_2h \Vert Gu\Vert _{L^2}^2 + Ch^2\Vert B_1 u\Vert _{H_h^{s - \frac{1}{2}}}^2 + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}^2. \end{aligned}$$

  Step 4. Adding the estimates in Steps 3. and 4., we obtain

  $$\begin{aligned} {{\,\mathrm{Im}\,}}\langle {f, G^*Gu}\rangle \le (C_1 + C_2 - \beta )h \Vert Gu\Vert _{L^2}^2 + Ch\Vert Bu\Vert _{H_h^s}^2 + Ch^2 \Vert B_1u\Vert _{H_h^{s - \frac{1}{2}}}^2 + \mathcal {O}(h^\infty ) \Vert u\Vert ^2_{H_h^{-N}}. \end{aligned}$$

  By ellipticity of B on $\mathrm{WF}_h(G)$, there is $Q \in \Psi _h^{s}({{\,\mathrm{\mathcal {M}}\,}})$ such that $G = QB_1 + R$, where $R \in h^\infty \Psi _h^{-\infty }({{\,\mathrm{\mathcal {M}}\,}})$, thus

  $$\begin{aligned} |\langle {f, G^*Gu}\rangle | \le |\langle {QB_1f, Gu}\rangle | + |\langle {Rf, Gu}\rangle | \le C\Vert B_1f\Vert _{H_h^s} \Vert Gu\Vert _{L^2} + \mathcal {O}(h^\infty ) \Vert u\Vert ^2_{H_h^{-N}}. \end{aligned}$$

  Now choose $\beta = C_1 + C_2 + 1$ to get

  $$\begin{aligned} \Vert Gu\Vert _{L^2}^2 \le C\Vert Bu\Vert ^2_{H_h^s} + Ch^{-1} \Vert B_1f\Vert _{H_h^s} \Vert Gu\Vert _{L^2} + Ch\Vert B_1u\Vert ^2_{H_h^{s-\frac{1}{2}}} + \mathcal {O}(h^\infty )\Vert u\Vert _{H_h^{-N}}^2. \end{aligned}$$

  We can absorb the $\Vert Gu\Vert _{L^2}$ term to the left hand side, at the cost of the additional term $Ch^{-2} \Vert B_1f\Vert _{H_h^s}^2$ on the right hand side. Next, use the condition $\mathrm{WF}_h(A) \subset {{\,\mathrm{ell}\,}}_h(G)$ and elliptic estimates to derive

  $$\begin{aligned} \Vert Au\Vert _{H^s_h} \le C \Vert Bu\Vert _{H_h^s} + Ch^{-1} \Vert B_1f\Vert _{H_h^s} + Ch^{\frac{1}{2}} \Vert B_1 u\Vert _{H_h^{s - \frac{1}{2}}} + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}.\nonumber \\ \end{aligned}$$

  (A.5)

  Step 5. Here, one can use the same induction procedure as in [DyZw, Proof of Th. E.47] to show that for each $\ell \in \mathbb {N}$

  $$\begin{aligned} \Vert Au\Vert _{H^s_h} \le C \Vert Bu\Vert _{H_h^s} + Ch^{-1} \Vert B_1f\Vert _{H_h^s} + Ch^{\frac{\ell }{2}} \Vert B_1 u\Vert _{H_h^{s - \frac{\ell }{2}}} + \mathcal {O}(h^\infty ) \Vert u\Vert _{H_h^{-N}}, \end{aligned}$$

  where the first step of the induction is exactly where we arrived in (A.5).   $\square$

Next we discuss semiclassical defect measures in the setting of the exotic calculus.

Proposition A.2

Assume that $u_h\in L^2({{\,\mathrm{\mathcal {M}}\,}})$ is a family satisfying $\Vert u_h\Vert _{L^2} = \mathcal {O}(1)$. Then there exists a Radon measure $\mu$, called semiclassical measure, and a sequence $h_j \rightarrow 0$, such that for any $A \in \Psi ^{{{\,\mathrm{comp}\,}}}_{h, \rho , 0}({{\,\mathrm{\mathcal {M}}\,}})$ with $\lim _{h\rightarrow 0} \sigma (A)(h;x,\xi ) = a_0(x,\xi )$ in $C_c^0(T^*{{\,\mathrm{\mathcal {M}}\,}})$,

$$\begin{aligned} \lim _{j\rightarrow \infty }\langle {Au_{h_j}, u_{h_j}}\rangle _{L^2} = \int _{T^*{{\,\mathrm{\mathcal {M}}\,}}} a_0\, d\mu . \end{aligned}$$

Proof

We follow the proof of [DyZw, Theorem E.42]. By Proposition A.1, we have that $A-\mathrm{Op}_h(a_h)\in h\Psi ^{\mathrm{comp}}_{h,\rho ,0}(\mathcal {M})$ for some symbol $a_h\in S_{h,\rho ,0}^{\mathrm{comp}}(T^*\mathcal {M})$ so that $a_{h}\rightarrow a_0$ in $C_c^0(T^*{{\,\mathrm{\mathcal {M}}\,}})$; it suffices to prove the claim for $A = {{\,\mathrm {Op}\,}}_{h_j}(a_{h_j})$. We write $I_h(a_h) := \langle {\mathrm{Op}(a_h) u_h, u_h}\rangle _{L^2}$ and claim

$$\begin{aligned} \limsup _{h \rightarrow 0} |I_h(a_h)| \le C \limsup _{h \rightarrow 0} \Vert a_h\Vert _\infty \le C\sup _{h} \Vert a_h\Vert _\infty . \end{aligned}$$

(A.6)

Indeed, by Cauchy-Schwarz and Proposition A.1 (item 6), we have

$$\begin{aligned} |I_h(a_h)| \le C \Vert a_h\Vert _{\infty } + \mathcal {O}_{A}(h^{\infty }), \end{aligned}$$

(A.7)

where $C = C(\Vert u_h\Vert _{L^2}) > 0$. Take a countable, dense subset $(a_h^{\ell })_{\ell \in \mathbb {N}} \subset S^{{{\,\mathrm{comp}\,}}}_{h, \rho , 0}(T^*\mathcal {M})$, where $S^{{{\,\mathrm{comp}\,}}}_{h, \rho , 0}(T^*\mathcal {M})$ is equipped with the inductive limit topology from the seminorms in (A.1). By a diagonal argument and since by (A.7) $I_h(a_h^\ell )$ is bounded for all $\ell$, we may extract a sequence $h_j \rightarrow 0$ such that $I_{h_j}(a_{h_j}^\ell )$ converges for all $\ell$. For each $a_h \in S^{{{\,\mathrm{comp}\,}}}_{h, \rho , 0}(T^*\mathcal {M})$ and $\ell$, we get by (A.7)

$$\begin{aligned} \limsup _{j, j' \rightarrow \infty } |I_{h_j}(a_{h_j}) - I_{h_{j'}}(a_{h_{j'}})| \le \limsup _{j, j' \rightarrow \infty } |I_{h_j}(a_{h_j}^\ell ) - I_{h_{j'}}(a_{h_{j'}}^\ell )| + C\sup _{h} \Vert a_h - a_{h}^\ell \Vert _{\infty }. \end{aligned}$$

Using the density of $a^\ell _h$, we obtain that $I_{h_j}(a_{h_j})$ is a Cauchy sequence and we may define for $a_h\in S^{{{\,\mathrm{comp}\,}}}_{h, \rho , 0}(T^*\mathcal {M})$

$$\begin{aligned} I(a_h) := \lim _{j \rightarrow \infty } I_{h_j}(a_{h_j}). \end{aligned}$$

By (A.7), the map I satisfies for each $a_h\in S^{\mathrm{comp}}_{h,\rho ,0}(T^*\mathcal {M})$

$$\begin{aligned} |I(a_h)| \le C \limsup _{j \rightarrow \infty } \Vert a_{h_j}\Vert _{\infty } \le C \sup _{h \in (0, h_0')} \Vert a_h\Vert , \end{aligned}$$

(A.8)

for any $h_0' > 0$. In particular, I restricts to a continuous linear functional on $C_c^0(T^*\mathcal {M})$, i.e. on h-independent functions. Given $a_h \in S^{\mathrm{comp}}_{h,\rho ,0}(T^*\mathcal {M})$ with $\lim _{h\rightarrow 0} a_{h}=a_0 \in C_c^\infty (T^*{{\,\mathrm{\mathcal {M}}\,}})$ in the $C_c^0(T^*\mathcal {M})$ topology, we get by (A.8)

$$\begin{aligned} |I(a_0 - a_h)| \le C\sup _{h \in (0, h_0')} \Vert a_0 - a_h\Vert _{\infty } \rightarrow 0 \text { as }h_0'\rightarrow 0, \end{aligned}$$

and thus $I(a_h)=I(a_0)$. By [DyZw, Theorem E.42] there is a Radon measure $\mu$ such that for each $a_0 \in C_c^\infty (T^*\mathcal {M})$

$$\begin{aligned} I(a_0) = \lim _{j\rightarrow \infty }\langle {{{\,\mathrm{Op}\,}}_{h_j}(a_0) u_{h_j}, u_{h_j}}\rangle _{L^2}= \int _{T^*\mathcal {M}} a_0\, d\mu . \end{aligned}$$

The main claim follows from this, by using $I(a_h) = I(a_0)$ under the given assumptions. $\square$

Now we prove a version of a propagation estimate for the semiclassical measure in the exotic calculus. This is a slight extension of [DyZw, Theorem E.44]. Note that if $\sigma (P)$ is real valued and $k\ge 1$, then ${{\,\mathrm{Im}\,}}P \in h\Psi _{h, \rho , k-1}^{m-1}(\mathcal {M})$ if $P \in \Psi _{h, \rho , k}^m(\mathcal {M})$ by Proposition A.1 (item 5).

Proposition A.3

Assume $\Vert u_h\Vert _{L^2} = \mathcal {O}(1)$ and $u_h$ converges to a semiclassical measure $\mu$. Let $P \in \Psi ^m_{h, \rho , k}({{\,\mathrm{\mathcal {M}}\,}})$ with $k\ge 1$, denote $p:= \sigma (P)$ and assume that p is real-valued for all h, and define $b := \sigma (h^{-1}{{\,\mathrm{Im}\,}}P)$. Assume that for each $a \in C_c^\infty (T^*{{\,\mathrm{\mathcal {M}}\,}})$,

$$\begin{aligned} (H_p a)_0 = \lim _{h \rightarrow 0} H_p a \text { and } b_0 = \lim _{h \rightarrow 0} b \text { exist in } C^0(T^*\mathcal {M}). \end{aligned}$$

Then there is $C>0$ such that for all $a \in C_c^\infty (T^*{{\,\mathrm{\mathcal {M}}\,}})$ and $Y \in \Psi _h^{{{\,\mathrm{comp}\,}}}({{\,\mathrm{\mathcal {M}}\,}})$ with $Y = 1 + \mathcal {O}(h^\infty )$ microlocally on ${{\,\mathrm{supp}\,}}(a)$,

$$\begin{aligned} \Big |\int _{T^*{{\,\mathrm{\mathcal {M}}\,}}} ((H_{p}a)_0 + 2b_0a) d\mu \Big | \le C \Vert a\Vert _\infty \limsup _{h \rightarrow 0} (h^{ - 1} \Vert YPu_h\Vert _{L^2} \Vert Yu_h\Vert _{L^2}). \end{aligned}$$

Proof

Assume without loss of generality that a is real valued. Let $A \in \Psi _h^{{{\,\mathrm{comp}\,}}}(M)$ be such that $\sigma (A) = a$ and $A^* = A$. We compute

$$\begin{aligned} h^{-1} {{\,\mathrm{Im}\,}}\langle {Pu_h, Au_h}\rangle&= (2i)^{-1}h^{- 1}\langle {Pu_h, Au_h}\rangle - (2i)^{-1} h^{- 1}\langle {P^*Au_h, u_h}\rangle \\&=(2i)^{-1} \langle {h^{ - 1}[A, P]u_h, u_h}\rangle + \langle {(h^{-1}{{\,\mathrm{Im}\,}}P) Au_h, u_h}\rangle . \end{aligned}$$

Now by Proposition A.1 (item 3), we have $h^{-1}[A, P] \in \Psi ^{{{\,\mathrm{comp}\,}}}_{h, \rho , k-1}({{\,\mathrm{\mathcal {M}}\,}})$ with $-i\sigma (h^{- 1}[A, P]) = H_p a$ and by assumptions $\sigma ((h^{-1}{{\,\mathrm{Im}\,}}P) A) = ba$. Thus by Proposition A.2 the right hand side converges to, after possibly extracting a subsequence $h_j \rightarrow 0$

$$\begin{aligned} \int _{T^*{{\,\mathrm{\mathcal {M}}\,}}} \Big (\frac{1}{2}(H_{p}a)_0 + b_0a\Big ) d\mu . \end{aligned}$$

Moreover, the left hand side equals

$$\begin{aligned} (2ih)^{-1}(\langle {PYu_h, AYu_h}\rangle - \langle {AYu_h, PYu_h}\rangle ) + \mathcal {O}(h^\infty ), \end{aligned}$$

from which we easily deduce the main estimate. $\quad \square$