Pseudodifferential operators on non-quasianalytic classes of Beurling type

Lemma 1.3. (1) For every $\lambda, s, t > 0$ we have $$2\lambda\varphi^\left(\frac{s+t}{2\lambda}\right) \leq \lambda\varphi^\left(\frac{s}{\lambda}\right) + \lambda\varphi^\left(\frac{t}{\lambda}\right) \leq \lambda\varphi^\left(\frac{s+t}{\lambda}\right).$$ (2) Let $L \in \mathbb{N}$ be such that $\omega(et) \leq L(1 + \omega(t))$ . Then $$kt + L^k \varphi^ \left(\frac{t}{L^k}\right) \le \varphi^(t) + \sum_{j=1}^k L^j$$ for all $t \geq 0$ and $k \in \mathbb{N}$ . Let $L \in \mathbb{N}$ be such that $\omega(et) \leq L(1+\omega(t))$ . Then $|\alpha| + nL\varphi^(|\alpha|/nL) \leq$ $nL + n\varphi^(|\alpha|/n)$ . Therefore, if $$q_{K,n}(f) := \sup_{x \in K} \sup_{\alpha \in \mathbb{N}_0^p} e^{-|\alpha|} |f^{(\alpha)}(x)| \exp(-n\varphi^*(|\alpha|/n))$$ then $$q_{K,n}(f) \le |f|_{K,n} \le e^{nL} q_{K,nL}(f),$$ so the topology of $\mathcal{E}_{(\omega)}(\Omega)$ can also be described by the system of seminorms $\{q_{K,n}\}.$

Lemma 1.4. For every $n, k \in \mathbb{N}$ and $t \geq 1$ we have - $\begin{array}{l} (1) \ t^k \leq e^{n\varphi^(k/n)} e^{n\omega(t)}, \\ (2) \ \inf_{j \in \mathbb{N}_0} t^{-j} e^{k\varphi^(j/k)} \leq e^{-k\omega(t) + \log t}. \end{array}$ The following result permits us to split $\mathbb{R}$ into intervals in which the infimum in 1.4 is attained in a finite set.

Lemma 1.5. Fix $k, N \in \mathbb{N}$ and assume $$\frac{k}{N}\,\varphi^\bigg(\frac{N}{k}\bigg) \leq \log t < \frac{k}{N+1}\,\varphi^\bigg(\frac{N+1}{k}\bigg).$$ Then - $(1) \min_{0 \le j \le N} t^{-j} e^{k\varphi^*(j/k)} \le e^{-k\omega(t) + \log t},$ - (2) $t^{-N} e^{2k\varphi^*(N/2k)} < e^{-k\omega(t) + \log t}$

Lemma 1.8. Let $a(x, y, \xi)$ be an amplitude in $S_{\varrho, \delta}^{m, \omega}(\Omega)$ . Then for every compact set $Q \subset \Omega \times \Omega$ there exists a sequence $C_n > 0$ , $n \in \mathbb{N}$ , such that $$|D_x^{\alpha} D_y^{\gamma} a(x, y, \xi)| \le C_n e^{(\varrho - \delta)n\varphi^(|\alpha|/n)} e^{(\varrho - \delta)n\varphi^(|\gamma|/n)} |\xi|^{\delta|\alpha + \gamma|} e^{m\omega(\xi)}$$ for every $(x, y) \in Q$ and $|\xi| \ge 1$ .

Proposition 1.9. Let $a(x, y, \xi)$ be an amplitude in $S_{\varrho, \delta}^{m, \omega}(\Omega)$ and let $f \in \mathcal{D}_{(\omega)}(\Omega)$ . For every compact set $K \subset \Omega$ and $n, \lambda \in \mathbb{N}$ there is a constant C > 0 such that $$\left| \int D_x^{\alpha} a(x, y, \xi) f(y) e^{-iy\xi} \, dy \right| \le C e^{-\lambda \omega(\xi)} e^{n\varphi^*(|\alpha|/n)}$$ for every $x \in K$ , $\xi \in \mathbb{R}^p$ . Moreover, a similar estimate holds if we replace $a(x, y, \xi)$ by $b(x, y, \xi) := a(x, y, \xi)e^{ix\xi}$ .

Lemma 2.1. Let $a(x,y,\xi)$ be an amplitude in $S^{m,\omega}_{\varrho,\delta}(\Omega)$ and let $\Psi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ . Then - (1) $K(x,y) := \int a(x,y,\xi)e^{i(x-y)\xi}\Psi(\xi) d\xi$ belongs to $\mathcal{E}_{(\omega)}(\Omega \times \Omega)$ , - (2) $B: \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega), \ B(f)(x) := \int K(x,y)f(y) dy, \ is \ a \ continuous \ linear \ operator.$ Let $\Psi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ be a test function such that $\Psi(\xi) = 1$ for $|\xi| \leq 1$ and $\Psi(\xi) = 0$ for $|\xi| \geq 2$ . We put $$(A_{\delta}f)(x) := \iint a(x, y, \xi)e^{i(x-y)\xi}f(y)\Psi(\delta\xi) dy d\xi.$$

Theorem 2.2. Let $a(x,y,\xi)$ be an amplitude in $S_{\varrho,\delta}^{m,\omega}(\Omega)$ . Then - (1) For every $f \in \mathcal{D}_{(\omega)}(\Omega)$ the limit $A(f) := \mathcal{E}_{(\omega)}(\Omega)$ - $\lim_{\delta \to 0+} A_{\delta}(f)$ exists and $A : \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ is a continuous linear operator, - (2) $(Af)(x) = \int (\int a(x, y, \xi)e^{i(x-y)\xi}f(y) dy) d\xi.$

Proposition 2.4. The operator P(x, D) associated to a symbol $p(x, \xi)$ in $S_{\varrho,\delta}^{m,\omega}(\Omega)$ can be extended to $\mathcal{D}_{L_1,(\omega)}(\mathbb{R}^p)$ and the extension is linear and continuous taking values in $\mathcal{E}_{(\omega)}(\Omega)$ .

Theorem 2.5. The pseudodifferential operator A associated to an amplitude $a(x, y, \xi)$ in $S_{\varrho, \delta}^{m, \omega}(\Omega)$ admits a continuous linear extension $\mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{D}'_{(\omega)}(\Omega)$ .

Corollary 2.6. Let $A: \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ be the pseudodifferential operator with amplitude $a(x,y,\xi)$ . Then $A^t|_{\mathcal{D}_{(\omega)}(\Omega)}: \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ is a pseudodifferential operator with amplitude $a(y,x,-\xi)$ .

Theorem 2.7. The extension $\overline{P}(x,D): \mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{D}'_{(\omega)}(\Omega)$ of the pseudodifferential operator P(x,D) is given by $$\langle \overline{P}(x,D)\mu,\psi\rangle = \int \widehat{\mu}(\xi) \Big( \int e^{ix\xi} p(x,\xi)\psi(x) \, dx \Big) \, d\xi.$$ Proof. Since $\langle P(x,D)\varphi,\psi\rangle=\int\widehat{\varphi}(\xi)(\int e^{ix\xi}p(x,\xi)\psi(x)\,dx)\,d\xi$ for every $\varphi,\psi\in\mathcal{D}_{(\omega)}(\Omega)$ , we only have to prove that $\overline{P}(x,D):\mathcal{E}'_{(\omega)}(\Omega)\to\mathcal{D}'_{(\omega)}(\Omega)$ is a well defined, continuous linear operator. To do this we first fix $\mu\in\mathcal{E}'_{(\omega)}(\Omega)$ and we let B be a bounded set in $\mathcal{D}_{(\omega)}(\Omega)$ . By the Paley-Wiener theorem ([8, 7.4]) there are constants A,D>0 such that $|\widehat{\mu}(\xi)|\leq De^{A\omega(\xi)}$ for every $\xi\in\mathbb{R}^p$ . Now we take k:=A+1 and we apply (the proof of) Proposition 1.9 to find a constant C>0 such that $|\int e^{ix\xi}p(x,\xi)\psi(x)\,dx|\leq Ce^{-k\omega(\xi)}$ for every $\psi\in B$ . Then $$\left|\widehat{\mu}(\xi)\left(\int e^{ix\xi}p(x,\xi)\psi(x)\,dx\right)\right| \le CDe^{-\omega(\xi)},$$ which implies that $\overline{P}(x,D)\mu: \mathcal{D}_{(\omega)}(\Omega) \to \mathbb{C}$ is bounded on bounded sets and, consequently, $\overline{P}(x,D)\mu \in \mathcal{D}'_{(\omega)}(\Omega)$ . Moreover the estimates just obtained also show that $\overline{P}(x,D): \mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{D}'_{(\omega)}(\Omega)$ transforms bounded sets in $\mathcal{E}'_{(\omega)}(\Omega)$ into weakly bounded sets in $\mathcal{D}'_{(\omega)}(\Omega)$ , and we conclude that $\overline{P}(x,D)$ is continuous. The correspondence between amplitudes and operators is not one-to-one, that is, two different amplitudes may define the same operator (see Example 2.11(4) below). The situation is much better for symbols.

Corollary 2.8. Let $p(x,\xi)$ and $q(x,\xi)$ be symbols in $S^{m,\omega}_{\varrho,\delta}(\mathbb{R}^p)$ defining the same pseudodifferential operator. Then $p(x,\xi)=q(x,\xi)$ . Proof. We put $r(x,\xi) := p(x,\xi) - q(x,\xi)$ . By hypothesis R(x,D) is the null operator and hence $\langle \overline{R}(x,D)\delta_y,\psi\rangle = 0$ for every $y \in \mathbb{R}^p$ and $\psi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ . For a fixed $\psi \in \mathcal{D}_{(\omega)}(\Omega)$ we set $I(\xi) := \int e^{ix\xi} r(x,\xi)\psi(x) dx$ and we deduce from Theorem 2.7 that the Fourier transform of the continuous function $I(\xi)$ vanishes. It follows that $\int e^{ix\xi} r(x,\xi)\psi(x) dx = 0$ for every $\psi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ , and so $r(x,\xi) = 0$ everywhere. In some cases it is possible to recover the symbol from the pseudodifferential operator it defines.

Proposition 2.9. Let $p(x,\xi)$ a symbol in $S_{\varrho,\delta}^{m,\omega}(\mathbb{R}^p)$ and assume that P(x,D) admits a continuous linear extension $A: \mathcal{E}_{(\omega)}(\mathbb{R}^p) \to \mathcal{E}_{(\omega)}(\mathbb{R}^p)$ . Then $$p(x,\xi) = \frac{1}{(2\pi)^p} e^{-ix\xi} A(e^{i(\cdot)\xi})(x).$$

Proposition 2.12. Let $\omega$ be a weight with $\omega(t) = o(t^d)$ , d < 1, and let G(D) be an $(\omega)$ -ultradifferential operator with constant coefficients such that $G(\xi)$ does not vanish on $\mathbb{R}^p$ . If one of the following two conditions is satisfied: - (1) G(D) is elliptic, or - (2) $G(D): \mathcal{E}_{(\omega)}(\mathbb{R}^p) \to \mathcal{E}_{(\omega)}(\mathbb{R}^p)$ is surjective and $\{t^d\}$ -hypoelliptic, then there exists a pseudodifferential operator of $(\omega)$ -class $P: \mathcal{D}_{(\omega)}(\mathbb{R}^p) \to \mathcal{E}_{(\omega)}(\mathbb{R}^p)$ such that $G(D) \circ P$ is the identity on $\mathcal{D}_{(\omega)}(\mathbb{R}^p)$ .

Proposition 2.14. Let P(x,D) be the pseudodifferential operator associated to $p(x,\xi) \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ . Then there is an ultradifferential operator G(D) of $(\omega)$ -class and a symbol $q(x,\xi) \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ of finite order such that if Q(x,D) is the corresponding pseudodifferential operator, then $P(x,D) = Q(x,D) \circ G(D)$ . Proof. We take D > 0 such that $D\omega(\xi/2) > m\omega(\xi)$ . Let G be an even entire function satisfying $\log |G(z)| = O(\omega(z))$ as $|z| \to \infty$ and $|G(z)| \ge e^{D\omega(z)}$ whenever $|\operatorname{Im} z| \le |\operatorname{Re} z|/D$ (the existence of such a function follows from [19, Corollary 1.4]). Then 1/G is a symbol as in 2.10. Indeed, it is clear that $|1/G(\xi)| \le e^{-D\omega(\xi)}$ for $\xi \in \mathbb{R}^p$ , and since it is holomorphic in $\{z \in \mathbb{C}^p : |\operatorname{Im} z| < |\operatorname{Re} z|/D\}$ , we conclude from the Cauchy integral formula that, for some C > 0 and $\xi$ large enough, $$\left| \left( \frac{1}{G(\xi)} \right)^{(\beta)} \right| \le C^{|\beta|} \beta! \frac{e^{-D\omega(\xi/2)}}{|\xi|^{|\beta|}} \le C^{|\beta|} \beta! \frac{e^{-D\omega(\xi/2)}}{|\xi|^{\varrho|\beta|}}.$$ We define $q(x,\xi) = p(x,\xi)/G(\xi)$ . It is easy to see that q is a symbol of finite order. Moreover, for every $\varphi \in \mathcal{D}_{(\omega)}(\Omega)$ we have $$\begin{split} (Q(x,D)\circ G(D))(\varphi) &= \int q(x,\xi) e^{ix\xi} \widehat{G(D)(\varphi)}(\xi) \, d\xi \\ &= \int q(x,\xi) e^{ix\xi} G(\xi) \widehat{\varphi}(\xi) \, d\xi = P(x,D)(\varphi). \ \blacksquare \end{split}$$ REMARK 2.15. The ultradifferential operator $G(D): \mathcal{D}'_{(\omega)}(\Omega) \to \mathcal{D}'_{(\omega)}(\Omega)$ in the proposition above has the property that $G(D)f \in \mathcal{E}_{(\omega)}(\Omega)$ if, and only if, $f \in \mathcal{E}_{(\omega)}(\Omega)$ [3, 2.1]. Hence the decomposition given in Proposition 2.14 could be useful in order to study hypoellipticity. We observe that each ultradifferential operator of $(\omega)$ -class acts continuously from $\mathcal{D}_{(\sigma)}(\Omega)$ into $\mathcal{E}_{(\sigma)}(\Omega)$ for any weight $\sigma \geq \omega$ , whereas each pseudodifferential operator of $(\omega)$ -class and finite order is also a pseudodifferential operator of $(\tau)$ -class for $\tau \leq \omega$ . However, one cannot expect that pseudodifferential operators of infinite order of $(\omega)$ -class be pseudodifferential operators of a different class. Roughly speaking, the p.d.o. (of infinite order) of $(\omega)$ -class are strongly tied to the Beurling space $\mathcal{D}_{(\omega)}(\Omega)$ . Next, we analyze the behaviour of a pseudodifferential operator when it is defined by an amplitude which does not depend on the x-variable. A combination of the next result and Proposition 2.4 will be useful to study the composition of p.d.o.'s in Theorem 3.18.

Proposition 2.16. Let $b(y,\xi)$ be an amplitude in $S_{\varrho,\delta}^{m,\omega}(\Omega)$ , and let B be the associated pseudodifferential operator. Then $Bf \in \mathcal{D}_{L_1,(\omega)}(\mathbb{R}^p)$ for every $f \in \mathcal{D}_{(\omega)}(\Omega)$ , and $B : \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{D}_{L_1,(\omega)}(\mathbb{R}^p)$ is continuous.

Theorem 2.17. The $(\omega)$ -singular support of the kernel K of a pseudo-differential operator A is contained in $\Delta := \{(x,y) \in \Omega \times \Omega : x = y\}$ .

Theorem 2.18. Let $A: \mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{D}'_{(\omega)}(\Omega)$ be the pseudodifferential operator associated to an amplitude $a(x,y,\xi)$ in $S^{m,\omega}_{o,\delta}(\Omega)$ . Then $$\operatorname{sing}_{(\omega)}\operatorname{supp}(A\mu)\subset\operatorname{sing}_{(\omega)}\operatorname{supp}(\mu)$$ for every $\mu \in \mathcal{E}'_{(\omega)}(\Omega)$ . If a convolution operator $\psi \mapsto \psi * S$ , $\psi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ , $S \in \mathcal{D}'_{(\omega)}(\mathbb{R}^p)$ , is a pseudodifferential operator, by 2.18 the $(\omega)$ -singular support of S reduces to $\{0\}$ . Therefore, not every convolution operator is a pseudodifferential operator. 3. Symbolic calculus. One of the problems one has to face is how to determine the class of symbols in order that the theory for the operators can be converted into an algebraic theory for the corresponding symbols. Moreover, the class of operators should be closed under products. This leads to the necessity of developing the classical symbolic calculus in our setting. The definitions below are motivated by [23, 25].

Proposition 3.5. Let A be the pseudodifferential operator defined by an amplitude $a \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ which is equivalent to zero. Then A is an $(\omega)$ -smoothing operator.

Theorem 3.7. Let $\sum a_j \in FAS_{\varrho,\delta}^{m,\omega}(U)$ and let $\Omega$ be a relatively compact open subset of U. Then there is an amplitude $a \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ such that $a \sim \sum a_j$ on $\Omega$ . Proof. We put $$\Psi_{j,n}(\xi) := 1 - \Phi_j \left( \frac{\xi}{Re^{(n/j)\varphi^*(j/n)}} \right),$$ where R will be determined later. We set $R_1 := (2R)^{\varrho-\delta}$ and we observe that $\Psi_{j,n}(\xi) \neq 0$ implies $e^{(\varrho-\delta)n\varphi^(j/n)} \leq |\xi|^{j(\varrho-\delta)}(1/R_1)^j$ , whereas if $\xi$ is in the support of any derivative we also have $(|\xi|/3)^{j(\varrho-\delta)}(1/R_1)^j \leq e^{(\varrho-\delta)n\varphi^(j/n)}$ . According to Definition 3.1, we can select R large enough so that $\sum je^{Djp}/R_1^j < \infty$ and, for some sequence $(C_n)$ , $$|D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} a_j(x, y, \xi)|$$ $$\leq C_n B^{|\beta|} \beta! e^{(\varrho - \delta)2n\varphi^(|\alpha + \gamma|/2n)} e^{m\omega(\xi)} |\xi|^{|\alpha + \gamma|\delta - |\beta|\varrho} e^{(\varrho - \delta)2n\varphi^(j/2n)} |\xi|^{-(\varrho - \delta)j\varrho}$$ whenever $(x,y) \in \Omega$ and $\log(|\xi|/R) \ge \frac{n}{|\beta|+j} \varphi^* (\frac{|\beta|+j}{n})$ . We first assume $(x,y) \in \Omega$ , $n \in \mathbb{N}$ , $\log(|\xi|/3R) \ge (n/|\beta|)\varphi^*(|\beta|/n)$ and $\Psi_{j,n}(\xi) \ne 0$ . Then $$\log\left(\frac{|\xi|}{R}\right) \ge \max\left(\frac{n}{|\beta|}\,\varphi^\!\left(\frac{|\beta|}{n}\right), \frac{n}{j}\,\varphi^\!\left(\frac{j}{n}\right)\right) \ge \frac{2n}{|\beta|+j}\,\varphi^*\!\left(\frac{|\beta|+j}{2n}\right).$$ Moreover from $D_{\xi}^{i}\Psi_{j,n}(\xi) \neq 0$ we deduce that $\log(|\xi|/3R) \leq (n/j)\varphi^{}(j/n)$ , and consequently, $|\beta| \leq j$ since $\varphi^{}(t)/t$ is increasing. Since $|\xi| \geq 1$ and $0 \leq \varrho \leq 1$ , it follows that $|\xi|^{-|i|} \leq |\xi|^{-|i|}\varrho$ and we can estimate $$\begin{split} |D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta}(a_j(x,y,\xi) \varPsi_{j,n}(\xi))| e^{-m\omega(\xi)} \\ &\leq \sum_{i \leq \beta} \binom{\beta}{i} |D_{\xi}^i \varPsi_{j,n}(\xi) D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta-i} a_j(x,y,\xi)| e^{-m\omega(\xi)} \end{split}$$ $$\leq CC_n e^{2n\varphi^*(|\alpha+\gamma|/2n)} |\xi|^{|\alpha+\gamma|\delta-(\varrho-\delta)j} e^{n\omega(\xi)} \sum_{i\leq\beta} \frac{B^{|\beta-i|}D^{|i|}j^{|i|+1}}{i!} |\xi|^{-|i|-\varrho|\beta-i|}$$ $$\leq CC_n B^{|\beta|}\beta! e^{(\varrho-\delta)2n\varphi^(|\alpha+\gamma|/2n)} |\xi|^{|\alpha+\gamma|\delta-|\beta|\varrho} e^{(\varrho-\delta)2n\varphi^(j/2n)} |\xi|^{-(\varrho-\delta)j}$$ $$\times \sum_{i<\beta} \frac{D^{|i|}j^{|i|+1}}{i!}$$ $$\leq CC_n B^{|\beta|} \beta! e^{(\varrho - \delta)2n\varphi^*(|\alpha + \gamma|/2n)} |\xi|^{|\alpha + \gamma|\delta - |\beta|\varrho} \left(\frac{1}{R_1}\right)^j j e^{Djp}.$$ By induction we select a sequence $(j_n)$ of natural numbers such that $j_1 := 0$ , $j_n < j_{n+1}$ , $\lim_{n\to\infty} j_n/n = \infty$ and $$C_{n+1} \sum_{j=j_{n+1}+1}^{\infty} \frac{je^{Djp}}{R_1^j} \le \frac{C_n}{2} \sum_{j=j_n+1}^{j_n+2} \frac{je^{Djp}}{R_1^j}.$$ Then $$\overline{D}_n := C_n \sum_{j=j_n+1}^{j_{n+1}} \frac{j e^{Djp}}{R_1^j}$$ satisfies $\overline{D}_{n+1} \leq \overline{D}_n/2$ . We now prove that $$a(x, y, \xi) := a_0(x, y, \xi) + \sum_{n=1}^{\infty} \sum_{j=i-1}^{j_{n+1}} \Psi_{j,n}(\xi) a_j(x, y, \xi)$$ is an amplitude. Since $j_n \leq j$ and $\Psi_{j,n}(\xi) \neq 0$ implies $$\frac{n}{j_n} \varphi^ \left( \frac{j_n}{n} \right) \le \frac{n}{j} \varphi^ \left( \frac{j}{n} \right) \le \log \left( \frac{|\xi|}{2R} \right)$$ the condition $\lim_{n\to\infty}(n/j_n)\varphi^(j_n/n)=\infty$ permits us to conclude that the sum defining a is locally finite. Hence a is a well defined $C^{\infty}$ function. Let us assume $\log(|\xi|/3R) \geq (n/|\beta|)\varphi^(|\beta|/n)$ . Then, for every $n \in \mathbb{N}$ , $$\begin{split} \left| D_x^\alpha D_y^\gamma D_\xi^\beta \bigg( \sum_{k=n}^\infty \sum_{j=j_k+1}^{j_{k+1}} \varPsi_{j,k}(\xi) a_j(x,y,\xi) \bigg) \right| \\ & \leq C B^{|\beta|} \beta! e^{(\varrho-\delta)n\varphi^*(|\alpha+\gamma|/n)} |\xi|^{|\alpha+\gamma|\delta-|\beta|\varrho} e^{m\omega(\xi)} \sum_{k=0}^\infty \overline{D}_k. \end{split}$$ Since $a_0 + \sum_{k=1}^{n-1} \sum_{j=j_k+1}^{j_{k+1}} \Psi_{j,k} a_j$ is a finite sum of amplitudes we conclude that $a(x,y,\xi) \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ . To finish we have to show that $a \sim \sum a_j$ on $\Omega$ . In order to do this, we assume that $(x,y) \in \Omega$ and $\log(|\xi|/3R) \ge \frac{n}{|\beta|+N} \varphi^*(\frac{|\beta|+N}{n})$ and we estimate the derivatives of $$a - \sum_{j < N} a_j = \sum_{k=1}^{\infty} \sum_{j=j_k+1}^{j_{k+1}} \Psi_{j,k} a_j - \sum_{j=1}^{N-1} a_j.$$ We will only consider the case $N > nj_n$ . For every $j \in \mathbb{N}$ there is $k \in \mathbb{N}$ with $j_k < j \le j_{k+1}$ . Then k < n implies $j \le j_n (< N)$ and $$\log\left(\frac{|\xi|}{3R}\right) \ge \frac{n}{N} \, \varphi^\!\left(\frac{N}{n}\right) \ge \frac{1}{j_n} \, \varphi^(j_n) \ge \frac{k}{j} \, \varphi^*\!\left(\frac{j}{k}\right).$$ On the other hand, $k \geq n$ and N > j also imply $$\log \left(\frac{|\xi|}{3R}\right) \geq \frac{n}{N} \, \varphi^ \! \left(\frac{N}{n}\right) \geq \frac{k}{j} \, \varphi^ \! \left(\frac{j}{k}\right)$$ and $\Psi_{j,k}(\xi) = 1$ . Consequently, $a - \sum_{j < N} a_j$ can be expressed as a sum of functions $\Psi_{j,k}a_j$ with $j \ge N$ and $k \ge n$ . It follows from the above estimates that $$\begin{split} &|D_{x}^{\alpha}D_{y}^{\gamma}D_{\xi}^{\beta}\left(a_{j}(x,y,\xi)\Psi_{j,k}(\xi)\right)|e^{-m\omega(\xi)} \\ &\leq C\overline{D}_{k}B^{|\beta|}\beta!e^{(\varrho-\delta)2k\varphi^{}(|\alpha+\gamma|+N/2k)}|\xi|^{|\alpha+\gamma|\delta-|\beta|\varrho-(\varrho-\delta)N} \\ &\qquad \times e^{(\varrho-\delta)k\varphi^{}((j-N)/k)}|\xi|^{-(\varrho-\delta)(j-N)} \\ &\leq C\overline{D}_{k}B^{|\beta|}\beta!e^{(\varrho-\delta)2k\varphi^{*}(|\alpha+\gamma|+N/2k)}|\xi|^{|\alpha+\gamma|\delta-|\beta|\varrho-(\varrho-\delta)N}(1/R_{1})^{j-N}. \end{split}$$ where the last inequality follows from the fact that $\Psi_{i,k}(\xi) \neq 0$ implies that $$\log\left(\frac{|\xi|}{2R}\right) \ge \frac{k}{j} \, \varphi^\!\left(\frac{j}{k}\right) \ge \frac{k}{j-N} \, \varphi^\!\left(\frac{j-N}{k}\right).$$ Finally, we get $$\begin{split} \left| D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} \bigg( a - \sum_{j < N} a_j \bigg) \right| &\leq \sum_{k \geq n} \sum_{j \geq N} |D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} (\Psi_{j,k} a_j)| \\ &\leq C B^{|\beta|} \beta! e^{(\varrho - \delta)n\varphi^*(|\alpha + \gamma| + N/n)} |\xi|^{|\alpha + \gamma|\delta - |\beta|\varrho - (\varrho - \delta)N} \\ &\times e^{m\omega(\xi)} \sum_{k \geq n} \overline{D}_k \bigg( \sum_{j \geq N} \bigg( \frac{1}{R_1} \bigg)^{j - N} \bigg). \quad \blacksquare \end{split}$$ We observe that any other choice of the sequence $(j_n)$ and the constant R satisfying the estimates in the proof of the previous result would give another amplitude defining the same pseudodifferential operator, modulo an $(\omega)$ -smoothing one. Hence, in what follows we can assume without loss of generality that $(j_n)$ and R are as large as necessary. Our next aim is to give an asymptotic expansion formula. From now on, we will always assume that $\mathcal{E}_{(\omega)}(\Omega)$ contains the Gevrey class $\Gamma^{\{s\}}(\Omega)$ for some s > 1. Then, for $\sigma(t) := t^{1/s}$ , $\mathcal{E}_{(\sigma)}(\Omega)$ is contained (and dense) in $\mathcal{E}_{(\omega)}(\Omega)$ (see [8]). We assume that $(n/j)\varphi^*(j/n) \ge n$ for every $j \ge j_n$ . We put $\varphi_j := \Psi_{j,n}$ if $j_n < j \le j_{n+1}$ , $\varphi_0(\xi) = 1$ . As in the proof of 2.2 we have the following

Lemma 3.8. Let $a \in S_{\varrho,\delta}^{m,\omega}(\Omega)$ and let A be the pseudodifferential operator defined by a. Then, for every $u \in \mathcal{D}_{(\omega)}(\Omega)$ , $$A(u) = \sum_{j=0}^{\infty} A_j(u),$$ where $A_j$ is the pseudodifferential operator with amplitude $a_j(x, y, \xi) := (\varphi_j - \varphi_{j+1})(\xi)a(x, y, \xi)$ .

Lemma 3.9. Let $\sum_{j=0}^{\infty} p_j(x,\xi)$ be a formal sum in $FAS_{\varrho,\delta}^{m,\omega}(U)$ , $\Omega$ a relatively compact open subset of U, and $(j_n)$ as in the proof of 3.7 and satisfying the additional assumption $(n/j)\varphi^*(j/n) \ge \max(n, \log C_n)$ for $j \ge j_n$ , $(C_n)$ being the constants of Definition 3.1 relative to the closure of $\Omega$ . Let $$p(x,\xi) := \sum_{j=0}^{\infty} \varphi_j(\xi) p_j(x,\xi),$$ which is a symbol in $AS_{\varrho,\delta}^{m,\omega}(\Omega)$ . Then the corresponding pseudodifferential operator P(x,D) is the limit in $L(\mathcal{D}_{(\omega)}(\Omega),\mathcal{D}'_{(\omega)}(\Omega))$ of the sequence of operators $P_N:\mathcal{D}_{(\omega)}(\Omega)\to\mathcal{E}_{(\omega)}(\Omega)$ , where each $P_N$ is the pseudodifferential operator with symbol $\sum_{j=0}^N (\varphi_j-\varphi_{j+1})(\xi)(\sum_{l=0}^j p_l(x,\xi))$ .

Lemma 3.10. For every $n \in \mathbb{N}$ we have $$\lim_{N \to \infty} \frac{N}{e^{(n/N)\varphi^*(N/n)}} = 0.$$

Lemma 3.11. Let $m \ge n$ and $$\frac{1}{e}e^{(m/j)\varphi^(j/m)} \le t \le e^{(n/j)\varphi^(j/n)}.$$ Then $$|t|^{j+1} \ge e^{n\omega(t)} e^{2m\varphi^*(j/2m)} e^{-j}.$$ In particular $$e^{n\varphi^(j/n)} \ge e^{(n-1)\omega(t)}e^{2n\varphi^(j/2n)}$$ for $j$ large enough.

Lemma 3.12. Let $\sigma(t) = t^d$ , 0 < d < 1, and let $\omega$ be a weight function such that $\omega(t) = o(\sigma(t))$ . Then there are $\lambda > 0$ and a sequence $(j_n)$ of natural numbers such that $$\lambda \sigma(e^{(n/j)\varphi^*(j/n)}) \ge j$$ for every $j \ge j_n$ .

Theorem 3.13. Let $\omega$ be a weight such that $\omega(t) = o(t^d)$ , $d \leq \varrho - \delta$ , d < 1. Let $a \in AS_{\varrho,\delta}^{m,\omega}(U)$ with associated pseudodifferential operator A and let $\Omega$ be a relatively compact open subset of U. Then there are a pseudodifferential operator $P(x,D): \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ and an $(\omega)$ -smoothing operator $R: \mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ such that $A\varphi = P(x,D)\varphi + R\varphi$ for every $\varphi \in \mathcal{D}_{(\omega)}(\Omega)$ . Moreover $$p(x,\xi) \sim \sum_{j=0}^{\infty} p_j(x,\xi), \quad \text{where} \quad p_j(x,\xi) = \sum_{|\alpha|=j} \frac{1}{\alpha!} D_{\xi}^{\alpha} \partial_y^{\alpha} a(x,y,\xi)|_{y=x}.$$

Proposition 3.14. Let P(x,D) be the operator associated to $p(x,\xi) \in AS_{\varrho,\delta}^{m,\omega}(U)$ and let $\Omega$ be a relatively compact open subset of U. Then the transposed operator (restricted to $\mathcal{D}_{(\omega)}(\Omega)$ ) can be decomposed as $P(x,D)^t = Q(x,D) + R$ , where R is $(\omega)$ -smoothing and Q(x,D) is defined by a symbol $q(x,\xi) \sim \sum q_j$ , and we have $q_j(x,\xi) := \sum_{|\alpha|=j} (1/\alpha!) \partial_{\xi}^{\alpha} D_x^{\alpha} p(x,-\xi)$ .

Proposition 3.16. (1) $$((\sum p_j)^t)^t \sim \sum p_j$$ . (2) If $\sum p_j \sim \sum p_j'$ and $\sum q_j \sim \sum q_j'$ , then $(\sum p_j) \circ (\sum q_j) \sim (\sum p_j') \circ (\sum q_j')$ .

Lemma 3.17. Let $\Omega \subset \mathbb{R}^p$ be an open bounded set, and let $p(x,\xi), q(x,\xi)$ $\in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ . Assume $b(x,\xi) \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ satisfies $b(x,\xi) \sim q^t(x,-\xi)$ and $r(x,\xi) \in AS_{\varrho,\delta}^{2m,\omega}(\Omega)$ is equivalent to $\sum_{j} \sum_{|\alpha|=j} \frac{1}{\alpha!} \partial_{\xi}^{\alpha} D_{y}^{\alpha}(p(x,\xi)b(y,\xi))|_{y=x}$ . Then $r(x,\xi) \sim p(x,\xi) \circ q(x,\xi)$ .

Theorem 3.18. Let $p(x,\xi), q(x,\xi) \in AS_{a,\delta}^{m,\omega}(U)$ and let $\Omega$ be an open set which is relatively compact in U. Denote by $\tilde{P}$ and Q the corresponding pseudodifferential operators and assume that either P or Q is properly supported. Then $P \circ Q : \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ coincides, modulo an $(\omega)$ -smoothing operator, with the pseudodiferential operator associated to $(2\pi)^p(p(x,\xi)\circ q(x,\xi))$ .

Definition 1.6. Let $\Omega$ be an open set in $\mathbb{R}^p$ , $0 \le \delta < \varrho \le 1$ , $d := \varrho - \delta$ and assume that $\omega(t) = o(t^d)$ as $t \to \infty$ . An amplitude in $S_{o,\delta}^{m,\omega}(\Omega)$ is a function $a(x, y, \xi)$ in $C^{\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ such that for every compact set $Q \subset \Omega \times \Omega$ there are $R \geq 1$ and a sequence $C_n > 0$ , $n \in \mathbb{N}$ , with $$() |D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} a(x, y, \xi)| \le C_n e^{(\varrho - \delta)n\varphi^(|\alpha + \beta + \gamma|/n)} e^{m\omega(\xi)} (1 + |\xi|)^{|\alpha + \gamma|\delta - |\beta|\varrho}$$ for every $n \in \mathbb{N}$ , $(x, y) \in Q$ , and $\xi$ with $\log(|\xi|/R) \ge (n/|\beta|)\varphi^*(|\beta|/n)$ . For $|\beta| = 0$ the estimate holds for every $\xi \in \mathbb{R}^p$ . In the case $a(x, y, \xi) = p(x, \xi)$ , the function $p(x, \xi)$ is usually called a symbol. Since the function $\varphi$ is convex, we may replace $e^{(\varrho-\delta)n\varphi^(|\alpha+\beta+\gamma|/n)}$ in (\) by $e^{(\varrho-\delta)n\varphi^(|\alpha+\gamma|/n)+(\varrho-\delta)n\varphi^*(|\beta|/n)}$ (see Lemma 1.3). Some examples will be given in 2.11. Remark 1.7. We make some comments on the requirement $\omega(t) = o(t^d)$ in the definition of amplitude. - (1) If $\varrho=1$ and $\delta=0$ , this does not mean any restriction on $\omega$ . For other values of $\varrho$ and/or $\delta$ , this extra assumption means that $\mathcal{E}_{(\omega)}(\Omega)$ contains, as a continuously and densely embedded subspace, the Gevrey class $\Gamma^{\{1/d\}}(\Omega)$ and it ensures that $j!=O(e^{(\varrho-\delta)n\varphi^*(j/n)})$ as $j\to\infty$ for every $n\in\mathbb{N}$ . As shown in [22], if $\omega$ is a weight function for which an extension of the classical Borel theorem holds, then $\mathcal{E}_{(\omega)}(\Omega)$ contains $\Gamma^{\{s\}}(\Omega)$ for some s>1. - (2) If $\omega(t) = (\log(1+t))^s$ , s > 1, for t large enough then $\omega(t) = o(t^d)$ for every 0 < d < 1, whereas if $\omega(t) = t(\log t)^{-a}$ (a > 1), then $\mathcal{E}_{(\omega)}(\Omega)$ does not contain any Gevrey class.

Definition 2.3. The operator $A: \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ introduced in Theorem 2.2 is called a pseudodifferential operator of $(\omega)$ -class associated to the amplitude $a(x, y, \xi)$ . In the case $a(x, y, \xi) = p(x, \xi)$ the pseudodifferential operator A is denoted by P(x, D) and we have $$P(x,D)f = \int p(x,\xi)e^{ix\xi}\widehat{f}(\xi) d\xi$$ for every $f \in \mathcal{D}_{(\omega)}(\Omega)$ . It is clear that the expression above makes sense for $f \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ , and even for a wider class of functions.

Definition 2.10. Let $\Omega$ be an open set in $\mathbb{R}^p$ , $0 \le \delta < \varrho \le 1$ , $d := \varrho - \delta$ and assume $\omega(t) = o(t^d)$ as $t \to \infty$ . An amplitude in $AS_{\rho,\delta}^{m,\omega}(\Omega)$ is a function $a(x, y, \xi)$ in $C^{\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ such that for every compact set $Q \subset \Omega \times \Omega$ there are $R, B \geq 1$ and a sequence $C_n > 0$ , $n \in \mathbb{N}$ , with $$|D_x^\alpha D_y^\gamma D_\xi^\beta a(x,y,\xi)| \leq C_n B^\beta \beta! e^{(\varrho-\delta)n\varphi^*(|\alpha+\gamma|/n)} e^{m\omega(\xi)} (1+|\xi|)^{|\alpha+\gamma|\delta-|\beta|\varrho}$$ for every $n \in \mathbb{N}$ , $(x, y) \in Q$ , and $\xi$ with $\log(|\xi|/R) \ge (n/|\beta|)\varphi^*(|\beta|/n)$ . An amplitude in $AS_{\varrho,\delta}^{m,\omega}(\Omega)$ is said to be of finite order if it satisfies the inequalities above with $(1+|\xi|)^m$ instead of $e^{m\omega(\xi)}$ . It follows from the Stirling formula and the condition $\omega(t) = o(t^d)$ , $d := \varrho - \delta$ , that $AS_{\varrho,\delta}^{m,\omega}(\Omega) \subset S_{\varrho,\delta}^{m,\omega}(\Omega)$ . We observe that given two weight functions $\sigma = O(\omega)$ , each amplitude of finite order with respect to $\omega$ is also an amplitude with respect to $\sigma$ , and thus the corresponding pseudodifferential operator admits a continuous linear extension to $\mathcal{D}_{(\sigma)}(\Omega)$ which takes values in $\mathcal{E}_{(\sigma)}(\Omega)$ . Now we give examples of amplitudes and of pseudodifferential operators. Example 2.11. (1) $\omega(t) := \log(1+t)$ . This is a limit case that we are not considering, since $\omega$ does not have property $(\gamma)$ in 1.1. Then $\mathcal{E}_{(\omega)}(\Omega) = C^{\infty}(\Omega)$ and $\varphi^*(t) = \infty$ for every t > 1. It follows that $a(x, y, \xi)$ is an amplitude of $(\omega)$ -class in $S_{\varrho,\delta}^{m,\omega}(\Omega)$ according to Definition 1.6 if, and only if, for every compact set $K \subset \Omega$ and every $\alpha, \beta, \gamma \in \mathbb{N}_0^p$ there is a constant C > 0 such that $$|D_x^{\alpha} D_y^{\gamma} D_{\varepsilon}^{\beta} a(x, y, \xi)| \le C(1 + |\xi|)^{m + \delta(|\alpha + \gamma|) - \varrho|\beta|}$$ for $x, y \in K$ . This means that $a(x, y, \xi)$ is a symbol in the sense of Grigis and Sjöstrand [13]. (2) $\omega(t) := t^d$ , 0 < d < 1. Then $\mathcal{E}_{(\omega)}(\Omega)$ is a Gevrey class of Beurling type. In this case $n\varphi^*(t/n) = (t/d)\log(t/nd) - t/d$ and it follows from the Stirling formula that for every $n \in \mathbb{N}$ there are positive constants $A_n$ and $B_n$ such that for every $\alpha \in \mathbb{N}_0^p$ , $$A_n(\alpha!)^{1/d} \left(\frac{1}{nd}\right)^{|\alpha|/d} \le e^{n\varphi^*(|\alpha|/n)} \le B_n(\alpha!)^{1/d} \left(\frac{2p}{nd}\right)^{|\alpha|/d}.$$ Thus, $a(x, y, \xi)$ is an amplitude in $AS_{1,0}^{m,\omega}(\Omega)$ if, and only if, for every compact set $K \subset \Omega$ there are $R, B \geq 1$ such that for every $\lambda > 0$ there is C > 0 satisfying $$|D_x^\alpha D_y^\gamma D_\xi^\beta a(x,y,\xi)| \leq C B^\beta \beta! (\alpha!)^{1/d} (\gamma!)^{1/d} \lambda^{|\alpha+\gamma|} e^{m\omega(\xi)} (1+|\xi|)^{-|\beta|}$$ for every $(x,y) \in K$ and $|\xi| \geq R(\beta\lambda)^{1/d}$ . This should be compared with the definition of amplitude of Gevrey class (of Roumieu type) which can be found for instance in Rodino [23]. (3) Linear partial differential operators with coefficients in $\mathcal{E}_{(\omega)}(\Omega)$ are examples of pseudodifferential operators defined by symbols of finite order. (4) Let $K(x,y) \in \mathcal{E}_{(\omega)}(\Omega \times \Omega)$ . The integral operator with kernel K, $(A\varphi)(x) = \int K(x,y)\varphi(y)\,dy$ , is a pseudodifferential operator. In fact, given any $\chi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ with $\int \chi = 1$ , $a(x,y,\xi) := K(x,y)e^{-i(x-y)\xi}\chi(\xi)$ is an amplitude in $AS_{1,0}^{0,\omega}(\Omega)$ . This can be easily deduced from the compactness of the support of $\chi$ and the fact that $K \in \mathcal{E}_{(\omega)}(\Omega \times \Omega)$ . The operators $T: \mathcal{D}_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ admitting a continuous linear extension $\widetilde{T}: \mathcal{E}'_{(\omega)}(\Omega) \to \mathcal{E}_{(\omega)}(\Omega)$ are called $(\omega)$ -smoothing. These are exactly the integral operators defined by kernels in $\mathcal{E}_{(\omega)}(\Omega \times \Omega)$ . (5) Every ultradifferential operator in the sense of Komatsu [17, p. 42] defines a pseudodifferential operator. We recall that an ultradifferential operator of $(\omega)$ -class in the sense of Komatsu is an operator $G(x,D) := \sum_{\alpha \in \mathbb{N}_0^p} a_{\alpha}(x) D^{\alpha}$ with $a_{\alpha} \in \mathcal{E}_{(\omega)}(\Omega)$ satisfying the following condition: there exists $m \in \mathbb{N}$ such that for every compact set $K \subset \Omega$ and every $n \in \mathbb{N}$ there is $C_n > 0$ with $$\sup_{x \in K} |D^{\beta} a_{\alpha}(x)| \le C_n e^{n\varphi^(|\beta|/n)} e^{-m\varphi^(|\alpha|/m)}$$ for every $\alpha, \beta \in \mathbb{N}_0^p$ . It is easy to prove that $p(x,\xi) := (2\pi)^{-p} \sum_{\alpha \in \mathbb{N}_0^p} a_{\alpha}(x) \xi^{\alpha}$ is a symbol in $AS_{1.0}^{k,\omega}(\Omega)$ for some $k \geq m$ . Moreover, for every $\varphi \in \mathcal{D}_{(\omega)}(\Omega)$ we have $$G(x,D)\varphi = \frac{1}{(2\pi)^p} \sum_{\alpha \in \mathbb{N}_0^p} a_{\alpha}(x) \int e^{ix\xi} \xi^{\alpha} \widehat{\varphi}(\xi) d\xi = P(x,D)\varphi.$$ Examples of ultradifferential operators in the sense of Komatsu are the partial differential operators with coefficients in the class $\mathcal{E}_{(\omega)}(\Omega)$ as well as the ultradifferential operators with constant coefficients ([7]). In this case $G(z) := \sum_{\alpha \in \mathbb{N}_0^p} a_{\alpha} z^{\alpha}$ is an entire function satisfying $\log |G(z)| = O(\omega(z))$ . Therefore an application of the Paley-Wiener theorem ([8, 7.3]) gives the existence of an element $\mu \in \mathcal{E}'_{(\omega)}(\mathbb{R}^p)$ with support $\{0\}$ such that $G(D)\varphi = \mu * \varphi$ for every $\varphi \in \mathcal{D}_{(\omega)}(\mathbb{R}^p)$ . - (6) If $f \in \mathcal{E}_{(\omega)}(\Omega)$ then $a(x, y, \xi) := (2\pi)^{-p} f(x) \in AS_{1,0}^{0,\omega}(\Omega)$ . Thus the operator $\varphi \mapsto f\varphi$ is a pseudodifferential operator. - (7) Let $f \in \mathcal{D}_{(\omega)}(\mathbb{R})$ be a test function with supp f = [-1, 1] and $f(0) \neq 0$ . We put $\varphi := f\chi_{[0,\infty[}$ , which is an ultradistribution with compact support and $\operatorname{sing}_{(\omega)} \operatorname{supp} \varphi = \{0\}$ . Then $\widehat{\varphi}(\xi)$ is a symbol and the associated pseudodifferential operator is the operator $\psi \mapsto 2\pi(\varphi * \psi)$ . (See the comment after Theorem 2.18.) We first observe that for every $N \in \mathbb{N}$ we have $$\widehat{\varphi}(\xi) = \sum_{k=0}^{N} \frac{f^{(k)}(0)}{(i\xi)^{k+1}} + \frac{1}{(i\xi)^{N+1}} \int_{0}^{1} f^{(N+1)}(t) e^{-it\xi} dt.$$ This follows after integrating by parts N times in $\widehat{\varphi}(\xi) = \int_0^1 f(t)e^{-it\xi} dt$ , using the fact that f and all its derivatives vanish at t = 1. Hence $$\begin{split} \widehat{\varphi}^{(N)}(\xi) &= \sum_{k=0}^{N} (-1)^{N} \frac{f^{(k)}(0)}{(i\xi)^{k+1}} N! \binom{N+k}{k} \frac{1}{\xi^{N}} \\ &+ \frac{1}{\xi^{N} i^{N+1}} \sum_{k=0}^{N} \binom{N}{k} \frac{(N+k)!}{N!} (-1)^{k} \frac{1}{\xi^{k+1}} \int\limits_{0}^{1} (-it)^{N-k} f^{(N+1)}(t) e^{-it\xi} \, dt. \end{split}$$ For every $n \in \mathbb{N}$ we put $C_n := |f|_n$ and assume that $\log |\xi| \ge (n/N)\varphi^*(N/n)$ . Then for every $k \le N$ we have $|f^{(k)}(0)|/|\xi|^k \le C_n$ and $$\left| \sum_{k=0}^{N} (-1)^{N} \frac{f^{(k)}(0)}{(i\xi)^{k+1}} N! \binom{N+k}{k} \frac{1}{\xi^{N}} \right| \le \frac{C_{n} 4^{N} N!}{|\xi|^{N+1}}.$$ Moreover, since $|\xi|^{k+1} \ge e^{n\varphi^*(k/n)} \ge \varepsilon k!$ for some $\varepsilon > 0$ , there is $D_n > 0$ with $$\left| \sum_{k=0}^{N} {N \choose k} \frac{(N+k)!}{N!} (-1)^k \frac{1}{\xi^{k+1}} \int_{0}^{1} (-it)^{N-k} f^{(N+1)}(t) e^{-it\xi} dt \right|$$ $$\leq D_n 8^N e^{n\varphi^*((N+1)/n)}.$$ We conclude that $$|\widehat{\varphi}^{(N)}(\xi)| \le A_n B^N |\xi|^{-N} e^{n\varphi^*(N/n)}$$ for some $A_n, B > 0$ and for every $\xi$ with $\log |\xi| \ge (n/N)\varphi^*(N/n)$ . Our next example is less obvious. It shows that the class of operators under consideration contains not only ultradifferential operators but also parametrices under some extra assumptions.

Definition 3.1. We denote by $FAS_{\varrho,\delta}^{m,\omega}(\Omega)$ the set of all formal sums $$\sum_{j\in\mathbb{N}_0} a_j(x,y,\xi)$$ such that $a_j(x, y, \xi) \in C^{\infty}(\Omega \times \Omega \times \mathbb{R}^p)$ and for every compact set $Q \subset \Omega \times \Omega$ there are $R, B \geq 1$ and a sequence $C_n > 0$ , $n \in \mathbb{N}$ , with $$|D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} a_j(x, y, \xi)|$$ $$\leq C_n B^{|\beta|} \beta! e^{(\varrho - \delta)n\varphi^*(|\alpha + \gamma| + j/n)} e^{m\omega(\xi)} (1 + |\xi|)^{|\alpha + \gamma|\delta - |\beta|\varrho - (\varrho - \delta)j}$$ for every $j \in \mathbb{N}_0$ , $(x,y) \in Q$ , and $\xi$ with $\log(|\xi|/R) \ge \frac{n}{|\beta|+j} \varphi^* (\frac{|\beta|+j}{n})$ . Let $a \in AS_{\varrho,\delta}^{m,\omega}(\Omega)$ and put $a_0 := a$ , $a_j := 0$ for $j \neq 0$ . Then we can identify a with the formal sum $\sum a_j$ . EXAMPLE 3.2. Let $a \in AS^{m,\omega}_{\varrho,\delta}(\Omega)$ . Then the series $\sum_{j=0}^{\infty} p_j(x,\xi)$ , where $p_j(x,\xi) := \sum_{|\alpha|=j} (1/\alpha!) D^{\alpha}_{\xi} \partial^{\alpha}_{y} a(x,y,\xi)_{|y=x}$ , is a formal sum in $FAS^{m,\omega}_{\varrho,\delta}(\Omega)$ .

Definition 3.3. Two formal sums $\sum a_j$ and $\sum b_j$ in $FAS_{\varrho,\delta}^{m,\omega}(\Omega)$ are said to be equivalent if for every compact set $Q \subset \Omega \times \Omega$ there are $R, B \geq 1$ and two sequences $C_n > 0$ and $N_n$ $(n \in \mathbb{N})$ with $$\begin{split} \left| D_x^{\alpha} D_y^{\gamma} D_{\xi}^{\beta} \sum_{j < N} (a_j - b_j) \right| \\ & \leq C_n B^{|\beta|} \beta! e^{(\varrho - \delta)n\varphi^(|\alpha + \gamma| + N/n)} e^{m\omega(\xi)} (1 + |\xi|)^{|\alpha + \gamma|\delta - |\beta|\varrho - (\varrho - \delta)N} \\ \text{for every } (x, y) \in Q, \ N \geq N_n, \ \text{and} \ \xi \ \text{with} \ \log(|\xi|/R) \geq \frac{n}{|\beta| + N} \varphi^ \left(\frac{|\beta| + N}{n}\right). \end{split}$$ Remark 3.4. If $a(x,y,\xi)=0$ for every $x,y\in\Omega$ and $|\xi|\geq M$ then $a\sim0$ . In particular, every $(\omega)$ -smoothing operator is associated to an amplitude equivalent to zero.

Definition 3.15. (1) For $\sum p_j \in FAS_{\varrho,\delta}^{m,\omega}(\Omega)$ we define $(\sum p_j)^t$ as the formal sum $\sum_{j} q_{j}$ , where $q_{j}$ is as before. (2) For $\sum p_{j} \in FAS_{\varrho,\delta}^{m_{1},\omega}(\Omega)$ and $\sum q_{j} \in FAS_{\varrho,\delta}^{m_{2}}$ we define $(\sum p_{j}) \circ$ $(\sum q_i) = \sum r_i$ , where $r_i$ is as above. The following two results are straightforward, therefore we omit their proof ([25]).