*Article* **Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and** *C***<sup>∞</sup> Semigroups**

**Bienvenido Barraza Martínez †, Jonathan González Ospino †, Rogelio Grau Acuña † and Jairo Hernández Monzón \*,†**

> Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla 081007, Colombia; bbarraza@uninorte.edu.co (B.B.M.); gjonathan@uninorte.edu.co (J.G.O.); graur@uninorte.edu.co (R.G.A.)

**\*** Correspondence: jahernan@uninorte.edu.co

† These authors contributed equally to this work.

**Abstract:** We consider Fourier multiplier systems on R*<sup>n</sup>* with components belonging to the standard Hörmander class *S<sup>m</sup>* 1,0(R*n*), but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector <sup>Λ</sup> <sup>⊂</sup> <sup>C</sup> (introduced by Denk, Saal, and Seiler) we show the generation of both *C*<sup>∞</sup> semigroups and analytic semigroups (in a particular case) on the Sobolev spaces *W<sup>k</sup> <sup>p</sup>*(R*n*, C*q*) with *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>0, 1 <sup>≤</sup> *<sup>p</sup>* < <sup>∞</sup> and *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces.

**Keywords:** *C*∞-semigroups; analytic semigroups; Fourier multipliers; Λ-ellipticity

**MSC:** 35J48; 35S05; 35S30; 47D03; 47D06

#### **1. Introduction**

Elliptic systems of partial differential equations were introduced in 1955 by A. Douglis and L. Nirenberg in [1]. Then, in 1973, R. Kramer formulated and solved in [2] several Cauchy problems for systems of partial differential equations which are elliptic in the sense given by Douglis and Nirenberg in [1]. In the same year, A. Koževnikov, in his study in [3] about spectral asymptotics for elliptic pseudodifferential systems with the structure of Douglis–Nirenberg, introduced an algebraic condition on the symbol (called the parameter– ellipticity condition) which permitted him to prove the similarity of the system satisfying this condition to an almost diagonal system up to a symbol of order −∞, but he did not consider questions of equation solvability for those operators. In 2009, R. Denk, J. Saal and J. Seiler considered in [4] pseudodifferential Douglis–Nirenberg systems on R*<sup>n</sup>* with components belonging to the standard Hörmander class *S*∗ 1,*δ*(R*<sup>n</sup>* <sup>×</sup> <sup>R</sup>*n*), 0 <sup>≤</sup> *<sup>δ</sup>* < 1. They introduced the formulation of parameter–ellipticity with respect to a subsector <sup>Λ</sup> <sup>⊂</sup> <sup>C</sup>, which is motivated by a notion of parameter–ellipticity introduced by Denk, Menniken, and Volevich in [5] and connected with the so-called Newton polygon associated with the system. They showed that their formulation of ellipticity is equivalent to the given by Koževnikov in [3] and that this condition implies the existence of a bounded *H*∞-calculus for their pseudodifferential systems in suitable scales of Sobolev spaces with 1 < *<sup>p</sup>* < <sup>∞</sup>, hence of *Lp*-maximal regularity. Furthermore, it is known that the maximal regularity implies the generation of an analytic semigroup, however the reverse implication is false.

In this paper, we will consider certain Fourier multiplier systems on R*n*, similar but not necessarily with the exact structure of a Douglis–Nirenberg system, with components belonging to the standard Hörmander class *S<sup>m</sup>* 1,0(R*n*), but with limited regularity (see Definition 2), and using the notion of parameter–ellipticity with respect to a subsector <sup>Λ</sup> <sup>⊂</sup> <sup>C</sup> given in [4], we will establish (in Theorem 1) the generation of *<sup>C</sup>*<sup>∞</sup> semigroups and

**Citation:** Barraza Martínez, B.; González Ospino, J.; Grau Acuña, R.; Hernández Monzón, J. Parameter– Elliptic Fourier Multipliers Systems and Generation of Analytic and *C*<sup>∞</sup> Semigroups. *Mathematics* **2022**, *10*, 751. https://doi.org/10.3390/ math10050751

Academic Editor: Patricia J. Y. Wong

Received: 25 January 2022 Accepted: 3 February 2022 Published: 26 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

analytic semigroups (in a particular case) on the Sobolev spaces *W<sup>k</sup> <sup>p</sup>*(R*n*, <sup>C</sup>*q*) with *<sup>k</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> and <sup>1</sup> <sup>≤</sup> *<sup>p</sup>* < <sup>∞</sup> giving a direct proof. For this direct proof of our main result we use the approach based on oscillatory integrals and kernel estimates for them (as in [6]), taking advantage of the fact that the associated symbols to the pseudodifferential operators are matrices valued and the entries of these matrices are symbols of order greater than 1/2 and are independent of the spatial variable. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution (see Theorem 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in suitable Sobolev spaces (see Section 5). Other applications of the theory of semigroups and its generalizations address the control and stablility theory for mechanical systems or the controllability of fractional evolution equations or inclusions (see [7–14] and the conclusions in Section 6).

The paper is organized as follows: In Section 2 we present the definition of our system of Fourier multipliers, which are defined in terms of suitable oscillatory integrals. Following [4], we give in Section 3 the notion of Λ ellipticity for this system of Fourier multipliers, with respect to a sector Λ of the complex plane. In order to allow that the correspondent estimate in the definition of Λ ellipticity for the characteristic polynomial of the matrix symbol of our system of Fourier multipliers hold for all values of the symbol variable *ξ* in R*n*, we consider a perturbation of the system by a constant, following again the ideas given in [4] (see Remark 2). Section 4 is the core of the paper. There we obtain the main result of the paper about generation, under suitable hypothesis, of *C*<sup>∞</sup> semigroups and analytic semigroups for a Sobolev space realization of the perturbed operator associated to a Λ-elliptic system (Theorem 1). We also present in that section, existence and uniqueness results for non-autonomous Cauchy problems based on the obtained results about generation of semigroups (Theorem 2 and corollary 2). In Section 5, as examples and as already mentioned above, the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation are considered. Finally, in the conclusions in Section 6, we summarize the results obtained in the paper and point out some possible future scope of this work.

#### **2. Fourier Multiplier Systems**

In the following, for *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>ρ</sup><sup>n</sup>* denotes the smallest even integer greater than *<sup>n</sup>*, *E* represents an arbitary Banach space, L(*E*) the space of linear and continuous maps of *<sup>E</sup>* into *<sup>E</sup>*, <sup>S</sup>(R*n*, *<sup>E</sup>*) the Schwartz space of rapidly decreasing functions and *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, *<sup>E</sup>*) the space of all functions *<sup>u</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> *<sup>E</sup>* such that *<sup>∂</sup>α<sup>u</sup>* is bounded and continuous on <sup>R</sup>*<sup>n</sup>* for all *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> . *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, *<sup>E</sup>*), for *<sup>k</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> and 1 <sup>≤</sup> *<sup>p</sup>* <sup>≤</sup> <sup>∞</sup>, are the usual Sobolev spaces equipped with their standard norm and it is well konwn that <sup>S</sup>(R*n*, *<sup>E</sup>*) <sup>⊂</sup> *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, *<sup>E</sup>*) <sup>∩</sup> *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, *<sup>E</sup>*) and that <sup>S</sup>(R*n*, *<sup>E</sup>*) is dense in *<sup>W</sup><sup>k</sup> <sup>p</sup>*(R*n*, *<sup>E</sup>*) if 1 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>. Also we will use the following notations throughout the paper: *Dξ<sup>j</sup>* := −*i∂ξ<sup>j</sup>* , *ξ* := (1 + |*ξ*| <sup>2</sup>)1/2, *<sup>ξ</sup>*, *<sup>t</sup>* := (<sup>1</sup> <sup>+</sup> <sup>|</sup>*ξ*<sup>|</sup> <sup>2</sup> <sup>+</sup> <sup>|</sup>*t*<sup>|</sup> <sup>2</sup>)1/2 and |*ξ*, *t*| := (|*ξ*| <sup>2</sup> <sup>+</sup> <sup>|</sup>*t*<sup>|</sup> <sup>2</sup>)1/2, for *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>t</sup>* <sup>∈</sup> <sup>R</sup>.

For the following definition, see Equation (1) in [6].

**Definition 1.** *Let m* <sup>∈</sup> <sup>R</sup> *and <sup>ρ</sup>* <sup>∈</sup> <sup>N</sup>0*.*

(a) *The symbol class <sup>S</sup>m*,*ρ*(R*n*,L(*E*)) :<sup>=</sup> *<sup>S</sup>m*,*<sup>ρ</sup>* 1,0 (R*n*,L(*E*)) *consists of all functions <sup>a</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>L</sup>(*E*) *of class <sup>C</sup><sup>ρ</sup> with the property that for each <sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> *with* |*α*| ≤ *ρ, there exists a positive constant Cα such that*

$$\left\| \partial\_{\xi}^{\alpha} a(\xi) \right\|\_{\mathcal{L}(E)} \leq \mathsf{C}\_{\mathfrak{a}} \langle \xi \rangle^{m - |\alpha|} \text{ for all } \xi \in \mathbb{R}^{n}.$$

(b) *In Sm*,*ρ*(R*n*,L(*E*)) *we define the norm*

$$||a||\_{S^{m,\emptyset}} := \max\_{|\boldsymbol{\alpha}| \le \boldsymbol{\rho}} \sup\_{\substack{\boldsymbol{\xi} \in \mathbb{R}^n \\ \boldsymbol{\rho}}} \langle \boldsymbol{\xi} \rangle^{|\boldsymbol{\alpha}|-m} \left|| \partial\_{\boldsymbol{\xi}}^{\boldsymbol{\alpha}} a(\boldsymbol{\xi}) \right||\_{\mathcal{L}(E)}.$$

(c) *For a* <sup>∈</sup> *<sup>S</sup>m*,*ρ*(R*n*,L(*E*)) *with <sup>ρ</sup>* <sup>≥</sup> *<sup>ρ</sup>n, the Fourier multiplier operator a*(*D*) *is defined by*

$$[a(D)u](\mathbf{x}) := \text{Os} - \int \int e^{i\overline{\xi}\cdot\eta} a(\overline{\xi}) \mu(\mathbf{x} - \eta) \frac{d(\overline{\xi}, \eta)}{(2\pi)^{\overline{n}}} \tag{1}$$

*for all x* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and u* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> *<sup>b</sup>* (R*n*, *<sup>E</sup>*)*, where the symbol* Os<sup>−</sup> *stands for oscillatory integrals.*

In the case that *<sup>E</sup>* <sup>=</sup> <sup>C</sup>*q*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, we identify <sup>L</sup>(C*q*) with <sup>C</sup>*q*×*q*, <sup>C</sup>1×<sup>1</sup> with <sup>C</sup> and we write *Sm*,*ρ*(R*n*) instead of *Sm*,*ρ*(R*n*, C).

**Remark 1.** (a) *The definition and some properties of the oscillatory integrals can be found in [15] for the scalar case and in [16] (Appendix A) for the vector valued case.*

(b) *For ρ* ≥ *ρn, Lemma A.4 and Remark A.5 in [16] imply that the oscillatory integral in* (1) *exists. Moreover, due to Lemma A.6 in [16] we have that a*(*D*) ∈ L(*C*<sup>∞</sup> *<sup>b</sup>* (R*n*, *<sup>E</sup>*))*.*

(c) *Fourier multipliers with limited regularity symbols were also studied in [17,18].*

**Definition 2** (Compare with [4] (Definition 2.3))**.** *The Fourier multipliers system we will consider in this paper is a q* × *q-matrix of Fourier multipliers*

$$A(D) = \left( a\_{ij}(D) \right)\_{1 \le i,j \le q}$$

*such that*

$$a\_{ij} \in S^{r\_{ij}\cdot\rho}(\mathbb{R}^n),$$

*where rij* <sup>∈</sup> <sup>R</sup>*, ri* :<sup>=</sup> *rii* <sup>≥</sup> <sup>0</sup>*, for all i*, *<sup>j</sup>* <sup>=</sup> 1, ..., *q, and <sup>ρ</sup>* <sup>∈</sup> <sup>N</sup> *is such that <sup>ρ</sup>* <sup>≥</sup> *<sup>ρ</sup>n.*

#### **3. Λ-Elliptic Fourier Multipliers Systems**

From now on we fix *<sup>θ</sup>*, with 0 < *<sup>θ</sup>* < *<sup>π</sup>*, and let <sup>Λ</sup>(*θ*) denote the closed subsector of the complex plane C, given by

$$\Lambda := \Lambda(\theta) := \left\{ n^{i\gamma} : r \ge 0, \theta \le \gamma \le 2\pi - \theta \right\}.$$

For the following definition we refer to [4] (Definition 3.1).

**Definition 3.** *Let A*(*D*) *be a Fourier multipliers system (as in Definition 2). We say that A*(*D*) *is* <sup>Λ</sup>-elliptic *(or* <sup>Λ</sup>(*θ*)-elliptic *to highlight the angle) , if there exist constants <sup>C</sup>* > <sup>0</sup> *and <sup>R</sup>* <sup>≥</sup> <sup>0</sup> *such that*

$$|p(\mathfrak{F};\lambda)| \ge \mathfrak{C} \left( \langle \mathfrak{F} \rangle^{r\_1} + |\lambda| \right) \cdots \left( \langle \mathfrak{F} \rangle^{r\_q} + |\lambda| \right).$$

*for all* (*ξ*, *<sup>λ</sup>*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup> *with* <sup>|</sup>*ξ*<sup>|</sup> <sup>≥</sup> *R, where p*(*ξ*; *<sup>λ</sup>*) :<sup>=</sup> det(*A*(*ξ*) <sup>−</sup> *<sup>λ</sup>*)*.*

**Remark 2.** *Let A*(*D*) *be a* Λ*-elliptic Fourier multipliers system. Due to Lemma 3.4 in [4], there exists a constant α*<sup>0</sup> ≥ 0 *such that*

$$|\det(A\_{\mathfrak{a}\_0}(\xi) - \lambda)| \ge \mathcal{C}(\langle \xi \rangle^{r\_1} + |\lambda|) \cdots (\langle \xi \rangle^{r\_q} + |\lambda|) \quad \forall \xi \in \mathbb{R}^n \ and \ \lambda \in \Lambda\_\tau.$$

*where Aα*<sup>0</sup> (*ξ*) := *A*(*ξ*) + *α*0*, i.e., Aα*<sup>0</sup> (*D*) *is* Λ*-elliptic with R* = 0*.*

**Lemma 1** ([4], Lemma 3.5)**.** *Let A*(*D*) *be* Λ*-elliptic and*

$$\left(\mathcal{g}\_{i\bar{j}}(\xi;\lambda)\right)\_{1\le i,j\le q} := (\mathcal{A}\_{\mathbb{A}\_0}(\xi) - \lambda)^{-1}.$$

*Then,*

$$\begin{aligned} \left| \partial\_{\boldsymbol{\xi}}^{\boldsymbol{a}} \mathcal{g}\_{\boldsymbol{i}\boldsymbol{j}} (\boldsymbol{\xi}; \boldsymbol{\lambda}) \right| &\leq \mathsf{C}\_{\boldsymbol{a}} \left( \langle \boldsymbol{\xi} \rangle^{r\_{\boldsymbol{i}}} + |\boldsymbol{\lambda}| \right)^{-1} \left( \langle \boldsymbol{\xi} \rangle^{r\_{\boldsymbol{j}}} + |\boldsymbol{\lambda}| \right)^{-1} \langle \boldsymbol{\xi} \rangle^{r\_{\boldsymbol{i}\boldsymbol{j}} - |\boldsymbol{a}|}, \ \ (\boldsymbol{i} \neq \boldsymbol{j}),\\ \left| \partial\_{\boldsymbol{\xi}}^{\boldsymbol{a}} \mathcal{g}\_{\boldsymbol{i}\boldsymbol{i}} (\boldsymbol{\xi}; \boldsymbol{\lambda}) \right| &\leq \mathsf{C}\_{\boldsymbol{a}} \left( \langle \boldsymbol{\xi} \rangle^{r\_{\boldsymbol{i}}} + |\boldsymbol{\lambda}| \right)^{-1} \langle \boldsymbol{\xi} \rangle^{-|\boldsymbol{a}|} \end{aligned}$$

*for all <sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> *, being the estimates uniform in* (*ξ*, *<sup>λ</sup>*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>*.*

Following the ideas of the proof of this lemma in [4], we note that the condition *r*<sup>1</sup> ≥···≥ *rq* ≥ 0 given there, is not necessary for the estimates above. However, we get another crucial estimate under the following additional assumption about the orders of the symbols in the system:

$$\sum\_{j=1}^{k} r\_{i\_j \pi(i\_j)} = \sum\_{j=1}^{k} r\_{i\_j} \tag{2}$$

for all subsets of indices {*i*1, ... , *ik*}⊂{1, ... , *q*} and all bijections *π* : {*i*1, ... , *ik*} → {*i*1,..., *ik*}.

The crucial estimate we mentioned above is given in the following lemma.

**Lemma 2.** *Let A*(*D*) *be* Λ*-elliptic,*

$$\left(\mathcal{g}\_{ij}(\xi;\lambda)\right)\_{1\le i,j\le q} := \left(\mathcal{A}\_{\mathbb{A}\_0}(\xi) - \lambda\right)^{-1},$$

*and suppose that the assumption* (2) *holds. Then, for all <sup>i</sup>* <sup>=</sup> 1, ... , *q, <sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> *with* <sup>0</sup> <sup>&</sup>lt; <sup>|</sup>*α*<sup>|</sup> <sup>≤</sup> *<sup>ρ</sup>, and* (*ξ*, *<sup>λ</sup>*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>*, we have*

$$\left|\partial\_{\xi}^{a}\mathcal{g}\_{ii}(\xi;\lambda)\right| \leq \mathbb{C}\_{a} \sum\_{j=1}^{q} \left( \langle \xi \rangle^{r\_{i}} + |\lambda| \right)^{-1} \left( \langle \xi \rangle^{r\_{j}} + |\lambda| \right)^{-1} \langle \xi \rangle^{r\_{j} - |a|} \tag{3}$$

*for some constant Cα.*

**Proof.** Let *i* ∈ {1, . . . , *q*} be fixed. It should first be noted that

$$\mathcal{g}\_{ii}(\vec{\xi};\lambda) = \frac{1}{\det(A\_{a\_0}(\vec{\xi}) - \lambda)} \text{Cof}\_{(i,i)}\left(A\_{a\_0}(\vec{\xi}) - \lambda\right),$$

where Cof(*i*,*i*) *Aα*<sup>0</sup> (*ξ*) − *λ* is the *cofactor* (*i*, *i*) of *Aα*<sup>0</sup> (*ξ*) − *λ*, that is, the determinant of the matrix obtained by removing the *i*-th row and *i*-th column of this matrix. With the convention *<sup>m</sup>* ∏ *l*=*k* (···)*<sup>l</sup>* :<sup>=</sup> 1 if *<sup>k</sup>* <sup>&</sup>gt; *<sup>m</sup>*, which we will use from now on in this proof, we have that Cof(*i*,*i*) *Aα*<sup>0</sup> (*ξ*) − *λ* is a linear combination of terms

$$\left(\prod\_{l=1}^{k} (a\_{i\_l i\_l} + a\_0 - \lambda) \right) \prod\_{l=k+1}^{q-1} a\_{i\_l \pi(i\_l) \nu}$$

where " *i*1,..., *iq*−<sup>1</sup> # <sup>=</sup> {1, ... , *<sup>q</sup>*}-{*i*}, 0 ≤ *k* ≤ *q* −1, and *π* : {*i*1, ... , *iq*−1}→{*i*1, ... , *iq*−1} is a bijection which have {*i*1,..., *ik*} as its set of fixed points. Therefore " *ik*+1,..., *iq*−<sup>1</sup> # <sup>=</sup> " *π*(*ik*<sup>+</sup>1),..., *π*(*iq*−1) # and, in virtue of assumption (2), it holds

$$r\_{i\_{k+1}\pi(i\_{k+1})} + \dots + r\_{i\_{q-1}\pi(i\_{q-1})} = r\_{i\_{k+1}} + \dots + r\_{i\_{q-1}}.\tag{4}$$

If *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with 0 <sup>&</sup>lt; <sup>|</sup>*α*<sup>|</sup> <sup>≤</sup> *<sup>ρ</sup>*, the Leibniz' formula implies that *<sup>∂</sup><sup>α</sup> <sup>ξ</sup> gii* is a linear combination of terms

$$\partial^{\beta} \left( \frac{1}{p\_0} \right) \underbrace{\left( \prod\_{l=1}^k \partial^{\gamma\_l} (a\_{i\_l i\_l} + a\_0 - \lambda) \right)}\_{=:H} \prod\_{l=k+1}^{q-1} \partial^{\gamma\_l} a\_{i\_l \pi(i\_l) \prime} \tag{5}$$

where *<sup>β</sup>*, *<sup>γ</sup>*1, ... , *<sup>γ</sup>q*−<sup>1</sup> <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> with *β* + *γ*<sup>1</sup> + ··· + *γq*−<sup>1</sup> = *α*, *k* ∈ {0, 1, ... , *q*}, and *p*<sup>0</sup> = *p*0(*ξ*, *λ*) := det(*Aα*<sup>0</sup> (*ξ*) − *λ*). Note that the term *aii* + *α*<sup>0</sup> − *λ* is not in *H* (see (5)), and also we can estimate *∂γl ailil* + *α*<sup>0</sup> − *λ* from above by *ξ ri <sup>l</sup>* + |*λ*| if *γ<sup>l</sup>* = 0 and by *ξ ri l* −|*γ<sup>l</sup>* | if *γ<sup>l</sup>* = 0.

If *β* = 0, then *γ<sup>j</sup>* = 0 for some *j* = 1, ... , *q* − 1. Therefore, the term related to *aijij* which appears in *<sup>H</sup>* is equal to *<sup>∂</sup>γ<sup>j</sup> aijij* , and then, due to the Λ-ellipticity condition (together with Remark 2) and (4), the expression (5) can be estimated from above by *ξ ri* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> −1 *ξ ri <sup>j</sup>* + |*λ*| −1 *ξ ri j* −|*α*| .

In order to consider the case *<sup>β</sup>* <sup>=</sup> 0, we will prove first that for each *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> , 0 <sup>&</sup>lt; <sup>|</sup>*α*| ≤ *<sup>ρ</sup>*, there exists *<sup>C</sup>* > 0 such that

$$|\partial^{\mathfrak{a}}p\_0(\xi,\lambda)| \le \mathbb{C} \left( \sum\_{j=1}^q \left( \prod\_{\substack{i=1 \\ i \neq j}}^q (\langle \xi \rangle^{r\_i} + |\lambda|) \right) \langle \xi \rangle^{r\_j} \right) \langle \xi \rangle^{-|a|} \tag{6}$$

for all *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Let <sup>Z</sup> :<sup>=</sup> {1, ... , *<sup>q</sup>*}. Note that *<sup>p</sup>*<sup>0</sup> is a linear combination of terms of the form

$$\left(\prod\_{l=1}^{k} (a\_{i\_l i\_l} + a\_0 - \lambda)\right) \prod\_{l=k+1}^{q} a\_{i\_l \pi(i\_l)} \qquad (k = 0, \ldots, q)\_{\prime}$$

where *π* : Z→Z is a bijection with fixed points *i*1, ... , *ik*, and therefore {*ik*+1, ... , *iq*} = {*π*(*ik*<sup>+</sup>1),..., *π*(*iq*)} which, again due to the assumption (2), yields

$$r\_{i\_{k+1}\pi(i\_{k+1})} + \cdots + r\_{i\_{\emptyset}\pi(i\_{\emptyset})} = r\_{k+1} + \cdots + r\_{\emptyset}.\tag{7}$$

Indeed, if P*k*, *k* = 0, 1, ... , *q*, denotes the set of all bijections *π* : Z→Z with exactly *k* fixed points, then *p*<sup>0</sup> can be written as

$$p\_0 = \sum\_{k=0}^{q-1} \sum\_{\substack{\pi \in \mathcal{P}\_k \\ i\_1 < \dots < i\_k \\ i\_{k+1} < \dots < i\_q}} \pm \left( \prod\_{l=1}^k (a\_{i\_l i\_l} + a\_0 - \lambda) \right) \prod\_{l=k+1}^q a\_{i\_l \pi(i\_l) \tau}$$

where in each summand, *i*1,..., *ik* are the fixed points of *π*.

If *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>0</sup> , 0 <sup>&</sup>lt; <sup>|</sup>*α*| ≤ *<sup>ρ</sup>*, then

$$\partial^{a}p\_{0} = \sum\_{k=0}^{q-1} \sum\_{\substack{n \in \mathbb{P}\_{k} \\ i\_{1} < \cdots < i\_{l} \\ i\_{k+1} < \cdots < i\_{l}}} \sum\_{a\_{1}, \ldots, a\_{q} \in \mathbb{N}\_{0}^{q}} \mathbb{C}\_{a\_{1} \ldots a\_{q}} \left( \prod\_{l=1}^{k} \partial^{a\_{l}}(a\_{l;i\_{l}} + a\_{0} - \lambda) \right) \prod\_{l=k+1}^{q} \partial^{a\_{l}} a\_{i\_{l} \pi(i\_{l})} \cdots$$

Now,

$$\left|\partial^{\alpha\_l} (a\_{i\_l i\_l} + a\_0 - \lambda)\right| \le C \begin{cases} \langle \mathfrak{E} \rangle^{r\_{i\_l}} + |\lambda|, & \mathfrak{a}\_l = \mathfrak{O}, \\ \langle \mathfrak{E} \rangle^{r\_{i\_l}} \langle \mathfrak{E} \rangle^{-|\mathfrak{a}\_l|}, & \mathfrak{a}\_l \ne \mathfrak{O}, \end{cases}$$

and

$$|\partial^{\alpha\_l} a\_{i\_l \pi(i\_l)}| \le \mathcal{C} \langle \mathfrak{F} \rangle^{r\_{i\_l \pi(i\_l)}} \langle \mathfrak{F} \rangle^{-|\alpha|}.$$

Since *α* = 0, *α<sup>j</sup>* = 0 for some *j* and, therefore, taking (7) in account, it holds

$$\left|Q\_k\right| \le \mathbb{C} \left(\prod\_{\substack{l=1\\l\neq j}}^q \left( \langle \mathfrak{F} \rangle^{r\_{i\_l}} + |\lambda| \right) \langle \mathfrak{F} \rangle^{r\_{i\_j} - |\alpha|} .\right.$$

Then, we have

$$\begin{split} \left| \partial^{\boldsymbol{\alpha}} p\_{0} \right| &\leq \overline{\mathbb{C}} \sum\_{k=0}^{q-1} \sum\_{\pi \in \mathcal{P}\_{k}} \sum\_{j=1}^{q} \left( \prod\_{\substack{l=1 \\ l \neq j}}^{q} \left( \langle \xi \rangle^{r\_{lj}} + |\lambda| \right) \right) \langle \xi \rangle^{r\_{j}} \langle \xi \rangle^{-|\boldsymbol{\alpha}|} \\ &\leq \hat{\mathsf{C}} \left( \sum\_{j=1}^{q} \left( \prod\_{\substack{l=1 \\ l \neq j}}^{q} (\langle \xi \rangle^{r\_{l}} + |\lambda|) \right) \langle \xi \rangle^{r\_{j}} \right) \langle \xi \rangle^{-|\boldsymbol{\alpha}|}, \end{split}$$

which shows (6). Thus, we can estimate *<sup>∂</sup>β*(1/*p*0) for *<sup>β</sup>* <sup>=</sup> 0. Indeed, if *<sup>β</sup>* <sup>=</sup> 0 it holds that (see [19], Lemma 10.4, p. 74)

$$
\partial^{\mathfrak{G}}\left(\frac{1}{p\_0}\right) = \sum\_{k=1}^{|\mathcal{S}|} \sum\_{\substack{\mathfrak{F}\_1,\dots,\mathfrak{F}\_k \in \mathbb{F}\_0^n \times \{0\} \\ \mathfrak{F}\_1 + \dots + \mathfrak{F}\_k = \mathfrak{F}}} \mathbb{C}\_{\mathfrak{F}\_1 \dots \mathfrak{F}\_k} \frac{\left(\partial^{\mathfrak{G}\_1} p\_0\right) \cdot \dots \cdot \left(\partial^{\mathfrak{G}\_k} p\_0\right)}{p\_0^{1+k}}.
$$

Due to (6) and the Λ-ellipticity condition we obtain

*∂β* 1 *p*0 <sup>≤</sup> *<sup>C</sup><sup>β</sup>* |*β*| ∑ *k*=1 ∑ *<sup>β</sup>*1,...,*βk*∈N*<sup>n</sup>* 0-{0} *β*1+···+*βk*=*β <sup>q</sup>* ∑ *j*=1 *<sup>q</sup>* ∏ *i*=1 *i*=*j <sup>ξ</sup> ri* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> *ξ rj k <sup>ξ</sup>*−|*β*<sup>|</sup> |*p*0|1+*<sup>k</sup>* <sup>≤</sup> *<sup>C</sup>*<sup>ˆ</sup> *β* |*β*| ∑ *k*=1 *<sup>q</sup>* ∑ *j*=1 *<sup>q</sup>* ∏ *i*=1 *i*=*j <sup>ξ</sup> ri* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> *k ξ krj <sup>ξ</sup>*−|*β*<sup>|</sup> *q* ∏ *i*=1 *ξ ri* + |*λ*| <sup>1</sup>+*<sup>k</sup>* <sup>=</sup> *<sup>C</sup>*<sup>ˆ</sup> *β q* ∏ *i*=1 *ξ ri* + |*λ*| |*β*| ∑ *k*=1 *q* ∑ *j*=1 *ξ krj<sup>ξ</sup>*−|*β*<sup>|</sup> *ξ rj* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> *k* <sup>=</sup> *<sup>C</sup>*<sup>ˆ</sup> *β q* ∏ *i*=1 *ξ ri* + |*λ*| |*β*| ∑ *k*=1 *q* ∑ *j*=1 *ξ rj ξ rj* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> *k*−<sup>1</sup> *ξ rj<sup>ξ</sup>*−|*β*<sup>|</sup> *ξ rj* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> ≤ *C*∗ *β q* ∏ *i*=1 *ξ ri* + |*λ*| *<sup>q</sup>* ∑ *j*=1 *ξ rj ξ rj* <sup>+</sup> <sup>|</sup>*λ*<sup>|</sup> *<sup>ξ</sup>*−|*β*<sup>|</sup> .

Now, since

 

$$|H| \le \left(\prod\_{\substack{l=1 \\ l \ne i}}^q \left( \langle \xi \rangle^{\gamma\_l} + |\lambda| \right) \right) \langle \xi \rangle^{-|\gamma\_1 + \dots + \gamma\_{q-1}|} \zeta$$

if 0 < <sup>|</sup>*β*| ≤ *<sup>ρ</sup>*, we can estimate (5) from above by

$$\begin{split} \frac{\mathsf{C}\_{\beta}^{\*}}{\prod\_{i=1}^{q} \left( \langle \xi \rangle^{r\_{i}} + |\lambda| \right)} \Big( \sum\_{j=1}^{q} \frac{\langle \xi \rangle^{r\_{j}}}{\langle \xi \rangle^{r\_{j}} + |\lambda|} \Big) \Big( \prod\_{l=1 \atop l \neq i}^{q} \left( \langle \xi \rangle^{r\_{l}} + |\lambda| \right) \big( \langle \xi \rangle^{-|a|} \Big) \\ = \mathsf{C}\_{\beta}^{\*} \sum\_{j=1}^{q} \frac{\langle \xi \rangle^{r\_{j} - |a|}}{\left( \langle \xi \rangle^{r\_{i}} + |\lambda| \right) \left( \langle \xi \rangle^{r\_{j}} + |\lambda| \right)}. \end{split}$$

With the estimates from above for (5), in both cases *β* = 0 and *β* = 0, we obtain the estimate (3) for 0 <sup>&</sup>lt; <sup>|</sup>*α*| ≤ *<sup>ρ</sup>* and (*ξ*, *<sup>λ</sup>*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>.

Under the assumption (2) on the order of the symbols in the system, Lemma 1, estimate (3), and the equivalence

$$
\left< \xi \right>^r + \left| \lambda \right| \sim \left< \xi' \left| \lambda \right|^{1/r} \right>^r \qquad (r \ge 0),
$$

lead to the following assertion.

**Corollary 1.** *Let A*(*D*) *be* Λ*-elliptic,*

$$b\_{\lambda}(\cdot) := \left(A\_{\mathfrak{a}\_0}(\cdot) - \lambda\right)^{-1} = \left(\mathfrak{g}\_{ij}(\cdot;\lambda)\right)\_{1 \le i,j \le q'}$$

*and suppose that the assumption* (2) *holds. Then for each i*, *j* = 1, . . . , *q, we have*

$$(b\_{\lambda}(\cdot))\_{ij} = \mathbb{g}\_{ij}(\cdot;\lambda) \in S^{-r\_{ji}\rho}(\mathbb{R}^n, \mathcal{L}(\mathbb{C}^q)), \quad \forall \lambda \in \Lambda$$

*with*

$$\begin{split} \left| \mathcal{g}\_{ii}(\overleftarrow{\xi};\lambda) \right| &\leq \mathcal{C} \Big\langle \overline{\xi}, \left| \lambda \right|^{1/r\_{i}} \right\rangle^{-r\_{i}}, \qquad (i = 1, \ldots, q) \\ \left| \partial\_{\overline{\xi}}^{a} \mathcal{g}\_{ii}(\overleftarrow{\xi};\lambda) \right| &\leq \mathcal{C}\_{a} \sum\_{j=1}^{q} \left\langle \overline{\xi}, \left| \lambda \right|^{1/r\_{i}} \right\rangle^{-r\_{i}} \left\langle \overline{\xi}, \left| \lambda \right|^{1/r\_{j}} \right\rangle^{-r\_{j}} \left\langle \overline{\xi} \right|^{r\_{j} - |a|}, \quad ((i, a) \in \mathcal{Z}\_{1}) \\ \left| \partial\_{\overline{\xi}}^{a} \mathcal{g}\_{ij}(\overleftarrow{\xi};\lambda) \right| &\leq \mathcal{C} \left\langle \overline{\xi}, \left| \lambda \right|^{1/r\_{i}} \right\rangle^{-r\_{i}} \left\langle \overline{\xi}, \left| \lambda \right|^{1/r\_{j}} \right\rangle^{-r\_{j}} \left\langle \overline{\xi} \right| \mathcal{Y}^{1:r\_{i}} \right\rangle^{-r\_{j}}, \quad ((i, j, a) \in \mathcal{Z}\_{2}) \end{split}$$

*for all* (*ξ*, *<sup>λ</sup>*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>Λ</sup>*, where <sup>C</sup> is a positive constant independent on <sup>α</sup>*, *<sup>ξ</sup> and <sup>λ</sup>,* <sup>Z</sup><sup>1</sup> :<sup>=</sup> {(*i*, *<sup>α</sup>*) : 1 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>q</sup>*, 0 <sup>&</sup>lt; <sup>|</sup>*α*<sup>|</sup> <sup>≤</sup> *<sup>ρ</sup>*}*, and* <sup>Z</sup><sup>2</sup> :<sup>=</sup> {(*i*, *<sup>j</sup>*, *<sup>α</sup>*) : 1 <sup>≤</sup> *<sup>i</sup>*, *<sup>j</sup>* <sup>≤</sup> *<sup>q</sup>*, *<sup>i</sup>* <sup>=</sup> *<sup>j</sup>*, <sup>|</sup>*α*<sup>|</sup> <sup>≤</sup> *<sup>ρ</sup>*}*.*
