2.1 Pseudodifferential operators having the transmission property

Let $(\tilde {M},\tilde {g})$ be a closed d-dimensional Riemannian manifold, endowed with a volume form $d\mathrm {Vol}_{\tilde {g}}\in \Omega ^d(\tilde {M})$ ; our study can be extended to noncompact manifolds, but for simplicity, we restrict our attention to compact ones. Let $M \hookrightarrow \tilde {M}$ be a compact embedded submanifold of the same dimension having a smooth boundary. Since every compact Riemannian manifold with smooth boundary can be embedded in its closed double [Reference LeeLee12, p. 226], we will henceforth view every compact Riemannian manifold with smooth boundary M as smoothly embedded in a closed ambient Riemannian manifold $\tilde {M}$ .

Let $\tilde {{\mathbb E}},\tilde {{\mathbb F}}\to \tilde {M}$ be Riemannian vector bundles over $\tilde {M}$ ; denote by ${\mathbb E}=\tilde {{\mathbb E}}|_M$ and ${\mathbb F}=\tilde {{\mathbb F}}|_M$ the pullback bundles, which are vector bundles over M. Let ${\mathbb J},{\mathbb G}\to {\partial M}$ be Riemannian vector bundles over ${\partial M}$ .

For $\psi ,\eta \in L^2\Gamma (\tilde {{\mathbb E}})$ , we denote their $L^2$ -inner product by

$$\begin{align*}\langle\psi,\eta\rangle = \int_{\tilde{M}} (\psi,\eta)_{\tilde{{\mathbb E}}}\,d\mathrm{Vol}_{\tilde{g}}. \end{align*}$$

We use the same notation for the $L^2$ -inner product associated with sections over M. Likewise, for $\rho ,\tau \in L^2\Gamma ({\mathbb G})$ , we denote the induced $L^2$ -inner product on the boundary ${\partial M}$ by

$$\begin{align*}\langle\rho,\tau\rangle = \int_{\partial M} (\rho,\tau)_{{\mathbb G}}\,\, d\mathrm{Vol}_{\jmath^*g}, \end{align*}$$

where $\jmath :{\partial M}\hookrightarrow M$ is the inclusion map of the boundary, and $d\mathrm {Vol}_{\jmath ^*g}$ is the volume form associated with the pullback metric at the boundary, obtained by inserting the unit normal vector into the first entry of $d\mathrm {Vol}_g$ .

A differential operator $\Gamma (\tilde {{\mathbb E}})\to \Gamma (\tilde {{\mathbb F}})$ is a linear map that can be represented as an ${\mathbb {R}}^{N_1}\to {\mathbb {R}}^{N_2}$ differential operator in any local trivializations of $\tilde {{\mathbb E}}$ and $\tilde {{\mathbb F}}$ . Since this definition is local, it extends to linear maps $\Gamma ({\mathbb E})\to \Gamma ({\mathbb F})$ and boundary differential operators $\Gamma ({\mathbb E})\to \Gamma ({\mathbb G})$ .

Differential operators are the prominent example of a larger class of linear operators, known as pseudodifferential operators. In ${\mathbb {R}}^d$ , pseudodifferential operators are defined through their action on the Fourier transform $\hat {u}(\xi )$ of their argument $u(x)$ via a so-called symbol matrix $A(x,\xi )$ . On a manifold, their definition is based on their definition in ${\mathbb {R}}^d$ via the pullback by coordinate charts. A pseudodifferential operator is said to be of order $m\in {\mathbb R}$ if its associated symbol matrix satisfies a growth condition with exponent m. We adopt the notation of [Reference Rempel and SchulzeRS82] and denote by $L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ the space of pseudodifferential operators of order m. By definition, a pseudodifferential operator of order m is also of any order greater than m. We denote by

$$\begin{align*}L(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb F}})=\bigcup_{m\in{\mathbb Z}} L^m(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb F}}) \end{align*}$$

the space of all pseudodifferential operators and by

$$\begin{align*}L^{-\infty}(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb F}})=\bigcap_{m\in{\mathbb Z}}L^m(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb F}}) \end{align*}$$

the space of so-called smoothing operators. We denote by $\operatorname {ord}(A)$ the set of $m\in {\mathbb Z}\cup \{-\infty \}$ such that $A\in L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ ; we say that $\operatorname {ord}(A) < \operatorname {ord}(Q)$ if $\operatorname {ord}(Q) \subsetneq \operatorname {ord}(A)$ .

Pseudodifferential operators were introduced as a class of operators, rich enough to encompass both differential operators and singular integral operators arising as inverse operators (parametrices) for elliptic differential systems. We refer the reader to the abundant literature on the subject [Reference HörmanderHör94, Reference Rempel and SchulzeRS82, Reference Wloka, Rowley and LawrukWRL95, Reference GrubbGru96, Reference TaylorTay11b, Reference TaylorTay11c]; in the following, we will only list those properties of pseudodifferential operators that are of relevance to the present work.

Every pseudodifferential operator $A\in L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ is associated with a symbol, generalizing the principal symbol of a differential operator [Reference TaylorTay11a, pp. 176–178] or [Reference Rempel and SchulzeRS82, Sec. 1.2.4.1]. On a manifold, unlike in ${\mathbb {R}}^d$ , the symbol is an equivalence class of smooth bundle maps $\sigma _A:T^*\tilde {M}\otimes \tilde {{\mathbb E}}\to \tilde {{\mathbb F}}$ ,

$$\begin{align*}\sigma_A: (x,\xi,v)\mapsto \sigma_A(x,\xi)v \qquad \text{for } x\in\tilde{M}, \xi\in T^*_x\tilde{M} \text{and} v\in \tilde{{\mathbb E}}_x. \end{align*}$$

The primary role of symbols is to reduce analytical properties of pseudodifferential operators into algebraic properties of their symbols; notably, it allows for a functional classification of pseudodifferential operators. Adopting the notation of [Reference Rempel and SchulzeRS82], the space of all symbols of order m is denoted by $S^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ , where $\sigma _A\in S^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ implies m-growth conditions with respect to the variable $\xi $ . Symbol are defined up to lower-order terms, which is to say that if $A,Q\in L(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ with $\operatorname {ord}(A)>\operatorname {ord}(Q)$ , then

$$\begin{align*}\sigma_{A+Q}(x,\xi)=\sigma_A(x,\xi). \end{align*}$$

An operator $A\in L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ is said to be homogeneous if for every $\lambda> 0$ and $|\xi |$ large enough,

$$\begin{align*}\sigma_A(x,\lambda\xi) = \lambda^m\,\sigma_A(x,\xi). \end{align*}$$

Homogeneity holds trivially for differential operators, but does not for general pseudodifferential operators. Operators having symbols that possess locally, as $\xi \to \infty $ , an asymptotic expansion of homogeneous symbols are called classical. Sticking with the notation of [Reference Rempel and SchulzeRS82], we denote by $L^m_{\mathrm {cl}}(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ the space of classical pseudodifferential operators of order m. The importance of this class is in the homomorphism properties satisfied by their symbols, which is used repeatedly in this work.

A pseudodifferential operator $A\in L(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ is first and foremost a continuous linear map between Fréchet spaces,

$$\begin{align*}A:\Gamma(\tilde{{\mathbb E}})\to \Gamma(\tilde{{\mathbb F}}), \end{align*}$$

with the topology induced by the uniform convergence of sections along with all their derivatives. Pseudodifferential operators are closed under composition

(2.1) $$ \begin{align} AQ \in L^{m_A+m_Q}(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb F}}') \end{align} $$

for every $A\in L^{m_A}(\tilde {M},\tilde {{\mathbb F}},\tilde {{\mathbb F}}')$ and $Q\in L^{m_Q}(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ . Moreover, every $A\in L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ admits a formal adjoint $A^*\in L^m(\tilde {M},\tilde {{\mathbb F}},\tilde {{\mathbb E}})$ , given by the property that

(2.2) $$ \begin{align} \langle A\psi,\eta\rangle = \langle \psi,A^*\eta\rangle \qquad \text{for every } \psi \in \Gamma(\tilde{{\mathbb E}}) \text{ and } \eta\in\Gamma(\tilde{{\mathbb F}}). \end{align} $$

The class $L_{\mathrm {cl}}(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ of classical operators is closed under composition and adjointness as well, with the additional property that classical symbols satisfy the homomorphism properties [Reference Rempel and SchulzeRS82, p. 74],

$$\begin{align*}\sigma_{AQ}(x,\xi)=\sigma_A(x,\xi)\circ \sigma_Q(x,\xi) \qquad\text{ and }\qquad \sigma_{A^*}(x,\xi)=(\sigma_A(x,\xi))^*. \end{align*}$$

This work is concerned with Sobolev sections of vector bundles, $W^{s,p}\Gamma (\tilde {{\mathbb E}})$ defined for $s\in {\mathbb R}$ and $1<p<\infty $ . The definition goes through first defining scalar-valued Sobolev functions on ${\mathbb R}^d$ , then on domains $\Omega \subset {\mathbb R}^d$ , and then on closed manifolds by means of partitions of unity and coordinate charts. Finally, Sobolev sections of vector bundles over closed manifolds are defined [Reference Rempel and SchulzeRS82, Sec. 1.2.1.2].

There are several variants of Sobolev spaces. The spaces $H^{s,p}\Gamma (\tilde {{\mathbb E}})$ (also known as Bessel-potential spaces) are defined for every $s\in {\mathbb R}$ and $1<p<\infty $ by means of the Fourier transform [Reference Rempel and SchulzeRS82, pp. 42–46], [Reference GrubbGru90, pp. 291–293]. For $s\in {{\mathbb N}_0}$ , $H^{s,p}\Gamma (\tilde {{\mathbb E}})$ is the completion of $\Gamma (\tilde {{\mathbb E}})$ with respect to the Sobolev norm,

$$\begin{align*}\|\psi\|_{s,p}=\sum_{{{\mathbb N}_0}\ni \alpha\leq s}\|\nabla^{\alpha}\psi\|_{L^{p}}, \end{align*}$$

where $\nabla $ is any connection on $\tilde {{\mathbb E}}$ .

Our eventual goal is to pass to manifolds with boundary, where trace theorems are being invoked. For $s\in {\mathbb R}_{+}\setminus {{\mathbb N}_0}$ , the spaces $H^{s,p}\Gamma (\tilde {{\mathbb E}})$ are insufficient for these theorems to hold. This is where Besov spaces $B^{s,p}\Gamma (\tilde {{\mathbb E}})$ , $s\in {\mathbb R}$ and $1<p<\infty $ , come in [Reference GrubbGru90, p. 293], [Reference Rempel and SchulzeRS82, p. 45–46]. As in [Reference GrubbGru90], we set

$$\begin{align*}W^{s,p}\Gamma(\tilde{{\mathbb E}})=\begin{cases} H^{s,p}\Gamma(\tilde{{\mathbb E}}) & s\in{\mathbb Z}\\ B^{s,p}\Gamma(\tilde{{\mathbb E}}) & s\in {\mathbb R}\setminus {\mathbb Z}. \end{cases} \end{align*}$$

Our results will be formulated for $W^{s,p}$ -spaces for $s\in {{\mathbb N}_0}$ (i.e., for ‘standard’ Sobolev sections). References to noninteger s are needed because of the mapping properties of trace operators.

Pseudodifferential operators satisfy various mapping properties with respect to these Sobolev spaces. Most prominently, $A\in L^m(\tilde {M};\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ extends to a continuous linear map [Reference GrubbGru90, p. 312],

(2.3) $$ \begin{align} A:W^{s,p}\Gamma(\tilde{{\mathbb E}})\to W^{s-m,p}\Gamma(\tilde{{\mathbb F}}) \end{align} $$

for every $s\in {\mathbb R}$ and $1< p<\infty $ . In particular, every $A\in L^{-\infty }(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ extends into a map

$$\begin{align*}A: {\mathcal{D}}'\Gamma(\tilde{{\mathbb E}})\to \Gamma(\tilde{{\mathbb F}}). \end{align*}$$

A pseudodifferential operator $E\in L(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ is called elliptic if $\sigma _E(x,\xi ):\tilde {{\mathbb E}}_x\to \tilde {{\mathbb F}}_x$ is an isomorphism for every $x\in \tilde {M}$ and for every $|\xi |$ large enough. A parametrix (also known as an approximate inverse) for E is an operator $P:\Gamma (\tilde {{\mathbb F}})\to \Gamma (\tilde {{\mathbb E}})$ satisfying

$$\begin{align*}PE-{\operatorname{Id}}\in L^{-\infty}(\tilde{M},\tilde{{\mathbb E}},\tilde{{\mathbb E}}) \qquad\text{ and }\qquad EP-{\operatorname{Id}}\in L^{-\infty}(\tilde{M},\tilde{{\mathbb F}},\tilde{{\mathbb F}}). \end{align*}$$

Every elliptic $E\in L^m(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ admits a parametrix $P\in L^{-m}(\tilde {M},\tilde {{\mathbb F}},\tilde {{\mathbb E}})$ , which is unique modulo $L^{-\infty }(\tilde {M},\tilde {{\mathbb F}},\tilde {{\mathbb E}})$ [Reference Rempel and SchulzeRS82, p. 76].

Pseudodifferential operators are generally defined on a manifold without boundary. We are interested in a subclass of pseudodifferential operators over $\tilde {M}$ that truncate ‘nicely’ to M while retaining the closure of the calculus to adjoints, compositions and parametrices. Such operators were introduced by Hörmander [Reference HörmanderHör94, p. 105]; our exposition is based on a combination of [Reference GrubbGru96, p. 23], [Reference Rempel and SchulzeRS82, Sec. 2.3] and [Reference Wloka, Rowley and LawrukWRL95, p. 512].

Let $r_+:{\mathcal {D}}'\Gamma (\tilde {{\mathbb F}})\to {\mathcal {D}}'\Gamma ({\mathbb F})$ be the restriction operator,

$$\begin{align*}r_+\psi=\psi|_M \end{align*}$$

(i.e., the restriction of $\psi $ acting on test functions with support in M), and let $e_+:\Gamma ({\mathbb E})\to {\mathcal {D}}'\Gamma (\tilde {{\mathbb E}})$ be the extension-by-zero operator,

$$\begin{align*}e_+\psi=\begin{cases} \psi & \text{in } M \\ 0 & \text{in } \tilde{M}\setminus M. \end{cases} \end{align*}$$

A pseudodifferential operator $A\in L(\tilde {M},\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ is said to have the transmission property with respect to M when its truncation,

$$\begin{align*}A_+=r_+Ae_+:\Gamma({\mathbb E})\to {\mathcal{D}}'\Gamma({\mathbb F}), \end{align*}$$

is a continuous map $A_+:\Gamma ({\mathbb E})\to \Gamma ({\mathbb F})$ .

We adopt the notation of [Reference Rempel and SchulzeRS82] and denote the space of all classical operators of order m over $\tilde {M}$ having the transmission property with respect to M by $\mathrm {OP}(\mathfrak {A}^m)(\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ , or by $\mathrm {OP}(\mathfrak {A}^m)$ when there is no ambiguity, and let

$$\begin{align*}\mathrm{OP}(\mathfrak{A})=\bigcup_{m\in{\mathbb Z}} \mathrm{OP}(\mathfrak{A}^m). \end{align*}$$

There is ample discussion in the cited literature on sufficient conditions for a pseudodifferential operator to have the transmission property. For our purposes, we will only mention that every differential operator is in $\mathrm {OP}(\mathfrak {A})$ , and that $\mathrm {OP}(\mathfrak {A})$ is closed under adjoints, compositions and parametrices [Reference Rempel and SchulzeRS82, p. 136] (Proposition 2 requires the elements to be properly-supported [Reference HörmanderHör94, p. 86], but every pseudodifferential operator is properly supported in a compact manifold).

Truncations of operators in $\mathrm {OP}(\mathfrak {A}^m)$ satisfy mapping properties as well, which requires defining Sobolev spaces on manifolds with boundaries. For $s<0$ , we note that the spaces $W^{s,p}\Gamma (\tilde {{\mathbb E}})$ consist of distributions. We define [Reference GrubbGru90, pp. 294–297],

$$\begin{align*}W^{s,p}\Gamma({\mathbb E}) = W^{s,p}\Gamma(\tilde{{\mathbb E}}) / \{\omega \in W^{s,p}\Gamma(\tilde{{\mathbb E}}) ~:~ \operatorname{supp} \omega \subseteq \overline{\tilde{M}\setminus M}\}, \end{align*}$$

and

$$\begin{align*}W_0^{s,p}\Gamma({\mathbb E}) = \{\omega\in W^{s,p}\Gamma(\tilde{{\mathbb E}}) ~:~ \operatorname{supp}\omega \subseteq M\}. \end{align*}$$

For $1/p+1/q=1$ ,

$$\begin{align*}(W^{s,q}_0\Gamma({\mathbb E}))^*=W^{-s,p}\Gamma({\mathbb E}). \end{align*}$$

The mapping properties of $A\in \mathrm {OP}(\mathfrak {A}^m)$ are given in [Reference GrubbGru90, p. 312],

(2.4) $$ \begin{align} A_+:W^{s,p}\Gamma({\mathbb E})\to W^{s-m,p}\Gamma({\mathbb F}) \end{align} $$

for every ${\mathbb Z}\ni s\ge 1/p-1$ and $1< p< \infty $ . Henceforth, we remove the ‘ $+$ ’ subscript from the truncation of a differential operator. Since these always act locally, this should cause no confusion.

2.2 Integration by parts, trace operators and normal conditions

As mentioned in the previous section, the space $\mathrm {OP}(\mathfrak {A})$ of classical operators having the transmission property is closed under adjoints (i.e., if $A\in \mathrm {OP}(\mathfrak {A})(\tilde {{\mathbb E}},\tilde {{\mathbb F}})$ then $A^*\in \mathrm {OP}(\mathfrak {A})(\tilde {{\mathbb F}},\tilde {{\mathbb E}})$ , where $A^*$ is defined by (2.2)). Consider the truncation of the adjoint,

$$\begin{align*}(A^*)_+=r_+A^*e_+. \end{align*}$$

The question is whether the truncation $(A^*)_+$ of $A^*$ is in some sense adjoint to the truncation $A_+$ of A; for example, does it hold that

$$\begin{align*}\langle A_+\psi,\eta\rangle = \langle \psi,(A^*)_+\eta\rangle \qquad \text{for every } \psi \in \Gamma({\mathbb E}) \text{ and } \eta\in\Gamma({\mathbb F})\, ? \end{align*}$$

[Reference GrubbGru96, p. 36] shows that every $A\in \mathrm {OP}(\mathfrak {A}^m)$ can be written as a sum

$$\begin{align*}A = D + Q, \end{align*}$$

where $D\in \mathrm {OP}(\mathfrak {A}^m)$ is a differential operator and $Q\in \mathrm {OP}(\mathfrak {A}^m)$ satisfies

$$\begin{align*}\langle Q_+\psi,\eta\rangle = \langle \psi,(Q^*)_+\eta\rangle \qquad \text{for every } \psi \in \Gamma({\mathbb E}) \text{ and } \eta\in\Gamma({\mathbb F}). \end{align*}$$

Since D is a differential operator,

$$\begin{align*}\langle D\psi,\eta\rangle =\langle \psi, D^*\eta\rangle \qquad \text{for every } \psi\in\Gamma({\mathbb E}) \text{ and } \eta\in\Gamma_c({\mathbb F}), \end{align*}$$

from which it follows that

$$\begin{align*}\langle A_+\psi,\eta\rangle =\langle \psi, (A^*)_+\eta\rangle \qquad \text{for every } \psi\in\Gamma({\mathbb E}) \text{ and } \eta\in\Gamma_c({\mathbb F}). \end{align*}$$

This formula holds also for noncompactly supported sections when $\operatorname {ord}(A) \leq 0$ since in this case, $A_+:L^2\Gamma ({\mathbb E})\to L^2\Gamma ({\mathbb F})$ continuously, and hence admits an $L^2$ -adjoint, $(A_+)^*$ , and by the uniqueness of the adjoint, $(A_+)^* = (A^*)_+$ . Throughout this work, compactly supported in a manifold with boundary means compactly supported in its interior. We henceforth denote $(A^*)_+$ simply by $A^*_+$ , recalling that the adjointness property is only with respect to compactly supported sections.

[Reference GrubbGru96, pp. 37–38] denotes by $\rho _N:\Gamma ({\mathbb E})\to (\Gamma (\jmath ^*{\mathbb E}))^N$ the Cauchy-boundary operator,

$$\begin{align*}\rho_N\psi=(D_{\mathfrak{n}}^0\psi,D_{\mathfrak{n}}\psi,...,D_{\mathfrak{n}}^{N-1}\psi), \end{align*}$$

where $D_{\mathfrak {n}}$ is the normal covariant derivative (which is well-defined in a collar neighborhood of ${\partial M}$ , and hence can be iterated) evaluated at the boundary, and $D_{\mathfrak {n}}^0$ is the trace on the boundary; the choice of connection on ${\mathbb E}$ is immaterial. Given $A\in \mathrm {OP}(\mathfrak {A})$ , there exists a unique matrix of tangential differential operators $U_A=(U_\alpha ^\beta )_{\alpha ,\beta =0,...,m-1}$ of orders $\le m-\alpha -1$ such that the following Green’s formula holds [Reference GrubbGru96, pp. 37–38]:

(2.5) $$ \begin{align} \langle A_+\psi,\eta\rangle =\langle \psi, A^*_+\eta\rangle+\langle U_A\rho_m\psi,\rho_m\eta\rangle \end{align} $$

for every $\psi \in \Gamma ({\mathbb E})$ and $ \eta \in \Gamma ({\mathbb F})$ .

In the sequel (e.g., Definition 3.1), we will encounter integration by parts formulas such as (2.5), where the operator A belongs to a class of operators larger than $\mathrm {OP}(\mathfrak {A})$ , which requires the expansion of the class of differential boundary operators. A trace operator T of order $m \in {\mathbb {R}}$ and class $r\in {{\mathbb N}_0}$ is a linear map $T:\Gamma ({\mathbb E})\to \Gamma ({\mathbb G})$ of the form

(2.6) $$ \begin{align} T = \sum_{j=0}^{r-1} S_jD_{\mathfrak{n}}^j + \jmath^*Q_+, \end{align} $$

where $S_j\in L^{m-j}_{\mathrm {cl}}({\partial M},\jmath ^*{\mathbb E},{\mathbb G})$ is a pseudodifferential operator on the boundary (which is a closed manifold) and $Q\in \mathrm {OP}(\mathfrak {A}^m)$ [Reference GrubbGru96, pp. 27–28, 33]. The operator $\rho _m$ is an instance of a trace operator of order $m-1$ and class m with ${\mathbb G} = (\jmath ^*{\mathbb E})^m$ , $S_j(\xi ) = (0,\dots ,0,\xi ,0,\dots ,0)$ and $Q=0$ .

The order of a trace operator is an extension of the order of a pseudodifferential operator (by (2.1), $\operatorname {ord}(S_jD_{\mathfrak {n}}^j) \le m$ for every j), whereas its class retains (one more than) the number of normal derivatives. We denote the set of trace operators of order $m\in {\mathbb R}$ and class $r\in {{\mathbb N}_0}$ by $\mathrm {OP}(\mathfrak {T}^{m,r})$ , as in [Reference Rempel and SchulzeRS82]. The class of trace operators can be extended to negative values [Reference GrubbGru90, pp. 309–311]. In simple terms, $T\in \mathrm {OP}(\mathfrak {T}^{m,-r})$ if $T\in \mathrm {OP}(\mathfrak {T}^{m,0})$ and $T D_{\mathfrak {n}}^r\in \mathrm {OP}(\mathfrak {T}^{m,0})$ . We denote the union of all $\mathrm {OP}(\mathfrak {T}^{m,r})$ of order $m\in {\mathbb R}$ and class $r\in {\mathbb Z}$ by $\mathrm {OP}(\mathfrak {T})$ . Trace operators have well-defined symbols, much like pseudodifferential operators. However, unlike operators with the transmission property, the mapping properties of trace operators depend on both the order and the class; for every $T\in \mathrm {OP}(\mathfrak {T}^{m,r})$ [Reference GrubbGru90, p. 312],

(2.7) $$ \begin{align} T:W^{s,p}\Gamma({\mathbb E})\to W^{s-m-1/p,p}\Gamma({\mathbb G}) \end{align} $$

for every ${\mathbb Z}\ni s>r+1/p-1$ and $1<p<\infty $ (for example, for $p=2$ , the restrictions to the boundary, which is an operator of order zero involves a loss of regularity of $1/2$ ). Note how the class r limits the mapping properties: a trace operator of order m reduces the regularity of a $W^{s,p}$ -section by $m+1/p$ , as expected, but only for s large enough. However, a negative class allows mapping between negative Sobolev spaces.

We next specify a particular class of trace operators: A system of trace operators associated with order m is a trace operator of the form $T=T_0\oplus T_1\oplus \cdots \oplus T_{m-1}\in \mathrm {OP}(\mathfrak {T}^{m-1,m})$ , where $T_i:\Gamma ({\mathbb E})\to \Gamma ({\mathbb J}_i)$ is in $\mathrm {OP}(\mathfrak {T}^{i,m})$ and ${\mathbb J}_i\to {\partial M}$ is a vector bundle [Reference GrubbGru96, pp. 45–46]. As stated above, every component $T_i$ can be written as

(2.8) $$ \begin{align} T_i = \sum_{j=0}^{m-1} S_{ij}D_{\mathfrak{n}}^j + \jmath^*(Q_i)_+, \end{align} $$

where $S_{ij}\in L^{i-j}_{\mathrm {cl}}({\partial M},\jmath ^*{\mathbb E},{\mathbb J}_i)$ and $Q_i\in \mathrm {OP}(\mathfrak {A}^i)$ . Since $T_i\in \mathrm {OP}(\mathfrak {T}^{i,m})$ , and since the mapping property (2.7) applies to each $T_i$ separately, systems of trace operators associated with order m satisfy the compound mapping property

$$\begin{align*}T:W^{s,p}\Gamma({\mathbb E})\to \bigoplus_{i=0}^{m-1} W^{s-i-1/p,p}\Gamma({\mathbb J}_i) \end{align*}$$

for every ${\mathbb Z}\ni s> m+1/p-1$ .

The normality of a system of trace operators implies surjectivity [Reference GrubbGru96, p. 80]:

The canonical example of a normal system of trace operators associated with order m is $\rho _m$ defined above.

Consider the integration by parts formula (2.5) for $A\in \mathrm {OP}(\mathfrak {A}^m)$ . We are interested in a setting where there exist differential operators, $B_A:\Gamma ({\mathbb E})\to \Gamma ({\mathbb G})$ and $B_{A^*}:\Gamma ({\mathbb F})\to \Gamma ({\mathbb G})$ , which are normal systems of trace operators associated with order m, such that

(2.9) $$ \begin{align} \langle A_+\psi,\eta\rangle = \langle\psi,A_+^*\eta\rangle + \langle B_A\psi, B_{A^*}\eta\rangle \end{align} $$

for every $\psi \in \Gamma ({\mathbb E})$ and $\eta \in \Gamma ({\mathbb F})$ . Not every $A\in \mathrm {OP}(\mathfrak {A})$ has this property. In our work, however, formulas such as (2.9) emerge naturally; hence, we omit this discussion.

2.3 Green operators and elliptic boundary-value problems

Let $E\in \mathrm {OP}(\mathfrak {A}^m)$ be elliptic and let $T\in \mathrm {OP}(\mathfrak {T}^{m-1,r})$ . Consider the problem of finding a linear map $R:\Gamma ({\mathbb F})\to \Gamma ({\mathbb E})$ satisfying

(2.10) $$ \begin{align} \begin{cases} E_+R={\operatorname{Id}} & \text{in } M \\ T R=0 & \text{on } {\partial M} \end{cases} \end{align} $$

(i.e., a solution operator for a pseudodifferential boundary-value problem

$$\begin{align*}\begin{cases} E_+ \psi= \rho & \text{in } M \\ T \psi= 0 & \text{on } {\partial M}, \end{cases} \end{align*}$$

with $\rho \in \Gamma ({\mathbb F})$ ). Since $E\in \mathrm {OP}(\mathfrak {A}^m)$ is elliptic, it has a parametrix $P\in \mathrm {OP}(\mathfrak {A}^{-m})$ . However, its truncation $P_+$ is generally not useful for finding R for two reasons: first, the boundary condition $T R=0$ has to be taken into account; second, for general $A,Q\in \mathrm {OP}(\mathfrak {A})$ ,

(2.11) $$ \begin{align} (AQ)_+-A_+Q_+\ne0, \end{align} $$

and in particular, $E_+P_+\ne {\operatorname {Id}} $ (modulo a smoothing operator). In fact, the expression (2.11) is not necessarily the truncation of an element in $\mathrm {OP}(\mathfrak {A})$ , and hence is not subject to the theory surveyed in Section 2.2.

This motivates the introduction of an even larger class of operators, which allows among other things to classify operators such as $A_+Q_+$ , and solution operators for (2.10). This new class retains some of the desirable properties of pseudodifferential operators. The idea, originating in work by Boutet de Monvel [Reference Boutet de MonvelBou71], is to construct a class of operators representing boundary-value problems, which is closed under its own algebra (the so-called Boutet de Monvel algebra).

A Green operator of order $m\in {\mathbb R}$ and class $r\in {\mathbb Z}$ is a system of operators ${\mathcal {A}}$ , which can be written in matrix form as

(2.12) $$ \begin{align} {\mathcal{A}}=\begin{pmatrix} A_++G & K_1 \\ T & K_2 \end{pmatrix}: \begin{pmatrix} \Gamma({\mathbb E}) \\ \Gamma({\mathbb J}) \end{pmatrix} \longrightarrow \begin{pmatrix} \Gamma({\mathbb F}) \\ \Gamma({\mathbb G}) \end{pmatrix}. \end{align} $$

Here, $A\in \mathrm {OP}(\mathfrak {A}^m)$ , $T\in \mathrm {OP}(\mathfrak {T}^{m-1,r})$ and $K_2\in L^m_{\mathrm {cl}}({\partial M},{\mathbb J},{\mathbb G})$ , which all belong to classes of operators that have already been introduced. The operator $K_1$ is known as a potential operator (of order m); it maps boundary sections into interior sections. The operator G is known as a singular Green operator. Singular Green operators are non-pseudodifferential operators, which are associated with a principal symbol much like pseudodifferential operators [Reference GrubbGru96, pp. 30–32]. They can also be characterized as classical in the sense of possessing an asymptotic expansion of homogeneous terms. Just like trace operators, they possess both an order and a class. They were introduced in order to obtain good composition rules (e.g., to rectify elements such as $A_+Q_+$ and possibly their approximate inverses). Specifically, if $A\in \mathrm {OP}(\mathfrak {A}^{m_A})$ and $Q\in \mathrm {OP}(\mathfrak {A}^{m_Q})$ , then $(AQ)_+ - A_+Q_+$ is a singular Green operator of order $m_A + m_Q - 1$ and class $m_Q$ [Reference Rempel and SchulzeRS82, p. 152].

The singular Green operator G in (2.12) is assumed to be of order $m-1$ and class $r\in {\mathbb Z}$ , in which case [Reference GrubbGru90, p 312],

(2.13) $$ \begin{align} G:W^{s,p}\Gamma({\mathbb E})\to W^{s-m,p}\Gamma({\mathbb F}) \end{align} $$

for every ${\mathbb Z}\ni s> r +1/p-1$ and $1<p<\infty $ . If $r=0$ , since a singular Green operator is $L^p$ -continuous, it has an adjoint of the same order [Reference GrubbGru96, p. 32]. For $r>0$ , this is, however, not true, so we have to keep track of the class of singular Green operators as we compose them with other operators.

Green operators of the form (2.12) satisfy the following mapping properties: If ${\mathcal {A}}$ is of order m and class r, then

(2.14) $$ \begin{align} {\mathcal{A}}: \begin{pmatrix} W^{s,p}\Gamma({\mathbb E}) \\ W^{s+1-1/p,p}\Gamma({\mathbb J}) \end{pmatrix} \longrightarrow \begin{pmatrix} W^{s-m,p}\Gamma({\mathbb F}) \\ W^{s-m+1-1/p,p}\Gamma({\mathbb G}) \end{pmatrix} \end{align} $$

for every ${\mathbb Z}\ni s> r+1/p-1$ . Green operators are associated with a pair of symbols,

(2.15) $$ \begin{align} \sigma({\mathcal{A}}) = \sigma_M({\mathcal{A}})\oplus \sigma_{\partial M}({\mathcal{A}}), \end{align} $$

where $\sigma _M({\mathcal {A}})(x,\xi )=\sigma _A(x,\xi ):{\mathbb E}_x\to {\mathbb F}_x$ , which is defined for every $x\in M$ and $\xi \in T_x^*M$ , is the interior symbol of $A\in \mathrm {OP}(\mathfrak {A})$ , and $\sigma _{\partial M}({\mathcal {A}})(x,\xi ')$ , which is defined for every $x\in {\partial M}$ and $\xi '\in T_x^*{\partial M}$ , is the boundary symbol of ${\mathcal {A}}$ ; the latter is a continuous linear map

(2.16) $$ \begin{align} \begin{aligned} &\sigma_{\partial M}({\mathcal{A}})(x,\xi'):\begin{pmatrix} \mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb E}_x) \\ {\mathbb C}\otimes {\mathbb J}_x \end{pmatrix}\longrightarrow \begin{pmatrix} \mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb F}_x) \\ {\mathbb C}\otimes {\mathbb G}_x \end{pmatrix}, \end{aligned} \end{align} $$

where for a vector bundle ${\mathbb U}\to M$ , $\mathscr {S}(\mkern 1.5mu\overline {\mkern -1.5mu{\mathbb {R}}\mkern -1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes {\mathbb U}_x)$ denotes the space of ${\mathbb C}\otimes {\mathbb U}_x$ -valued Schwartz functions on the half line $\mkern 1.5mu\overline {\mkern -1.5mu{\mathbb {R}}\mkern -1.5mu}\mkern 1.5mu_+=\left \{s\in {\mathbb {R}}~:~s\geq 0\right \}$ . We shall elaborate below upon how one obtains the map (2.16) from the Green operator ${\mathcal {A}}$ when both $K_1$ and $K_2$ are zero, in which case its domain is just $\mathscr {S}(\mkern 1.5mu\overline {\mkern -1.5mu{\mathbb {R}}\mkern -1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes {\mathbb E}_x)$ . A general definition is found in [Reference GrubbGru96, pp. 23–34].

Green operators form an algebra closed under composition, with their symbols satisfying a homomorphism property [Reference Rempel and SchulzeRS82, p. 175]:

Theorem 2.3. Let ${\mathcal {A}},{\mathcal {Q}}$ be Green operators of orders $m_A$ , $m_Q$ and classes $r_A$ , $r_Q$ . Then ${\mathcal {Q}}{\mathcal {A}}$ is a Green operator of order $m_A+m_Q$ and class $\max (m_A+r_Q,r_A)$ . The symbol of ${\mathcal {Q}}{\mathcal {A}}$ is given by

$$\begin{align*}\sigma({\mathcal{Q}}{\mathcal{A}})=\sigma({\mathcal{Q}})\circ\sigma({\mathcal{A}}) = (\sigma_M({\mathcal{Q}})\circ\sigma_M({\mathcal{A}})) \oplus (\sigma_{\partial M}({\mathcal{Q}})\circ\sigma_{\partial M}({\mathcal{A}})). \end{align*}$$

Moreover, if ${\mathcal {A}}$ is a Green operator of order m and ${\mathcal {Q}}$ is a Green operator of order $<m$ , then,

(2.17) $$ \begin{align} \sigma({\mathcal{A}}+{\mathcal{Q}})=\sigma({\mathcal{A}}). \end{align} $$

A Green operator ${\mathcal {A}}$ is called elliptic when $\sigma ({\mathcal {A}})$ is invertible. It should be noted that the notation $\sigma _M({\mathcal {A}})\oplus \sigma _{\partial M}({\mathcal {A}})$ is formal; the invertibility of the symbol amount to the separate invertibility of each component. Generalizing elliptic pseudodifferential operators on a closed manifold, an elliptic Green operator ${\mathcal {A}}$ benefits from the existence of a parametrix in the calculus, such that ${\mathcal {A}}{\mathcal {P}}-{\operatorname {Id}}$ and ${\mathcal {P}}{\mathcal {A}}-{\operatorname {Id}}$ are both Green operators of order $-\infty $ . If ${\mathcal {A}}$ is of order m and class r, then its parametrix ${\mathcal {P}}$ is of order $-m$ and class $r-m$ [Reference GrubbGru90, pp. 335–336].

Property (2.17) of the symbol raises a problem when considering systems of trace operators $T=T_0\oplus T_1\oplus \dots \oplus T_{m-1}$ associated with order m, since by the definition of the symbol, the only contribution to $\sigma (T)$ is that of $\sigma (T_{m-1})$ . The notion of ellipticity can be extended to encapsulate operators decomposing into direct sums ${\mathcal {E}}=\oplus _i {\mathcal {E}}_i$ and $T=\oplus _i T_i$ of operators having different orders (such systems are known as Douglis-Nirenberg boundary-value problems). The inclusion of such systems within the elliptic theory is justified by an order reduction argument, which will be elaborated below.

Consider the upper left term, $E_++G$ , of the Green operator. We denote all operators of this form where $E\in \mathrm {OP}(\mathfrak {A}^m)$ and G is a singular Green operator of order $m-1$ and class r by $\mathrm {OP}(\mathfrak {S}^{m,r})$ . We further introduce the class

$$\begin{align*}\mathrm{OP}(\mathfrak{S})=\bigcup_{m\in{\mathbb Z}, r\geq 0}\mathrm{OP}(\mathfrak{S}^{m,r}). \end{align*}$$

Note that an element $E_++G\in \mathrm {OP}(\mathfrak {S})$ can be identified with the Green operator

$$\begin{align*}{\mathcal{A}} = \begin{pmatrix} E_++G & 0 \\ 0 & 0 \end{pmatrix} : \begin{pmatrix} \Gamma({\mathbb E}) \\ 0 \end{pmatrix} \to \begin{pmatrix} \Gamma({\mathbb F}) \\ 0 \end{pmatrix}. \end{align*}$$

For conciseness, whenever there are no other nonzero entries, we will henceforth write ${\mathcal {A}} = E_++G$ .

We set $\mathrm {OP}(\mathfrak {S}^{-\infty ,r})=\bigcap _{m\in {\mathbb Z}} \mathrm {OP}(\mathfrak {S}^{m,r})$ as the operator class of smoothing operators of class r and set $\mathrm {OP}(\mathfrak {S}^{-\infty })=\bigcup _{r\geq 0}\mathrm {OP}(\mathfrak {S}^{-\infty ,r})$ [Reference Rempel and SchulzeRS82, p. 171]. As stated in the last reference, mappings in these classes map distributive sections into smooth ones and, as such, are always compact. When it comes to composition, since an $\mathrm {OP}(\mathfrak {S})$ operator can be viewed as a Green operator with all other terms equal zero, Theorem 2.3 implies the following:

Operators in $\mathrm {OP}(\mathfrak {S})$ benefit from Sobolev mapping properties with respect to their order and class, as inherited from (2.14). In particular, the mapping properties of operators in $\mathrm {OP}(\mathfrak {S})$ are limited by their class. Most importantly, for $r>0$ , elements in $\mathrm {OP}(\mathfrak {S}^{m,r})$ are not $L^p$ -continuous, and hence do not admit adjoints. Since $\mathrm {OP}(\mathfrak {A})$ operators are $L^p$ -continuous, this failure is due to the singular Green part. In fact (see the sharpness comment in [Reference GrubbGru90, p. 312]),

Proof. The $L^p$ -continuity for $r\le 0$ follows from the mapping property (2.14). In the other direction, suppose that ${\mathcal {E}}=E_++G\in \mathrm {OP}(\mathfrak {S}^{m,r})$ is $L^p$ -continuous. Since $E_+$ is $L^p$ -continuous, it follows that G is $L^p$ -continuous. By [Reference GrubbGru90, p. 306], every singular Green operator of order m and class r can be written as

$$\begin{align*}G=\sum_{j=0}^{r-1} K_jD^j_{\mathfrak{n}}+G', \end{align*}$$

where $G'$ is a singular Green operator of order m and class $\leq 0$ and $K_j$ are potential operators of order $m-j$ (the precise definition of $K_j$ is immaterial here).

Let $\psi $ be smooth, and let $\psi _n$ be a sequence of smooth, compactly supported sections converging to $\psi $ in $L^p$ . Since G is $L^p$ -continuous, and since $D^j_{\mathfrak {n}}\psi _n=0$ (as $\psi _n$ is compactly supported and $D^j_{\mathfrak {n}}$ are traces) for every j and n, it follows that

$$\begin{align*}G'\psi_n = G\psi_n \to G\psi \qquad\text{in } W^{-m,p}. \end{align*}$$

Since $G'$ has class zero, it follows from the mapping property (2.14) that

$$\begin{align*}G'\psi_n \to G'\psi \qquad\text{in } W^{-m,p}, \end{align*}$$

from which we conclude that $G = G'$ , and hence $r\leq 0$ .

A very important case is ${\mathcal {E}}\in \mathrm {OP}(\mathfrak {S}^{0,0})$ , in which case it is $L^p$ -continuous and as such has an adjoint ${\mathcal {E}}^*$ of the same order and class [Reference Rempel and SchulzeRS82, pp. 175–176]. We introduce the notation,

$$\begin{align*}\mathfrak{G}^0=\mathrm{OP}(\mathfrak{S}^{0,0}). \end{align*}$$

By Proposition 2.4, $\mathfrak {G}^0$ is also closed under compositions.

Consider now Green operators of the form

$$\begin{align*}\begin{pmatrix} {\mathcal{E}} & 0 \\ T & 0 \end{pmatrix}, \end{align*}$$

with ${\mathcal {E}} = E_+ + G$ , which for typographical reasons we denote $({\mathcal {E}},T)$ . As stated above, the symbol of $({\mathcal {E}},T)$ decomposes into

$$\begin{align*}\sigma({\mathcal{E}},T) = \sigma_M({\mathcal{E}},T) \oplus \sigma_{\partial M}({\mathcal{E}},T), \end{align*}$$

where $\sigma _M({\mathcal {E}},T)(x,\xi )=\sigma _E(x,\xi )$ . More specifically, we consider the case where

1. (a) the order and the class of G are strictly less than $m-1$ .

2. (b) the order of the $Q_+$ component of T in (2.6) is strictly less than m.

In this case, G and $Q_+$ do not contribute to the boundary symbol $\sigma _{\partial M}({\mathcal {E}},T)$ , which is only determined by the pseudodifferential operators E and $S_j$ : Let $x\in {\partial M}$ , and write $\xi \in T^*_xM$ in the form $\xi =\xi '+\xi _d\,dr$ , where $\xi '\in T_x^*{\partial M}$ and $dr$ is the unit covector normal to the boundary, so that $\xi _d\in {\mathbb {R}}$ is the normal component of $\xi $ . Consider the map

$$\begin{align*}\begin{pmatrix} \sigma_E(x,\xi'+\xi_d\,dr) \\ \sigma_T(x,\xi'+\xi_d\,dr) \end{pmatrix}:{\mathbb E}_x\longrightarrow \begin{pmatrix} {\mathbb F}_x \\ {\mathbb G}_x \end{pmatrix}, \end{align*}$$

where $\sigma _T(x,\xi '+\xi _ddr)$ is obtained from (2.6) by [Reference GrubbGru96, p. 27]

$$\begin{align*}\sigma_T(x,\xi'+\xi_ddr) = \sum_{0\leq j<m} \xi_d^j\, \sigma_{S_j}(x,\xi'). \end{align*}$$

(Since $S_j\in L^{m-j}_{\mathrm {cl}}({\partial M},\jmath ^*{\mathbb E},{\mathbb G})$ , it follows that $\sigma _{S_j}(x,\xi '):{\mathbb E}_x\to {\mathbb G}_x$ for $x\in {\partial M}$ and $\xi '\in T_x^*{\partial M}$ .) If one considers $\xi _d\in {\mathbb {R}}$ as an independent variable, then this map can be extended to operate on complexified vector-valued functions,

$$\begin{align*}F: \operatorname{Func}({\mathbb{R}};{\mathbb C}\otimes{\mathbb E}_x) \to \operatorname{Func}\left({\mathbb{R}};\begin{pmatrix} {\mathbb C}\otimes{\mathbb F}_x \\ {\mathbb C}\otimes{\mathbb G}_x \end{pmatrix}\right), \end{align*}$$

given by

$$\begin{align*}F(\psi)(t) = \begin{pmatrix} \sigma_E(x,\xi'+t\,dr)\psi(t) \\ \sigma_T(x,\xi'+t\,dr)\psi(t) \end{pmatrix}. \end{align*}$$

We then perform, formally, a one-dimensional Fourier transform, replacing $t \mapsto \iota \partial _s$ . This yields a differential map, $\hat {F}$ , given by

$$\begin{align*}\hat{F}(\psi)(s) = \begin{pmatrix} \sigma_E(x,\xi'+\imath\partial_s\,dr)\psi(s) \\ \sigma_T(x,\xi'+\imath\partial_s\,dr)\psi(s) \end{pmatrix}. \end{align*}$$

This map can be restricted to one-sided Schwartz functions, yielding a map

$$\begin{align*}\hat{F} : \mathscr{S}(\overline{{\mathbb{R}}}_+;{\mathbb C}\otimes{\mathbb E}_x) \to \mathscr{S}\left(\overline{{\mathbb{R}}}_+;\begin{pmatrix} {\mathbb C}\otimes{\mathbb F}_x \\ {\mathbb C}\otimes{\mathbb G}_x \end{pmatrix}\right). \end{align*}$$

The boundary symbol of $({\mathcal {E}},T)$ is the map

$$\begin{align*}\sigma_{\partial M}({\mathcal{E}},T)(x,\xi'):\mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb E}_x)\longrightarrow \begin{pmatrix} \mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb F}_x)\\{\mathbb C}\otimes{\mathbb G}_x \end{pmatrix}, \end{align*}$$

given by

$$\begin{align*}\sigma_{\partial M}({\mathcal{E}},T)(x,\xi') \psi = \begin{pmatrix} \{s\mapsto \sigma_E(x,\xi'+\iota\partial_s\,dr)\psi(s)\} \\ \sigma_T(x,\xi'+\iota\partial_s\,dr)\psi(0) \end{pmatrix}. \end{align*}$$

The notion of ellipticity for Green operators $({\mathcal {E}},T)$ reduces to two ingredients: the ellipticity of $E\in \mathrm {OP}(\mathfrak {A})$ as a pseudodifferential operator over $\tilde {M}$ , supplemented by the requirement that $\sigma _{\partial M}({\mathcal {E}},T)(x,\xi ')$ be a bijection [Reference GrubbGru96, p. 34]. This condition generalizes the classical Lopatinskii-Shapiro condition, which is a sufficient condition for differential systems. Indeed, when ${\mathcal {E}}$ and T are differential operators, then the boundary symbol $\sigma _{\partial M}({\mathcal {E}},T)(x,\xi ')$ can be viewed as the restriction of an ordinary differential operator,

$$\begin{align*}\sigma_E(x,\xi'+\iota\partial_s\,dr):C^{\infty}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb E}_x)\to C^{\infty}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb F}_x), \end{align*}$$

to Schwartz functions, supplemented by an initial condition map

$$\begin{align*}\Xi_{x,\xi'} = \sigma_T(x,\xi'+\iota\partial_s\,dr)|_{s=0}:C^{\infty}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb E}_x)\to {\mathbb C}\otimes{\mathbb G}_x. \end{align*}$$

These mappings coincide with the ones defined in [Reference HörmanderHör94, pp. 233–234] in the statement of the classical Lopatinskii-Shapiro condition. The following proposition demonstrates how the invertibility of $\sigma _{\partial M}({\mathcal {E}},T)(x,\xi ')$ extends this condition:

Proposition 2.6. Consider a system $({\mathcal {E}},T)$ , where E and T are differential operators. Suppose that $\sigma _E(x,\xi ):{\mathbb E}_x\to {\mathbb F}_x$ is injective for every $\xi \in T^*_xM\setminus \{0\}$ . For $\xi '\in T_x^*{\partial M}$ , let ${\mathbb M}_{x,\xi '}^+\subset C^{\infty }(\mkern 1.5mu\overline {\mkern -1.5mu{\mathbb {R}}\mkern -1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes {\mathbb E}_x)$ denote the space of decaying solutions of the linear ${\mathbb C}\otimes {\mathbb E}_x$ -valued ordinary differential equation

(2.18) $$ \begin{align} \sigma_E(x,\xi'+\iota\partial_s\,dr)\psi(s)=0. \end{align} $$

Then, the boundary symbol $\sigma _{\partial M}({\mathcal {E}},T)(x,\xi ')$ is injective if and only if the restriction of $\Xi _{x,\xi '}$ to ${\mathbb M}_{x,\xi '}^+$ is injective. If, in addition, $\dim {\mathbb E}_x=\dim {\mathbb F}_x$ , then the boundary symbol $\sigma _{\partial M}({\mathcal {E}},T)(x,\xi ')$ is a bijection if and only if $\dim _{{\mathbb C}}{\mathbb M}_{x,\xi '}^+=\dim {\mathbb G}_x$ .

2.4 Overdetermined elliptic systems

In the sequel, we invoke a degenerate form of ellipticity of Green operators [Reference Rempel and SchulzeRS82, p. 237], [Reference GrubbGru90, p. 315]:

By (2.15), the OD ellipticity of ${\mathcal {A}}$ amounts to the injectivity of the interior symbol $\sigma _A(x,\xi )$ for every $x\in M$ and $\xi \in T_x^*M\setminus \{0\}$ , and the injectivity of $\sigma _{\partial M}({\mathcal {A}})(x,\xi ')$ for every $x\in {\partial M}$ and $\xi '\in T_x^*{\partial M}\setminus \{0\}$ . By Proposition 2.6,

OD elliptic Green operators of the form $({\mathcal {E}},T)$ imply a priori Sobolev estimates and the existence of left-inverses [Reference GrubbGru90, pp. 335–336]:

Proposition 2.9. Let $({\mathcal {E}},T)$ be OD elliptic of order $m\in {\mathbb Z}$ and class $r\in {\mathbb Z}$ . Then there exists an a priori estimate

(2.19) $$ \begin{align} \|\psi\|_{s,p}\lesssim\|{\mathcal{E}}\psi\|_{s-m,p} + \|T\psi\|_{s-m+1-1/p,p}+ \|\psi\|_{0,p} \end{align} $$

for every ${\mathbb Z}\ni s> r +1-1/p$ . In particular, $\ker ({\mathcal {E}},T)\subseteq W^{s,p}\Gamma ({\mathbb E})$ is finite-dimensional, is independent of $s,p $ and consists of smooth sections. If, furthermore, $({\mathcal {E}},T)$ is injective, then it has a left-inverse of order $-m$ and class $r-m$ within the calculus of Green operators.

Note that OD ellipticity is defined as the injectivity of the symbol, which only implies the existence of an approximate left-inverse, yielding the a priori estimate. The injectivity of the operator is a stronger requirement.

Since $\ker ({\mathcal {E}},T)\subseteq L^2\Gamma ({\mathbb E}_k)$ , it admits an $L^2$ -orthogonal projection onto it, which we denote by ${\mathcal {S}}:L^2\Gamma ({\mathbb E})\to L^2\Gamma ({\mathbb E})$ . Since $\ker ({\mathcal {E}},T)$ consists solely of smooth sections, ${\mathcal {S}}:\Gamma ({\mathbb E})\to \Gamma ({\mathbb E})$ continuously as an integral operator with a smooth integral kernel. As such, ${\mathcal {S}}\in \mathrm {OP}(\mathfrak {S}^{-\infty })$ [Reference Rempel and SchulzeRS82, p. 196], and by continuity, the projection ${\mathcal {S}}:W^{s,p}\Gamma ({\mathbb E})\to W^{s,p}\Gamma ({\mathbb E})$ is a compact operator with a finite-dimensional range, ${\operatorname {Range}}({\mathcal {S}}) = \ker ({\mathcal {E}},T)\subseteq W^{s,p}\Gamma ({\mathbb E})$ . A standard procedure, using the finite-dimensionality of $\ker ({\mathcal {E}},T)$ and the Rellich compact embedding theorem [Reference BrezisBre11, p. 51] yields a refinement of the estimate (2.19), replacing $\|\psi \|_{0,p}$ by $\|{\mathcal {S}}\psi \|_{0,p}$ ,

$$\begin{align*}\|\psi\|_{s,p} \lesssim \|{\mathcal{E}}\psi\|_{s-m,p} + \|T\psi\|_{s-m+1-1/p,p} + \|{\mathcal{S}}\psi\|_{0,p} \end{align*}$$

for every ${\mathbb Z}\ni s> r +1- 1/p$ .

We next focus on systems ${\mathcal {E}}=\oplus _{i=1}^l{\mathcal {E}}_i$ and $T=\oplus _{j=0}^{m-1}T_j$ , where ${\mathcal {E}}_i$ and $T_j$ are of varying orders. This extension uses the so-called ‘simple reduction of order’ [Reference Rempel and SchulzeRS82, p. 208, pp. 234–235]; we provide all the details for self-containment.

We start with the boundary operators $T_j$ : Since ${\partial M}$ is a closed Riemannian manifold, there exists for every vector bundle ${\mathbb S}\to {\partial M}$ and every $t\in {\mathbb Z}$ an invertible pseudodifferential operator ${\mathcal {L}}_{{\mathbb S}}^t\in L^t_{\mathrm {cl}}({\partial M},{\mathbb S},{\mathbb S})$ , with inverse within the calculus (e.g., $({\operatorname {Id}} + \nabla ^*\nabla )^{t/2}$ , where $\nabla $ is any Riemannian connection on ${\mathbb S}$ ). Due to the mapping property (2.3), this operator extends to an isomorphism

$$\begin{align*}{\mathcal{L}}_{{\mathbb S}}^t:W^{s,p}\Gamma({\mathbb S})\to W^{s-t,p}\Gamma({\mathbb S}) \end{align*}$$

for every $s\in {\mathbb Z}$ and $1<p<\infty $ . On a vector bundle ${\mathbb U}\to M$ over a compact manifold with boundary, the existence of such an isomorphism is not trivial. This fact is proved in [Reference GrubbGru90, Secs. 4,5]: for every $m>0$ , there exists an $\mathrm {OP}(\mathfrak {S}^{m,0})$ operator ${\mathcal {L}}_{\mathbb U}^m$ , which extends to an isomorphism $W^{s,p}\Gamma ({\mathbb U})\to W^{s-m,p}\Gamma ({\mathbb U})$ for every $s>1/p-1$ . Its inverse is an $\mathrm {OP}(\mathfrak {S}^{-m,0})$ operator.

With these noted, the following is an adaptation of the construction in [Reference Rempel and SchulzeRS82, pp. 234–235] for elliptic systems of varying orders to OD systems of varying orders. This is essentially what is done in [Reference GrubbGru90, pp. 331–338], restated here to better suit the framework we develop later:

Proposition 2.11. Let $({\mathcal {E}},T)$ be OD elliptic of varying orders. Then there is an a priori estimate,

(2.20) $$ \begin{align} \|\psi\|_{s,p}\lesssim\sum_{i=1}^l\|{\mathcal{E}}_i\psi\|_{s-m_i,p}+\sum_{j=0}^q\|T_j\psi\|_{s-\gamma_j-1/p,p} + \|{\mathcal{S}}\psi\|_{0,p}, \end{align} $$

for every ${\mathbb Z}\ni s> r + 1/p-1$ , and $1<p<\infty $ . Here, ${\mathcal {S}}\in \mathrm {OP}(\mathfrak {S}^{-\infty })$ is the $L^2$ -orthogonal projection onto the finite-dimensional space $\ker ({\mathcal {E}},T)\subseteq W^{s,p}\Gamma ({\mathbb E})$ , which is independent of $s,p$ and consists of smooth sections. If, furthermore, $({\mathcal {E}},T)$ is injective, then it has a left-inverse of order $-m$ and class $r-m$ within the calculus of Green operators. Conversely, if $({\mathcal {E}},T)$ has a left-inverse of order $-m$ and class $r-m$ , then it is injective and OD elliptic of varying orders, with maximal order m and maximal class r.

Proof. Using the order reduction operators ${\mathcal {L}}_{{\mathbb F}_i}^a$ , ${\mathcal {L}}_{{\mathbb E}}^a$ and ${\mathcal {L}}_{{\mathbb J}_i}^a$ , consider the modified system:

$$\begin{align*}\begin{aligned} &\tilde{{\mathcal{E}}} = \bigoplus_{i=1}^l {\mathcal{L}}^{m-m_i}_{{\mathbb F}_i}{\mathcal{E}}_i {\mathcal{L}}_{{\mathbb E}}^{-m} \\ & \tilde{T}= \bigoplus_{j=0}^q {\mathcal{L}}^{m-\gamma_j-1}_{{\mathbb J}_j}T_j {\mathcal{L}}_{{\mathbb E}}^{-m}. \end{aligned} \end{align*}$$

By the homomorphism property of the symbols of Green operators,

$$\begin{align*}\begin{aligned} \sigma(\tilde{{\mathcal{E}}},\tilde{T}) &= \begin{pmatrix} \bigoplus_{i=1}^l \sigma({\mathcal{L}}^{m-m_i}_{{\mathbb F}_i}{\mathcal{E}}_i {\mathcal{L}}_{{\mathbb E}}^{-m},0) \\ \bigoplus_{j=0}^{m-1} \sigma(0,{\mathcal{L}}^{m-j-1}_{{\mathbb J}_j}T_j {\mathcal{L}}_{{\mathbb E}}^{-m}) \end{pmatrix} \\ &= \begin{pmatrix} \bigoplus_{i=1}^l \sigma({\mathcal{L}}^{m-m_i}_{{\mathbb F}_i}) \circ \sigma({\mathcal{E}}_i) \\ \bigoplus_{j=0}^{m-1} \sigma({\mathcal{L}}^{m-j-1}_{{\mathbb J}_j}) \circ \sigma(T_j ) \end{pmatrix}\circ \sigma({\mathcal{L}}_{{\mathbb E}}^{-m}). \end{aligned} \end{align*}$$

It is a straightforward yet tedious calculation to show that the fact that symbols of the order reduction operators are isomorphisms, and the fact that the direct sum of the symbols $\sigma ({\mathcal {E}}_i,0)$ and $\sigma (0,T_j)$ is injective, implies that $\sigma (\tilde {{\mathcal {E}}},\tilde {T})$ is injective. We conduct it only for the interior symbol, as the argument for the boundary symbol follows the same lines.

Let $x\in M$ , $\xi \in T_x^*M\setminus \{0\}$ , and suppose that $\psi \in \ker \sigma _M(\tilde {{\mathcal {E}}},\tilde {T})$ ; that is,

$$\begin{align*}\bigoplus_{i=1}^l \sigma_M({\mathcal{L}}^{m-m_i}_{{\mathbb F}_i})(x,\xi) \circ \sigma_M({\mathcal{E}}_i)(x,\xi) \circ \sigma_M({\mathcal{L}}_{{\mathbb E}}^{-m})(x,\xi) \psi = 0. \end{align*}$$

By the assumptions, $\sigma _M({\mathcal {E}}_i) = \sigma _{E_i}$ , and hence, the above identity for $\psi $ reads

$$\begin{align*}\bigoplus_{i=1}^l \sigma_M({\mathcal{L}}^{m-m_i}_{{\mathbb F}_i})(x,\xi) \circ \sigma_{E_i}(x,\xi) \circ \sigma_M({\mathcal{L}}^{-m}_{{\mathbb E}})(x,\xi) \psi=0. \end{align*}$$

This implies

$$\begin{align*}\sigma_M({\mathcal{L}}^{m-m_i}_{{\mathbb F}_i})(x,\xi) \circ \sigma_{E_i}(x,\xi) \left(\sigma_M({\mathcal{L}}^{-m}_{{\mathbb E}})(x,\xi) \psi\right) = 0 \qquad \text{for every } i. \end{align*}$$

Since $\sigma _M({\mathcal {L}}^{m-m_i}_{{\mathbb F}_i})(x,\xi )$ is bijective,

$$\begin{align*}\sigma_{E_i}(x,\xi) \left(\sigma_M({\mathcal{L}}^{-m}_{{\mathbb E}})(x,\xi) \psi\right) = 0 \qquad \text{for every } i. \end{align*}$$

Since $\bigoplus _{i=1}^l \sigma _{E_i}(x,\xi )$ is injective, it follows that

$$\begin{align*}\sigma_M({\mathcal{L}}^{-m}_{{\mathbb E}})(x,\xi) \psi = 0, \end{align*}$$

and since $\sigma _M({\mathcal {L}}^{-m}_{{\mathbb E}})(x,\xi )$ is bijective, $\psi =0$ , thus proving that $\sigma _{\tilde {E}}(x,\xi )$ is injective.

The fact that the kernel of the problem is finite-dimensional and admits a left-inverse follows from Proposition 2.9, by a simple composition with isomorphisms. We turn to the a priori estimate. Since $(\tilde {{\mathcal {E}}},\tilde {T})$ is of order zero and class r, the a priori estimate (2.19) assumes the form

$$\begin{align*}\|\psi\|_{s,p}\lesssim\sum_{i=1}^l \|{\mathcal{L}}^{m-m_i}_{{\mathbb F}_i}{\mathcal{E}}_i {\mathcal{L}}_{{\mathbb E}}^{-m}\psi\|_{s,p} +\sum_{j=0}^q\|{\mathcal{L}}^{m-\gamma_j-1}_{{\mathbb J}_j}T_j {\mathcal{L}}_{{\mathbb E}}^{-m}\psi\|_{s+1-1/p,p}+\|\tilde{{\mathcal{S}}}\psi\|_{0,p} \end{align*}$$

for every ${\mathbb Z}\ni s> r + 1/p-1$ , where $\tilde {{\mathcal {S}}}\in \mathrm {OP}(\mathfrak {S}^{-\infty })$ is the projection onto $\ker (\tilde {{\mathcal {E}}},\tilde {T})$ . By the isomorphism property of ${\mathcal {L}}_{{\mathbb E}}^{-m}$ ,

$$\begin{align*}\begin{aligned} \|\psi\|_{s,p} &\lesssim \|{\mathcal{L}}_{{\mathbb E}}^m\psi\|_{s-m,p} \\ & \lesssim \sum_{i=1}^l \|{\mathcal{L}}^{m-m_i}_{{\mathbb F}_i}{\mathcal{E}}_i\psi\|_{s-m,p} +\sum_{j=0}^q\|{\mathcal{L}}^{m-\gamma_j-1}_{{\mathbb J}_j}T_j \psi\|_{s-m+1-1/p,p}+\|\tilde{{\mathcal{S}}}{\mathcal{L}}_{{\mathbb E}}^m \psi\|_{0,p} \\ &\lesssim \sum_{i=1}^l \|{\mathcal{E}}_i\psi\|_{s-m_i,p} +\sum_{j=0}^q\|T_j \psi\|_{s-\gamma_j-1/p,p}+\|{\mathcal{S}}\psi\|_{m,p}. \end{aligned} \end{align*}$$

The replacement of the last term by $\|\psi \|_{0,p}$ follows from the equivalence of norms on a finite-dimensional vector space.

In the other direction, suppose that $({\mathcal {E}},T)$ has a left-inverse of order $-m$ and class $r-m$ . It is therefore injective. By using the order-reducing operators, we may assume that all the ${\mathcal {E}}_i$ and $T_j$ are of the same order. By the homomorphism property of the symbols, $\sigma ({\mathcal {E}},T)$ is invertible, and hence injective. The order and the classes are inferred by reverting the order reduction.

We next combine Proposition 2.6 and Proposition 2.11:

Proof. We argue that Items (a), (b) imply that $(E,T)$ satisfies the requirements of Definition 2.10 (i.e., that

$$\begin{align*}\bigoplus_{i=1}^l \sigma_{E_i}(x,\xi) : {\mathbb E}_x\to {\mathbb F}_x \end{align*}$$

and

$$\begin{align*}\begin{pmatrix} \bigoplus_{i=1}^l \sigma_{E_i}(x,\xi'+ \imath\partial_s\, dr) \\ \bigoplus_{j=0}^{m-1} \sigma_{T_j}(x,\xi'+ \imath\partial_s\, dr)|_{s=0} \end{pmatrix}: \mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb E}_x)\longrightarrow \begin{pmatrix} \mathscr{S}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu_+;{\mathbb C}\otimes{\mathbb F}_x)\\{\mathbb C}\otimes{\mathbb G}_x \end{pmatrix} \end{align*}$$

are injective). The first requirement is Item (a). The second requirement follows from Item (b), along with Corollary 2.8, which extends to operators in the form of direct sums.

3.1 Adapted Green operators and auxiliary decompositions

In the sequel, we study the splitting of spaces of sections into ranges and kernels of $\mathrm {OP}(\mathfrak {S})$ operators, reminiscent of a Fredholm alternative, which occurs in many elliptic boundary-value problems. Such operators satisfy integration by parts formula with surjective boundary operators, due to the noncharacteristic property satisfied by elliptic problems ([Reference TaylorTay11a, p. 470] and [Reference GrubbGru96, Sec. 1.4]). This motivates the following definition:

We remark that if ${\mathcal {A}}=A$ is a differential operator, it obviously satisfies Item (a) in Definition 3.1, but not necessarily Item (b).

Due to the closure of both $\mathfrak {G}^0$ and the class of differential operators to adjoints, and the symmetry between the requirements on $B_A$ and $B_{A^*}$ , ${\mathcal {A}}$ is an adapted Green operator if and only if ${\mathcal {A}}^*$ is an adapted Green operator. In particular, the mapping properties of Green operators in $\mathfrak {G}^0$ implies that $G:L^2\Gamma ({\mathbb E})\to L^2\Gamma ({\mathbb F})$ , and

$$\begin{align*}\langle G\psi,\eta\rangle =\langle \psi,G^*\eta\rangle \end{align*}$$

for every $\psi \in \Gamma ({\mathbb E})$ and $\eta \in \Gamma ({\mathbb F})$ . Thus, ${\mathcal {A}}$ and its adjoint ${\mathcal {A}}^*$ inherit the integration by parts formula (2.9),

(3.1) $$ \begin{align} \langle {\mathcal{A}}\psi,\eta\rangle = \langle\psi,{\mathcal{A}}^*\eta\rangle + \langle B_A\psi, B_{A^*}\eta\rangle. \end{align} $$

We introduce several definitions associated with adapted Green operators: Let ${\mathcal {A}}\in \mathrm {OP}(\mathfrak {S}^{m,0})$ be an adapted Green operator. The Banach dual of ${\mathcal {A}}:W^{m,q}\Gamma ({\mathbb E})\to L^q\Gamma ({\mathbb F})$ is the operator ${\mathcal {A}}_p^{\prime }:L^p\Gamma ({\mathbb F})\to (W^{m,q}\Gamma ({\mathbb E}))^*$ given by the pairing

$$\begin{align*}{\mathcal{A}}_p^{\prime}\eta(\psi) = \langle \eta, {\mathcal{A}}\psi\rangle \end{align*}$$

for every $\eta \in L^p\Gamma ({\mathbb F})$ and $\psi \in W^{m,q}\Gamma ({\mathbb E})$ , where $1/p+1/q=1$ . The kernel of ${\mathcal {A}}_p^{\prime }$ is the closed subspace of $L^p\Gamma ({\mathbb F})$ consisting of sections $\eta $ satisfying

$$\begin{align*}\langle\eta, {\mathcal{A}}\psi \rangle = 0 \end{align*}$$

for every $\psi \in W^{m,q}\Gamma ({\mathbb E})$ . For $\eta \in \ker {\mathcal {A}}_p^{\prime } \cap W^{m,p}\Gamma ({\mathbb F})$ , it follows from the integration by parts formula (3.1) that

$$\begin{align*}\langle {\mathcal{A}}^*\eta,\psi\rangle+\langle B_A\psi, B_{A^*}\eta\rangle=0 \end{align*}$$

for every $\psi \in W^{m,q}\Gamma ({\mathbb E})$ . Taking $\psi $ compactly supported, using the fact that $B_A$ is a differential operator, and hence $B_A\psi =0$ , yields

$$\begin{align*}\langle {\mathcal{A}}^*\eta,\psi\rangle = 0 \end{align*}$$

for every $\psi \in W_0^{m,q}\Gamma ({\mathbb E})$ . Since $W_0^{m,q}\Gamma ({\mathbb E})$ is dense in $L^q\Gamma ({\mathbb E})$ , it follows that ${\mathcal {A}}^*\eta =0$ . Using the surjectivity of $B_A$ to prescribe $B_A\psi $ arbitrary yields $B_{A^*}\eta =0$ . Thus, $\ker {\mathcal {A}}_p^{\prime }\cap W^{m_A,p}\Gamma ({\mathbb F})$ coincides with the intersection of the kernels of ${\mathcal {A}}^*$ and $B_{A^*}$ . This motivates the following notation for the kernel of ${\mathcal {A}}_p^{\prime }$ ,

$$\begin{align*}\mathscr{N}^{0,p}({\mathcal{A}}^*,B_{A^*})=\ker {\mathcal{A}}_p^{\prime}. \end{align*}$$

In a similar fashion, consider the Sobolev space,

$$\begin{align*}W^{s,p}_A\Gamma({\mathbb E})= W^{s,p}\Gamma({\mathbb E})\cap\ker B_A, \end{align*}$$

for $s\ge m$ and $1<p<\infty $ . By (3.1),

$$\begin{align*}\langle {\mathcal{A}}\psi,\eta\rangle=\langle\psi,{\mathcal{A}}^*\eta\rangle \end{align*}$$

for every $\psi \in W^{m,p}_A\Gamma ({\mathbb E})$ and $\eta \in W^{m,q}\Gamma ({\mathbb F})$ . Consider the restriction of ${\mathcal {A}}$ to $\ker B_A$ ,

$$\begin{align*}{\mathcal{A}}|_{\ker B_A}: W^{m,q}_A\Gamma({\mathbb E}) \to L^q\Gamma({\mathbb F}). \end{align*}$$

By definition, its Banach adjoint

$$\begin{align*}{\mathcal{A}}^{\prime}_{p,A} : L^p\Gamma({\mathbb F}) \to (W^{m,q}_A\Gamma({\mathbb E}))^* \end{align*}$$

is given by the pairing

$$\begin{align*}{\mathcal{A}}^{\prime}_{p,A}\eta(\psi) = \langle \eta, {\mathcal{A}}\psi\rangle \qquad \text{for every } \eta\in L^p\Gamma({\mathbb F}) \text{ and } \psi\in W^{m,q}_A\Gamma({\mathbb E}), \end{align*}$$

where we used again the isomorphism $L^p\Gamma ({\mathbb E})\simeq (L^q\Gamma ({\mathbb F}))^*$ . The kernel of ${\mathcal {A}}^{\prime }_{p,A} $ is the closed subspace of $L^p\Gamma ({\mathbb F})$ consisting of sections $\eta $ satisfying

$$\begin{align*}\langle\eta, {\mathcal{A}}\psi \rangle = 0 \end{align*}$$

for every $\psi \in W^{m,q}_A\Gamma ({\mathbb E})$ . We denote this space by

(3.2) $$ \begin{align} \mathscr{N}^{0,p}({\mathcal{A}}^*)=\ker {\mathcal{A}}^{\prime}_{p,A}. \end{align} $$

Comparing with (3.1), $\mathscr {N}^{0,p}({\mathcal {A}}^*)\cap W^{m,p}\Gamma ({\mathbb F})$ coincides with the classical kernel of ${\mathcal {A}}^*: W^{m,p}\Gamma ({\mathbb F})\to L^p\Gamma ({\mathbb E})$ .

For the next definitions, we recall the mapping properties (2.4) and (2.13).

Definition 3.2. Let $s\in {{\mathbb N}_0}$ and $1<p<\infty $ , and let ${\mathcal {A}}\in \mathrm {OP}(\mathfrak {S}^{m,0})$ be an adapted Green operator $\Gamma ({\mathbb E})\to \Gamma ({\mathbb F})$ . We define the following subspaces of $W^{s,p}\Gamma ({\mathbb E})$ and $W^{s,p}\Gamma ({\mathbb F})$ ,

$$\begin{align*}\begin{aligned} &\mathscr{R}^{s,p}{({\mathcal{A}})} = {\mathcal{A}}({W^{s+m,p}\Gamma({\mathbb E})}) &\qquad &\mathscr{N}^{s,p}({\mathcal{A}}) = W^{s,p}\Gamma({\mathbb E}) \cap \mathscr{N}^{0,p}({\mathcal{A}}), \\ &\mathscr{R}^{s,p}{({\mathcal{A}};B_A)} = {\mathcal{A}}({W^{s+m,p}_A\Gamma({\mathbb E})}) &\qquad &\mathscr{N}^{s,p}({\mathcal{A}},B_A) = W^{s,p}\Gamma({\mathbb E}) \cap \mathscr{N}^{0,p}({\mathcal{A}},B_A), \end{aligned} \end{align*}$$

along with their smooth versions,

$$\begin{align*}\begin{aligned} &\mathscr{R}{({\mathcal{A}})} = {\mathcal{A}}(\Gamma({\mathbb E})) &\qquad &\mathscr{N}({\mathcal{A}}) = \Gamma({\mathbb E}) \cap \ker {\mathcal{A}} \\ &\mathscr{R}{({\mathcal{A}};B_A)} = {\mathcal{A}}(\Gamma({\mathbb E})\cap\ker B_A) &\qquad &\mathscr{N}({\mathcal{A}},B_A) = \Gamma({\mathbb E}) \cap \ker({\mathcal{A}} \oplus B_A). \end{aligned} \end{align*}$$

The closed range theorem asserts that for a bounded linear map $T:V\to W$ between Banach spaces, $\ker T' = (T(V))^\bot $ , where $\bot $ is the Banach annihilator functor [Reference TaylorTay11a, p. 575]. Applying this theorem to ${\mathcal {A}}:W^{m,q}\Gamma ({\mathbb E})\to L^q\Gamma ({\mathbb F})$ and its Banach dual ${\mathcal {A}}_p^{\prime }:L^p\Gamma ({\mathbb F})\to (W^{m,q}\Gamma ({\mathbb E}))^*$ yields

(3.3) $$ \begin{align} (\mathscr{R}^{0,q}({\mathcal{A}}))^{\bot}=\mathscr{N}^{0,p}({\mathcal{A}}^*,B_{A^*}). \end{align} $$

Similarly, applying to closed range theorem to ${\mathcal {A}}: W^{m,q}_A\Gamma ({\mathbb E}) \to L^q\Gamma ({\mathbb F})$ and its Banach adjoint ${\mathcal {A}}^{\prime }_{p,A} : L^p\Gamma ({\mathbb F}) \to (W^{m,q}_A\Gamma ({\mathbb E}))^*$ yields

(3.4) $$ \begin{align} (\mathscr{R}^{0,q}({\mathcal{A}};B_A))^{\bot}=\mathscr{N}^{0,p}({\mathcal{A}}^*). \end{align} $$

Adapted operators were introduced for the purpose of the following definition:

Equation (3.5) can be viewed as a Fredholm alternative induced by the (generally non-elliptic) operator ${\mathcal {A}}$ , supplemented with a clause establishing its connection with the calculus of Green operators. In this setting, the composition rules in Proposition 2.4 imply that ${\mathcal {A}}{\mathcal {P}}_{{\mathcal {A}}}\in \mathfrak {G}^0$ , and hence,

$$\begin{align*}{\mathcal{A}}{\mathcal{P}}_{{\mathcal{A}}}:W^{s,p}\Gamma({\mathbb F})\to W^{s,p}\Gamma({\mathbb F}) \end{align*}$$

continuously for every $s\in {{\mathbb N}_0}$ and $1<p<\infty $ . Since $\Gamma ({\mathbb F})$ is dense in $W^{s,p}\Gamma ({\mathbb F})$ and ${\mathcal {A}}{\mathcal {P}}_{{\mathcal {A}}}$ is a projection, by an approximation/continuity argument, we deduce the existence of a $W^{s,p}$ -direct decomposition,

(3.6) $$ \begin{align} W^{s,p}\Gamma({\mathbb F})=\mathscr{R}^{s,p}({\mathcal{A}})\oplus \mathscr{N}^{s,p}({\mathcal{A}}^*,B_{A^*}), \end{align} $$

for every such $s,p$ . Conversely, if (3.6) holds for every $s,p$ , then the smooth version (3.5) also holds.

The existence of an auxiliary decomposition implies in particular that $\mathscr {R}^{s,p}({\mathcal {A}})$ and $\mathscr {N}^{s,p}({\mathcal {A}}^*,B_{A^*})$ are closed subspaces of $W^{s,p}\Gamma ({\mathbb F})$ . For $s=0$ and $p=2$ , the closedness of the ranges suffices for the existence of an $L^2$ -direct decomposition, but this is not true for general $s,p$ :

Proposition 3.4. Let ${\mathcal {A}}\in \mathrm {OP}(\mathfrak {S}^{m,0})$ be an adapted Green operator. There exist $L^2$ -orthogonal decompositions

$$\begin{align*}\begin{aligned} &L^2\Gamma({\mathbb F}) = \overline{\mathscr{R}^{0,2}({\mathcal{A}})} \oplus \mathscr{N}^{0,2}({\mathcal{A}}^*,B_{A^*}) \\ & L^2\Gamma({\mathbb F}) = \overline{\mathscr{R}^{0,2}({\mathcal{A}};B_A)} \oplus \mathscr{N}^{0,2}({\mathcal{A}}^*), \end{aligned} \end{align*}$$

where the overline stands for the closure in the $L^2$ -norm.

Proof. In view of the isomorphism $L^2\Gamma ({\mathbb F})\simeq (L^2\Gamma ({\mathbb F}))^*$ , the Banach annihilator of a closed subspace coincides with its orthogonal complement. Thus, (3.3) reads

$$\begin{align*}\mathscr{N}^{0,2}({\mathcal{A}}^*,B_{A^*}) = (\mathscr{R}^{0,2}({\mathcal{A}}))^\bot. \end{align*}$$

Since every closed subspace of a Hilbert space induces an orthogonal decomposition, we obtain

$$\begin{align*}L^2\Gamma({\mathbb F}) = (\mathscr{R}^{0,2}({\mathcal{A}}))^\bot \oplus ((\mathscr{R}^{0,2}({\mathcal{A}}))^\bot)^\bot = \mathscr{N}^{0,2}({\mathcal{A}}^*,B_{A^*}) \oplus \overline{\mathscr{R}^{0,2}({\mathcal{A}})}. \end{align*}$$

The proof of the second clause follows similar lines, starting with (3.4).

A decomposition of a Banach space is topologically direct if it is algebraically direct and both subspaces are closed. In a Banach space, unlike in a Hilbert space, a closed subspace may fail to induce a direct decomposition. It induces a direct decomposition if and only if the closed subspace admits a continuous projection onto it. As will be seen below, auxiliary decompositions are a first step towards a more refined Hodge-like decomposition, whence the adjective auxiliary.

3.2 Elliptic pre-complexes

We consider chains of operators between spaces of sections, as depicted below:

where $(A_{\bullet }) = (A_k)_{k\in {{\mathbb N}_0}}$ is a sequence of adapted differential operators $A_k:\Gamma ({\mathbb E}_k)\to \Gamma ({\mathbb E}_{k+1})$ of orders $m_k$ . We denote the corresponding normal systems of trace operators by $B_k:\Gamma ({\mathbb E}_k)\to \Gamma ({\mathbb G}_k)$ and $B_k^*:\Gamma ({\mathbb E}_{k+1})\to \Gamma ({\mathbb G}_k)$ – namely,

$$\begin{align*}\langle A_k\psi,\eta\rangle = \langle\psi,A_k^*\eta\rangle + \langle B_k\psi, B_k^*\eta\rangle \end{align*}$$

for every $\psi \in \Gamma ({\mathbb E}_k)$ and $\eta \in \Gamma ({\mathbb E}_{k+1})$ .

There are several distinctions between this definition and the classical notion of an elliptic complex. Most prominently, elliptic complexes are algebraic complexes (i.e., $A_k A_{k-1}=0$ ). In many applications, however, $A_k A_{k-1}$ does not vanish, satisfying instead Condition (b). This order reduction enables the ‘correction’ of $(A_{\bullet })$ by lower-order terms into a complex, whence the terminology of a pre-complex.

Secondly, in classical elliptic complexes, ellipticity is usually a property of the ‘Laplacian’ system ([Reference TaylorTay11b, pp. 460–465] and [Reference Schulze and SeilerSS19]),

$$\begin{align*}\begin{pmatrix} A_{k-1}A_{k-1}^* + A_k^* A_k \\ B_{k-1}^*\oplus B_k^*A_k \end{pmatrix}. \end{align*}$$

The notion of OD ellipticity of varying orders replaces this ellipticity and enables the consideration of chains in which the operators are of variable order, verifiable by a relatively simple criterion (Theorem 2.12). Elliptic complexes of variable orders have been considered in the literature, in particular in closed manifolds where an order reduction argument can be applied with relative ease [Reference Rempel and SchulzeRS82, p. 279-280]. In manifolds with a boundary, however, the picture is more involved, and this is where Theorem 2.12 provides a significant simplification.

3.3 The induced elliptic complex

The next theorem is our main result concerning elliptic pre-complexes.

Note that by convention, $\mathscr {N}({\mathcal {A}}_{-1}^*,B_{-1}^*)=\Gamma ({\mathbb E}_0)$ , which forces ${\mathcal {A}}_0=A_0$ . Moreover,

Proposition 3.7. In the setting of Theorem 3.6, each ${\mathcal {A}}_k$ induces an auxiliary decomposition,

(3.7) $$ \begin{align} \Gamma({\mathbb E}_{k+1})=\mathscr{R}({\mathcal{A}}_k)\oplus \mathscr{N}({\mathcal{A}}_k^*,B_k^*). \end{align} $$

We denote by ${\mathcal {P}}_k = {\mathcal {P}}_{{\mathcal {A}}_k}: \Gamma ({\mathbb E}_{k+1})\to \Gamma ({\mathbb E}_k)$ the operator associated with this decomposition (see Definition 3.3) (i.e., ${\mathcal {P}}_k\in \mathrm {OP}(\mathfrak {S}^{-m_k,0})$ , and ${\mathcal {A}}_k{\mathcal {P}}_k\in \mathfrak {G}^0$ is the projection onto $\mathscr {R}({\mathcal {A}}_k)$ ).

The following proposition shows that the induced elliptic complex $({\mathcal {A}}_{\bullet })$ is a ‘correction’ of the elliptic pre-complex $(A_{\bullet })$ by lower-order terms and gives an explicit formula for the difference:

Proposition 3.8. In the setting of Theorem 3.6, the induced elliptic complex $({\mathcal {A}}_{\bullet })$ satisfies

$$\begin{align*}G_k= {\mathcal{A}}_k-A_k\in\mathfrak{G}^0 \qquad \text{for every } k, \end{align*}$$

with $G_k$ given by the recursive formula,

(3.8) $$ \begin{align} \begin{aligned} &G_0=0 \\ &G_k=-A_kA_{k-1}{\mathcal{P}}_{k-1}-A_kG_{k-1}{\mathcal{P}}_{k-1}. \end{aligned} \end{align} $$

It follows from Proposition 3.8 that ${\mathcal {A}}_k$ and ${\mathcal {A}}_k^*$ satisfy the same integration by parts formula as $A_k$ and $A_k^*$ ,

(3.9) $$ \begin{align} \langle {\mathcal{A}}_k\psi,\eta\rangle = \langle\psi,{\mathcal{A}}_k^*\eta\rangle + \langle B_k\psi, B_k^*\eta\rangle. \end{align} $$

Theorem 3.6, Proposition 3.7 and Proposition 3.8 are proved simultaneously by induction on k in Section 4, Moreover, since ${\mathcal {A}}_k-A_k\in \mathfrak {G}^0$ , the systems $({\mathcal {A}}_{k-1}^*\oplus {\mathcal {A}}_k,B_{k-1}^*)$ are also OD elliptic.

3.4 Applications of the induced elliptic complex

The defining properties of the induced elliptic complex $({\mathcal {A}}_{\bullet })$ imply the following additional properties:

The auxiliary decomposition refines into a Hodge-like decomposition:

Theorem 3.10 (Hodge-like decomposition).

In the setting of Theorem 3.6, there exists a $W^{s,p}$ -direct decomposition,

(3.10) $$ \begin{align} W^{s,p}\Gamma({\mathbb E}_k) = \mathscr{R}^{s,p}({\mathcal{A}}_{k-1})\oplus\mathscr{R}^{s,p}({\mathcal{A}}_k^*;B_k^*)\oplus \mathscr{H}^k({\mathcal{A}}_{\bullet}) \end{align} $$

for every $s\in {{\mathbb N}_0}$ and $1<p<\infty $ , where the subspace

$$\begin{align*}\mathscr{H}^k({\mathcal{A}}_{\bullet})=\ker({\mathcal{A}}_k\oplus{\mathcal{A}}_{k-1}^*\oplus B_{k-1}^*) \end{align*}$$

is finite-dimensional, independent of s and p, and consists of smooth sections. In particular, comparing with the auxiliary decomposition (3.7),

(3.11) $$ \begin{align} \mathscr{N}^{s,p}({\mathcal{A}}_{k-1}^*,B_{k-1}^*)=\mathscr{R}^{s,p}({\mathcal{A}}_k^*;B_k^*)\oplus \mathscr{H}^k({\mathcal{A}}_{\bullet}). \end{align} $$

The smooth version of this decomposition follows immediately:

(3.12) $$ \begin{align} \Gamma({\mathbb E}_k)=\mathscr{R}({\mathcal{A}}_{k-1})\oplus\mathscr{R}({\mathcal{A}}_k^*;B_k^*)\oplus\mathscr{H}^k({\mathcal{A}}_{\bullet}). \end{align} $$

The proof of this theorem relies on some of the constructs developed in Section 4; hence, we present it in that same section, after the construction of the induced elliptic complex.

A particular instance of the $W^{s,p}$ -version is for $s=0$ , yielding the decomposition mentioned in (1.9):

$$\begin{align*}L^p\Gamma({\mathbb E}_k) = \mathscr{R}^{0,p}({\mathcal{A}}_{k-1}) \oplus \mathscr{R}^{0,p}({\mathcal{A}}_k^*; B_k^*) \oplus \mathscr{H}^k({\mathcal{A}}_{\bullet}). \end{align*}$$

We take a moment to compare these decompositions with other $L^p$ -Hodge decompositions in the literature, particularly those in [Reference Axelsson, Keith and McIntoshAKM06, Reference Hytönen, McIntosh and PortalHMP08]. These works aim to study rather general nilpotent operators $\mathcal {D}$ (denoted there by $\Gamma $ ) and rely on assumptions about their spectral properties to obtain Hodge-like decompositions of the form

(3.13) $$ \begin{align} L^p({\mathbb R}^{d};X^{N}) = \overline{\mathscr{R}^{0,p}(\mathcal{D})} \oplus \overline{\mathscr{R}^{0,p}(\mathcal{D}^*)} \oplus \ker(\mathcal{D}\oplus\mathcal{D}^*), \end{align} $$

where X is a reflexive Banach space, and $\overline {\mathscr {R}^{0,p}(\cdot )}$ denotes the closure of the range in the $L^p$ norm. Certain elliptic operators are known to satisfy the spectral conditions required for these constructions ([Reference Hytönen, McIntosh and PortalHMP08, App. A], [Reference TaylorTay11c, Ch. 13.7]).

The starting points of the approaches are quite similar, as both rely on an ‘auxiliary’ decomposition of the schematic form $L^p = \mathscr {N} \oplus \overline {\mathscr {R}}$ before refining it further into a full Hodge decomposition. However, whereas in [Reference Axelsson, Keith and McIntoshAKM06, Reference Hytönen, McIntosh and PortalHMP08] these auxiliary decompositions are obtained using spectral theory, here we use the additional structure provided by the encompassing framework of the elliptic pre-complex.

Thus, while the structural form of the decompositions is similar, the assumptions and techniques differ. Our results focus on overdetermined elliptic systems within the framework of elliptic pre-complexes rather than operators analyzed in isolation using spectral theory. This has both advantages and limitations. On the one hand, our approach provides a richer theory for operators that fall within the scope of this framework. Specifically, our Hodge-like decompositions are orthogonal, the ranges of the operators are always closed, and the results extend to all $W^{s,p}$ spaces, not just $s=0$ . On the other hand, the theory in [Reference Axelsson, Keith and McIntoshAKM06, Reference Hytönen, McIntosh and PortalHMP08] applies, in principle, to more general systems without requiring an encompassing framework, offering a non-orthogonal Hodge-like decomposition valid in $L^p$ – a generality that is applicable in other contexts.

The refinement of the auxiliary decomposition into a Hodge-like decomposition identifies $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ as the cohomology groups of the complex $({\mathcal {A}}_{\bullet })$ :

Theorem 3.11 (Cohomology groups).

Let $\psi \in W^{s,p}\Gamma ({\mathbb E}_k)$ , with $s\in {{\mathbb N}_0}$ and $1<p<\infty $ . Then,

$$\begin{align*}\psi\in \mathscr{R}^{s,p}({\mathcal{A}}_{k-1}) \end{align*}$$

if and only if

$$\begin{align*}\psi\in \mathscr{N}^{s,p}({\mathcal{A}}_k) \quad\text{ and }\quad \langle\psi,\zeta\rangle=0 \qquad \text{for every }\zeta\in\mathscr{H}^k({\mathcal{A}}_{\bullet}). \end{align*}$$

Equivalently,

(3.14) $$ \begin{align} \mathscr{N}^{s,p}({\mathcal{A}}_k)=\mathscr{R}^{s,p}({\mathcal{A}}_{k-1})\oplus\mathscr{H}^k({\mathcal{A}}_{\bullet}), \end{align} $$

or in the smooth case,

$$\begin{align*}\mathscr{N}({\mathcal{A}}_k)=\mathscr{R}({\mathcal{A}}_{k-1})\oplus\mathscr{H}^k({\mathcal{A}}_{\bullet}). \end{align*}$$

Proof. First, note that since elements in $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ are smooth, the coupling $\langle \psi ,\zeta \rangle $ is well-defined for every $\psi \in \mathscr {R}^{s,p}({\mathcal {A}}_{k-1})$ and $\zeta \in \mathscr {H}^k({\mathcal {A}}_{\bullet })$ . Let $\psi ={\mathcal {A}}_{k-1}\omega \in \mathscr {R}^{s,p}({\mathcal {A}}_{k-1})$ . Then, $\psi \in \mathscr {N}^{s,p}({\mathcal {A}}_k)$ by Lemma 3.9(b). Its $L^2$ -orthogonality to $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ follows from the Hodge-like decomposition (3.10).

In the other direction, let $\psi \in \mathscr {N}^{s,p}({\mathcal {A}}_k)$ be $L^2$ -orthogonal to $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ . Decompose $\psi $ according to (3.10). Its $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ component vanishes, whereas by Lemma 3.9(a), its $\mathscr {R}^{s,p}({\mathcal {A}}_k^*;B_k^*)$ component vanishes as well, remaining with $\psi \in \mathscr {R}^{s,p}({\mathcal {A}}_{k-1})$ (for $p<2$ , an additional approximation argument is needed, using the closedness of the subspaces).

To prove (3.14), we note that the inclusion

$$\begin{align*}\mathscr{R}^{s,p}({\mathcal{A}}_{k-1})\oplus\mathscr{H}^k({\mathcal{A}}_{\bullet}) \subseteq \mathscr{N}^{s,p}({\mathcal{A}}_k) \end{align*}$$

is trivial. The reverse inclusion is an immediate corollary of the decomposition (3.10) and Lemma 3.9(a) (see above comment for $p<2$ ).

Combining Theorems 3.10 and 3.11, we obtain the following compound decompositions:

Generalizing the technique introduced in [Reference SchwarzSch95], Hodge-like decompositions bestow us with the ability to solve nonhomogeneous, OD elliptic boundary-value problems:

Theorem 3.12 (Overdetermined boundary-value problem).

Given an elliptic pre-complex $(A_{\bullet })$ , consider the list of data,

$$\begin{align*}\chi\in W^{s-m_k,p}\Gamma({\mathbb E}_{k+1}) \qquad \xi\in W^{s-m_{k-1},p}({\mathbb E}_{k-1}) \quad\text{ and }\quad \phi\in W^{s,p}\Gamma({\mathbb E}_k), \end{align*}$$

where $s \ge m = \max (m_k,m_{k-1})$ . There exists a solution $\psi \in W^{s,p}\Gamma ({\mathbb E}_k)$ to the boundary-value problem

(3.15) $$ \begin{align} ({\mathcal{A}}_k\oplus {\mathcal{A}}_{k-1}^* \oplus B_{k-1}^*) \psi = (\chi,\xi,B_{k-1}^*\phi), \end{align} $$

if and only if the following integrability conditions are satisfied:

(3.16a) $$ \begin{align} \chi\in\mathscr{N}^{0,p}({\mathcal{A}}_{k+1}) \quad\text{ and }\quad \langle\chi,\zeta\rangle=0 \qquad \text{for every }\zeta\in\mathscr{H}^{k+1}({\mathcal{A}}_{\bullet}) \end{align} $$

(3.16b) $$ \begin{align} \xi - {\mathcal{A}}_{k-1}^*\phi\in \mathscr{N}^{0,p}({\mathcal{A}}_{k-2}^*,B_{k-2}^*) \end{align} $$

(3.16c) $$ \begin{align} \langle \xi,\nu\rangle =-\langle B_{k-1}^*\phi,B_{k-1}\nu\rangle \qquad \text{for every }\nu\in\mathscr{H}^{k-1}({\mathcal{A}}_{\bullet}). \end{align} $$

The solution is unique up to an arbitrary $\lambda \in \mathscr {H}^k({\mathcal {A}}_{\bullet })$ . Moreover, there is an a priori estimate

(3.17) $$ \begin{align} \begin{aligned} \|\psi\|_{s,p}&\lesssim \|{\mathcal{A}}_k\psi\|_{s-m_k,p}+ \|{\mathcal{A}}_{k-1}^*\psi\|_{s-m_{k-1},p} + \sum_{i=0}^{m_{k-1}-1} \|B_{i,k-1}^*\psi\|_{s-i-1/p,p} + \|\lambda_\psi\|_{0,p}, \end{aligned} \end{align} $$

where $B_{i,k-1}^*$ are the components of the normal system of trace operators $B_{k-1}^*$ , and $\lambda _\psi $ is the projection of $\psi $ onto $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ .

In the same spirit as the discussion following (3.10), comparing between the Hodge decomposition obtained in this work and in other works, it is important to note that, while the boundary value problems considered here encompass multi-order, nonlocal systems and boundary conditions, this work does not aim to provide a systematic characterization of the necessary setting for the solvability of boundary value problems in general. Such broader investigations are surveyed in [Reference GrubbGru96, Ch. 1] and studied in works such as [Reference Bär and BallmannBB12, Reference Bär and BandaraBB22], where, relevant to our approach, the interactions of solvability conditions with $L^2$ -direct decompositions of section spaces are analyzed in a general first-order setting.

Proof. The idea of the proof is as follows: the Hodge-decompositions provide us with an immediate solution, denoted below by $\omega $ , for the case $\xi =0$ and $\phi =0$ . The method adapted from [Reference SchwarzSch95] is to add to $\omega $ an ansatz of the form $\alpha +\phi $ , where the existence of an appropriate $\alpha $ hinges on the integrability conditions of the data.

We start by noting that for every $\nu \in \mathscr {H}^{k-1}({\mathcal {A}}_{\bullet })$ ,

$$\begin{align*}\langle B_{k-1}^*\phi,B_{k-1}\nu\rangle = \langle \phi, {\mathcal{A}}_{k-1}\nu\rangle - \langle {\mathcal{A}}_{k-1}^*\phi,\nu\rangle = - \langle {\mathcal{A}}_{k-1}^*\phi,\nu\rangle , \end{align*}$$

and hence, the integrability condition (3.16c) may take the alternative form

(3.16c-2) $$ \begin{align} \langle \xi- {\mathcal{A}}_{k-1}^*\phi,\nu\rangle = 0 \qquad \text{for every }\nu\in\mathscr{H}^{k-1}({\mathcal{A}}_{\bullet}). \end{align} $$

We first verify the necessity of the integrability conditions. Let $\psi \in W^{s,p}\Gamma ({\mathbb E}_k)$ be a solution to (3.15). Since $\chi \in \mathscr {R}^{s-m_k,p}({\mathcal {A}}_k) \subset \mathscr {R}^{0,p}({\mathcal {A}}_k)$ , then (3.16a) follows from Theorem 3.11. Since $\psi - \phi \in \ker B_{k-1}^*$ , then

$$\begin{align*}\xi - {\mathcal{A}}_{k-1}^* \phi = {\mathcal{A}}_{k-1}^*(\psi - \phi) \in \mathscr{R}^{s-m_{k-1},p}({\mathcal{A}}_{k-1}^*;B_{k-1}^*) \subset \mathscr{R}^{0,p}({\mathcal{A}}_{k-1}^*;B_{k-1}^*), \end{align*}$$

which by (3.11) implies both (3.16b) and (3.16c-2).

To prove sufficiency, it is enough to do it for smooth data, as the same claim in Sobolev regularity follows from the continuity of all the operators, along with an approximation argument.

The second integrability condition (3.16b) asserts that

$$\begin{align*}\xi - {\mathcal{A}}_{k-1}^*\phi \in \mathscr{N}({\mathcal{A}}_{k-2}^*,B_{k-2}^*) = \mathscr{R}({\mathcal{A}}_{k-1}^*;B_{k-1}^*) \oplus \mathscr{H}^{k-1}({\mathcal{A}}_{\bullet}), \end{align*}$$

where the equality follows from (3.11), whereas the third integrability condition, in its form (3.16c-2) implies that $\xi - {\mathcal {A}}_{k-1}^*\phi $ is $L^2$ -orthogonal to $\mathscr {H}^{k-1}({\mathcal {A}}_{\bullet })$ , and hence,

$$\begin{align*}\xi - {\mathcal{A}}_{k-1}^*\phi \in \mathscr{R}({\mathcal{A}}_{k-1}^*;B_{k-1}^*). \end{align*}$$

Let

$$\begin{align*}\xi - {\mathcal{A}}_{k-1}^*\phi = {\mathcal{A}}_{k-1}^*\alpha, \qquad \text{where } \alpha\in\ker B_{k-1}^*. \end{align*}$$

By Theorem 3.11, the first integrability condition (3.16a) implies that $\chi \in \mathscr {R}({\mathcal {A}}_k)$ . By the Hodge-like decomposition for $\Gamma ({\mathbb E}_k)$ , we may write

$$\begin{align*}\chi - {\mathcal{A}}_k(\alpha + \phi) = {\mathcal{A}}_k \omega, \qquad \text{where } \omega\in \mathscr{R}({\mathcal{A}}_k^*; B_k^*) \subset \mathscr{N}({\mathcal{A}}_{k-1}^*,B_{k-1}^*), \end{align*}$$

and the inclusion follows once again from (3.11). Let $\psi = \omega + \alpha + \phi $ . Then,

$$\begin{align*}\begin{aligned} {\mathcal{A}}_k\psi = {\mathcal{A}}_k \omega + {\mathcal{A}}_k(\alpha + \phi) = \chi \\ {\mathcal{A}}_{k-1}^*\psi = {\mathcal{A}}_{k-1}^*\omega + {\mathcal{A}}_{k-1}^*(\alpha + \phi) = \xi \\ B_{k-1}^*\psi = B_{k-1}^*\omega + B_{k-1}^*(\alpha + \phi) = B_{k-1}^*\phi \end{aligned} \end{align*}$$

(i.e., $\psi $ is a solution to (3.15)). The uniqueness clause is immediate.

Finally, the system $({\mathcal {A}}_{k-1}^*\oplus {\mathcal {A}}_k,B_{k-1}^*)$ is OD elliptic, and hence yields an a priori estimate (2.20), which takes the form (3.17).

4.1 Stage 1: Base and setup of induction step

For the base of the induction, it is convenient to append an extra level to the sequence $(A_{\bullet })$ , as in the diagram in Section 3.2, setting ${\mathbb E}_{-1}= M \times \{0\}$ , ${\mathbb G}_{-1}= {\partial M} \times \{0\}$ and $A_{-1}={\mathcal {A}}_{-1}=0$ . The base of the induction requires that

1. (a) ${\mathcal {A}}_{-1}$ induces an auxiliary decomposition,

   $$\begin{align*}\Gamma({\mathbb E}_0) = \mathscr{R}({\mathcal{A}}_{-1})\oplus \mathscr{N}({\mathcal{A}}_{-1}^*,B_{-1}^*). \end{align*}$$

2. (b) ${\mathcal {A}}_0{\mathcal {A}}_{-1}=0$ .

3. (c) ${\mathcal {A}}_0 = A_0$ on $\mathscr {N}({\mathcal {A}}_{-1}^*,B_{-1}^*)$ .

4. (d) ${\mathcal {A}}_0 = A_0 + G_0$ is an adapted Green operators with $G_0$ satisfying the properties specified in Proposition 3.8.

This is satisfied trivially by observing that $\mathscr {R}({\mathcal {A}}_{-1})=\{0\}$ and $\Gamma ({\mathbb E}_0)=\mathscr {N}({\mathcal {A}}_{-1}^*;B_{-1}^*)$ , and hence, Conditions (a) and (b) are satisfied. Condition (c) determines ${\mathcal {A}}_0 = A_0$ uniquely. Finally, Condition (d) is satisfied as ${\mathcal {A}}_0$ has a Green part $G_0=0$ .

4.2 Stage 2: Additional elliptic estimates

Since the system $(A_{k-1}^*\oplus A_k, B_{k-1}^*)$ is OD elliptic and ${\mathcal {A}}_{k-1}^*-{\mathcal {A}}_{k-1}^*,{\mathcal {A}}_k-A_k\in \mathfrak {G}^0$ , it follows that the system $({\mathcal {A}}_{k-1}^*\oplus {\mathcal {A}}_k, B_{k-1}^*)$ is also OD elliptic (we use here the closure of $\mathfrak {G}^0$ to adjoints). By Theorem 2.12, this implies the finite-dimensionality of the space

$$\begin{align*}\mathscr{H}^k({\mathcal{A}}_{\bullet})=\ker({\mathcal{A}}_{k-1}^*\oplus{\mathcal{A}}_k\oplus B_{k-1}^*), \end{align*}$$

along with the a priori estimate,

(4.2) $$ \begin{align} \begin{aligned} \|\psi\|_{s,p} &\lesssim \|{\mathcal{A}}_{k-1}^*\psi\|_{s-m_{k-1},p} + \|{\mathcal{A}}_k\psi\|_{s-m_k,p} + \sum_{i=0}^{m_{k-1}-1} \|B^*_{i,k-1}\psi\|_{s-i-1/p,p} + \|{\mathcal{S}}_k\psi\|_{0,p}, \end{aligned} \end{align} $$

for every ${{\mathbb N}_0}\ni s\ge m$ , where ${\mathcal {S}}_k\in \mathrm {OP}(\mathfrak {S}^{-\infty })$ is the $L^2$ -orthogonal projection onto $\mathscr {H}^k({\mathcal {A}}_{\bullet })$ , and $m=\max {(m_{k-1},m_k)}$ . As a consequence of (4.2), the system $({\mathcal {A}}_{k-1}^*\oplus {\mathcal {A}}_k\oplus {\mathcal {S}}_k, B_{k-1}^*)$ is OD elliptic and injective, and hence, by Proposition 2.11, admits a left-inverse or order $-m$ and class $m_{k-1}-m$ within the calculus of Green operators.