On Time-Periodic Solutions to Parabolic Boundary Value Problems of Agmon-Douglis-Nirenberg Type

TL;DR

This paper investigates time-periodic solutions to parabolic PDEs of Agmon-Douglis-Nirenberg type, providing explicit formulas and coercive Lp estimates that extend classical elliptic results to the time-periodic setting.

Contribution

It introduces explicit solution formulas and coercive Lp estimates for time-periodic parabolic problems, extending classical elliptic theory to this new context.

Findings

Explicit formulas for solutions in whole- and half-space cases

Coercive Lp estimates for time-periodic solutions

Extension of elliptic theory results to parabolic time-periodic problems

Abstract

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit formula for the solution and establish coercive Lp estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg for elliptic problems to the time-periodic case.

Equations

$$\{\begin{array}{rlrl}{\partial }_{t}u+Au& =f& & \text{in }\mathbb{R}\times \Omega ,\\ B_{j}u& =g_j& & \text{on }\mathbb{R}\times \partial \Omega ,\end{array}$$

$$\mathcal{P}u=\frac{1}{T}{\int }_0^{T}u(t,x)\mathrm{d}t,{\mathcal{P}}_{\perp }:=\mathrm{id}-\mathcal{P},$$

$$\{\begin{array}{rlrl}A\mathcal{P}u& =\mathcal{P}f& & \text{in }\Omega ,\\ B_j\mathcal{P}u& =\mathcal{P}{g}_j& & \text{on }\partial \Omega ,\end{array}$$

$$\{\begin{array}{rlrl}{\partial }_t{\mathcal{P}}_{\perp }u+A{\mathcal{P}}_{\perp }u& ={\mathcal{P}}_{\perp }f& & \text{in }\mathbb{R}\times \Omega ,\\ B_j{\mathcal{P}}_{\perp }u& ={\mathcal{P}}_{\perp }{g}_j& & \text{on }\mathbb{R}\times \partial \Omega .\end{array}$$

$$A(x,D):=\sum _{|\alpha |\le 2m}{a}_{\alpha }(x)D^{\alpha },B_j(x,D):=\sum _{|\alpha |\le m_j}{b}_{j,\alpha }(x)D^{\alpha }(j=1,\dots ,m)$$

$$A^H(x,D):=\sum _{|\alpha |=2m}{a}_{\alpha }(x)D^{\alpha },B_j^H(x,D):=\sum _{|\alpha |=m_j}{b}_{j,\alpha }(x)D^{\alpha }.$$

$$\{\begin{array}{rlrl}\lambda u+A^{H}u& =f& & \text{in }\Omega ,\\ B_j^{H}u& =0& & \text{on }\partial \Omega .\end{array}$$

$$\{\begin{array}{rlrl}{\partial }_{t}u+A^{H}u& =f& & \text{in }\mathbb{R}\times {\mathbb{R}}_{+}^n,\\ B_j^{H}u& =g_j& & \text{on }\mathbb{R}\times \partial {\mathbb{R}}_{+}^n.\end{array}$$

$$\displaystyle\begin{aligned} \lVert u\rVert_{W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})}\leq C\bigg{(}\lVert f\rVert_{L_{\mathrm{per}}^{p}(\mathbb{R}\times\mathbb{R}^{n}_{+})}+\sum_{j=1}^{m}\lVert g_{j}\rVert_{W^{\kappa_{j},2m\kappa_{j},p}_{\mathrm{per}}(\mathbb{R}\times\partial\mathbb{R}^{n}_{+})}\bigg{)}\end{aligned}$$

$$\begin{gathered} ({\partial }_t+A^H,B_1^H,\dots ,B_m^H): \\ {\mathcal{P}}_{\perp }{W}_{\mathrm{per}}^{1,2m,p}(\mathbb{R}\times {\mathbb{R}}_{+}^n)\to {\mathcal{P}}_{\perp }{L}_{\mathrm{per}}^p(\mathbb{R}\times {\mathbb{R}}_{+}^n)\times {\Pi }_{j=1}^m{\mathcal{P}}_{\perp }{W}_{\mathrm{per}}^{{\kappa }_j,2m{\kappa }_j,p}(\mathbb{R}\times \partial {\mathbb{R}}_{+}^n) \end{gathered}$$

$$\displaystyle\begin{aligned} &\lVert u\rVert_{W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)}\\ &\qquad\qquad\leq C\bigg{(}\lVert\partial_{t}u+Au\rVert_{L_{\mathrm{per}}^{p}(\mathbb{R}\times\Omega)}+\sum_{j=1}^{m}\lVert B_{j}u\rVert_{W^{\kappa_{j},2m\kappa_{j},p}_{\mathrm{per}}(\mathbb{R}\times\partial\Omega)}+\lVert u\rVert_{L_{\mathrm{per}}^{p}(\mathbb{R}\times\Omega)}\bigg{)}\end{aligned}$$

$$D(A_B^H):=\{u\in W^{2m,p}({\mathbb{R}}_{+}^n)|B_j^H(D)u=0,j=1,\dots ,m\}.$$

$$\{\begin{array}{rlrl}{\partial }_{t}u+A_B^{H}u& =f,& & t>0,\\ u(0)& =0.& & \end{array}$$

$$\Vert u,{\partial }_{t}u,D^{2m}u{\Vert }_{L^p(0,T;L^p({\mathbb{R}}_{+}^n))}\le c\Vert f{\Vert }_{L^p(0,T;L^p({\mathbb{R}}_{+}^n))}.$$

$$\tilde{u}:=u+{\int }_0^t{e}^{-(t-s)A_B^H}\mathcal{P}f\mathrm{d}s-e^{-t{A}_B^H}u(0)$$

$$\forall (\eta ,\xi )\in \mathbb{R}\times {\mathbb{R}}^n:|\eta ,\xi |:=(|\eta {|}^2+|\xi {|}^{4m}{)}^{\frac{1}{4m}}.$$

$$\underset{y<0}{\sup }{\int }_{\mathbb{R}}|\tilde{f}(x+iy){|}^2\text{d}x<\infty ,\underset{y\to 0-}{\lim }{\int }_{\mathbb{R}}|\tilde{f}(x+iy)-f(x){|}^2\text{d}x=0.$$

$$\displaystyle C^{\infty}_{0,\mathrm{per}}(\mathbb{R}\times\Omega):=\{f\in C^{\infty}(\mathbb{R}\times\Omega)\ \lvert\ f(t+T,x)=f(t,x),\ f\in C^{\infty}_{0}\big{(}[0,T]\times\Omega\big{)}\}$$

$$\displaystyle\lVert f\rVert_{p}:=\bigg{(}\frac{1}{T}\int_{0}^{T}\int_{\Omega}{\lvert f(t,x)\rvert}^{p}\,{\mathrm{d}}x{\mathrm{d}}t\bigg{)}^{\frac{1}{p}},$$

$$\displaystyle\lVert f\rVert_{1,2m,p}:=\bigg{(}\lVert\partial_{t}f\rVert_{p}^{p}+\sum_{0\leq{\lvert\alpha\rvert}\leq 2m}\lVert\partial_{x}^{\alpha}f\rVert_{p}^{p}\bigg{)}^{\frac{1}{p}}$$

$$\displaystyle L_{\mathrm{per}}^{p}(\mathbb{R}\times\Omega):=\overline{C^{\infty}_{0,\mathrm{per}}(\mathbb{R}\times\Omega)}^{\lVert\cdot\rVert_{p}},\quad W^{1,2m,p}_{\mathrm{per}}\big{(}\mathbb{R}\times\Omega\big{)}:=\overline{C^{\infty}_{0,\mathrm{per}}(\mathbb{R}\times\overline{\Omega})}^{\lVert\cdot\rVert_{1,2m,p}}.$$

$$L_{\mathrm{per}}^p(\mathbb{R}\times \Omega )=\{f\in L_{\mathrm{loc}}^p(\mathbb{R}\times \overline{\Omega })|f(t+T,x)=f(t,x)\text{ for almost every }(x,t)\}.$$

$$W_{\mathrm{per}}^{1,2m,p}(\mathbb{R}\times \Omega )=\{f\in W_{\mathrm{loc}}^{1,2m,p}(\mathbb{R}\times \overline{\Omega })|f(t+T,x)=f(t,x)\text{ for almost every }(x,t)\},$$

$$\displaystyle W^{s,2ms,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)=\big{(}L_{\mathrm{per}}^{p}(\mathbb{R}\times\Omega),W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)\big{)}_{s,p},\qquad s\in(0,1).$$

$$C^{\infty }(\mathbb{T}\times \Omega ):=\{u:\mathbb{T}\times \Omega \to \mathbb{C}|u\circ \pi \in C^{\infty }(\mathbb{R}\times \Omega )\}$$

$$C_0^{\infty }(\mathbb{T}\times \Omega ):=\{u\in C^{\infty }(\mathbb{T}\times \Omega )|\mathrm{supp}u\text{ is compact}\}$$

$$\displaystyle L^{p}({\mathbb{T}}\times\Omega):=\overline{C^{\infty}_{0}({\mathbb{T}}\times\Omega)}^{\lVert\cdot\rVert_{p}},\quad W^{1,2m,p}\big{(}{\mathbb{T}}\times\Omega\big{)}:=\overline{C^{\infty}_{0}({\mathbb{T}}\times\overline{\Omega})}^{\lVert\cdot\rVert_{1,2m,p}},$$

$$\displaystyle W^{1,2m,p}\big{(}{\mathbb{T}}\times\Omega\big{)}=\{u\in L^{p}({\mathbb{T}}\times\Omega)\ \lvert\ u,\partial_{t}u,\partial_{x}^{\alpha}u\in L^{p}({\mathbb{T}}\times\Omega)\text{ for all }{\lvert\alpha\rvert}\leq 2\},$$

$$\displaystyle W^{s,2ms,p}({\mathbb{T}}\times\Omega)=\big{(}L^{p}({\mathbb{T}}\times\Omega),W^{1,2m,p}({\mathbb{T}}\times\Omega)\big{)}_{s,p},$$

$$\displaystyle W^{1,2m,p}({\mathbb{T}}\times\mathbb{R}^{n}_{+})=L^{p}\big{(}\mathbb{R}_{+};W^{1,2m,p}({\mathbb{T}}\times\mathbb{R}^{n-1})\big{)}\cap W^{2m,p}\big{(}\mathbb{R}_{+};L^{p}({\mathbb{T}}\times\mathbb{R}^{n-1})\big{)}.$$

$$\displaystyle\begin{aligned} &\operatorname{Tr}_{l}:C^{\infty}_{0}({\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}})\rightarrow C^{\infty}_{0}({\mathbb{T}}\times{\mathbb{R}^{n-1}})^{l},\\ &\operatorname{Tr}_{l}(u)(t,x^{\prime}):=\big{(}u(t,x^{\prime},0),\partial_{n}u(t,x^{\prime},0),\ldots,\partial_{n}^{l-1}u(t,x^{\prime},0)\big{)}\end{aligned}$$

Full text

MSC2010: Primary 35B10, 35B45, 35K25.

Keywords: Time-periodic, parabolic, boundary value problem, a priori estimates.

1 Introduction

We investigate time-periodic solutions to parabolic boundary value problems

[TABLE]

where $A$ is an elliptic operator of order $2m$ and $B_{1},\ldots,B_{m}$ satisfy an appropriate complementing boundary condition. The domain $\Omega$ is either the whole-space, the half-space or a bounded domain, and $\mathbb{R}$ denotes the time-axis. The solutions $u(t,x)$ correspond to time-periodic data $f(t,x)$ and $g_{j}(t,x)$ of the same (fixed) period $T>0$. Using the simple projections

[TABLE]

we decompose (1.1) into an elliptic problem

[TABLE]

and a purely oscillatory problem

[TABLE]

The problem (1.2) is elliptic in the sense of Agmon-Douglis-Nirenberg, for which a comprehensive $L^{p}$ theory was established in [3]. In this article, we develop a complementary theory for the purely oscillatory problem (1.3). Employing ideas going back to Peetre [21] and Arkeryd [7], we are able to establish an explicit formula for the solution to (1.3) when the domain is either the whole- or the half-space. We shall then introduce a technique based on tools from abstract harmonic analysis to show coercive $L^{p}$ estimates. As a consequence, we obtain a time-periodic version of the celebrated theorem of Agmon, Douglis and Nirenberg [3].

The decomposition (1.2)–(1.3) is essential as the two problems have substantially different properties. In particular, we shall show in the whole- and half-space case that the principle part of the linear operator in the purely oscillatory problem (1.3) is a homeomorphism in a canonical setting of time-periodic Lebesgue-Sobolev spaces. This is especially remarkable since the elliptic problem (1.2) clearly does not satisfy this property. Another truly remarkable characteristic of (1.3) is that the $L^{p}$ theory we shall develop for this problem leads directly to a similar $L^{p}$ theory, sometimes referred to as maximal regularity, for the parabolic initial-value problem associated to (1.1).

We consider general differential operators

[TABLE]

with complex coefficients $a_{\alpha}:\Omega\to\mathbb{C}$ and $b_{j,\alpha}:\partial\Omega\to\mathbb{C}$. Here, $\alpha\in\mathbb{N}^{n}$ is a multi-index and $D^{\alpha}:=i^{|\alpha|}\partial_{x_{1}}^{\alpha_{1}}\ldots\partial_{x_{n}}^{\alpha_{n}}$. We denote the principle part of the operators by

[TABLE]

We shall assume that $A^{H}$ is elliptic in the following classical sense:

Definition 1.1** (Properly Elliptic).**

The operator $A^{H}$ is said to be properly elliptic if for all $x\in\Omega$ and all $\xi\in\mathbb{R}^{n}\setminus\{0\}$ it holds $A^{H}(x,\xi)\neq 0$, and for all $x\in\Omega$ and all linearly independent vectors ${\zeta},\xi\in\mathbb{R}^{n}$ the polynomial $P(\tau):=A^{H}(x,{\zeta}+\tau\xi)$ has $m$ roots in $\mathbb{C}$ with positive imaginary part, and $m$ roots in $\mathbb{C}$ with negative imaginary part.

Ellipticity, however, does not suffice to establish maximal $L^{p}$ regularity for the time-periodic problem. We thus recall Agmon’s condition, also known as parameter ellipticity.

Definition 1.2** (Agmon’s Condition).**

Let $\theta\in[-\pi,\pi]$. A properly elliptic operator $A^{H}$ is said to satisfy Agmon’s condition on the ray $\operatorname{e}^{i\theta}$ if for all $x\in\Omega$ and all $\xi\in\mathbb{R}^{n}\setminus\{0\}$ it holds $A^{H}(x,\xi)\notin\{r\operatorname{e}^{i\theta}\ \lvert\ r\geq 0\}$.

If $A^{H}$ satisfies Agmon’s condition on the ray $\operatorname{e}^{i\theta}$, then, since the roots of a polynomial depend continuously on its coefficients, the polynomial $Q(\tau):=-r\operatorname{e}^{i\theta}+A^{H}(x,{\zeta}+\tau\xi)$ has $m$ roots $\tau_{h}^{+}(r\operatorname{e}^{i\theta},x,{\zeta},\xi)\in\mathbb{C}$ with positive imaginary part, and $m$ roots $\tau_{h}^{-}(r\operatorname{e}^{i\theta},x,{\zeta},\xi)\in\mathbb{C}$ with negative imaginary part ($h=1,\ldots,m$). Consequently, the following assumption on the operator $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ is meaningful.

Definition 1.3** (Agmon’s Complementing Condition).**

Let $\theta\in[-\pi,\pi]$. If $A^{H}$ is a properly elliptic operator, then $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ is said to satisfy Agmon’s complementing condition on the ray $\operatorname{e}^{i\theta}$ if:

1. (i)

$A^{H}$* satisfies Agmon’s condition on the ray $\operatorname{e}^{i\theta}$.* 2. (ii)

For all $x\in\partial\Omega$, all pairs ${\zeta},\xi\in\mathbb{R}^{n}$ with ${\zeta}$ tangent to $\partial\Omega$ and $\xi\in\mathbb{R}^{n}$ normal to $\partial\Omega$ at $x$, and all $r\geq 0$, let $\tau_{h}^{+}(r\operatorname{e}^{i\theta},x,{\zeta},\xi)\in\mathbb{C}$ ($h=1,\ldots,m$) denote the $m$ roots of the polynomial ${Q(\tau):=-r\operatorname{e}^{i\theta}+A^{H}(x,{\zeta}+\tau\xi)}$ with positive imaginary part. The polynomials $P_{j}(\tau):=B^{H}_{j}(x,{\zeta}+\tau\xi)$ ($j=1,\ldots,m$) are linearly independent modulo the polynomial \Pi_{h=1}^{m}\big{(}\tau-\tau_{h}^{+}(r\operatorname{e}^{i\theta},x,{\zeta},\xi)\big{)}.

The property specified in Definition 1.3 was first introduced by Agmon in [2], and later by Agranovich and Vishik in [5] as parameter ellipticity. The condition was introduced in order to identify the additional requirements on the differential operators needed to extend the result of Agmon, Douglis and Nirenberg [3] from the elliptic case to the corresponding parabolic initial-value problem. The theorem of Agmon, Douglis and Nirenberg [3] requires $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ to satisfy Agmon’s complementing condition only at the origin (not on a full ray), in which case $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ is said to be elliptic in the sense of Agmon-Douglis-Nirenberg. Analysis of the associated initial-value problem relies heavily on properties of the resolvent equation

[TABLE]

It was shown by Agmon [2] that a necessary and sufficient condition for the resolvent of $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ to lie in the negative complex half-plane (and thereby for the generation of an analytic semi-group) is that Agmon’s complementing condition is satisfied for all rays with ${\lvert\theta\rvert}\geq\frac{\pi}{2}$. However, the step from analyticity of the semi-group to maximal $L^{p}$ regularity for the parabolic initial-value problem proved to be highly non-trivial. Although many articles were dedicated to this problem after the publication of [2], it was not until the celebrated work of Dore and Venni [12] that a framework was developed with which maximal regularity could be established comprehensively from the assumption that Agmon’s condition is satisfied for all rays with ${\lvert\theta\rvert}\geq\frac{\pi}{2}$. To apply [12], one has show that $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ admits bounded imaginary powers. Later, it was shown that maximal regularity is in fact equivalent to ${\mathcal{R}}$-boundedness of an appropriate resolvent family; see [11]. Remarkably, our result for the time-periodic problem (1.3) leads to a new and relatively short proof of maximal regularity for the parabolic initial-value problem without the use of either bounded imaginary powers or the notion of ${\mathcal{R}}$-boundedness; see Remark 1.6 below. Under the assumption that $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ generates an analytic semi-group, maximal regularity for the parabolic initial-value problem follows almost immediately as a corollary from our main theorem. We emphasize that our main theorem of maximal regularity for the time-periodic problem does not require the principle part of $(A,B_{1},\ldots,B_{m})$ to generate an analytic semi-group. As a novelty of the present paper, and in contrast to the initial-value problem, we establish that maximal $L^{p}$ regularity for the time-periodic problem requires Agmon’s complementing condition to be satisfied only on the two rays with $\theta=\pm\frac{\pi}{2}$, that is, only on the imaginary axis.

Our main theorem for the purely oscillatory problem (1.3) concerns the half-space case and the question of existence of a unique solution satisfying a coercive $L^{p}$ estimate in the Sobolev space $W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ of time-periodic functions on the time-space domain $\mathbb{R}\times\mathbb{R}^{n}_{+}$. We refer to Section 2 for definitions of the function spaces.

Theorem 1.4**.**

Let $p\in(1,\infty)$, $T>0$, $n\geq 2$. Assume that $A^{H}$ and $(B^{H}_{1},\ldots,B^{H}_{m})$ have constant coefficients. If $A^{H}$ is properly elliptic and $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ satisfies Agmon’s complementing condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$, then for all functions $f\in{\mathcal{P}}_{\bot}L_{\mathrm{per}}^{p}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ and $g_{j}\in{\mathcal{P}}_{\bot}W^{\kappa_{j},2m\kappa_{j},p}_{\mathrm{per}}(\mathbb{R}\times\partial\mathbb{R}^{n}_{+})$ with $\kappa_{j}:=1-\frac{m_{j}}{2m}-\frac{1}{2mp}$ ($j=1,\ldots,m$) there exists a unique solution $u\in{\mathcal{P}}_{\bot}W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ to

[TABLE]

Moreover,

[TABLE]

with $C=C(p,T,n)$.

Our proof of Theorem 1.4 contains two results that are interesting in their own right. Firstly, we establish a similar assertion in the whole-space case. Secondly, we provide an explicit formula for the solution; see (3.8) below. Moreover, our proof is carried out fully in a setting of time-periodic functions and follows an argument adopted from the elliptic case. This is remarkable in view of the fact that analysis of time-periodic problems in existing literature typically is based on theory for the corresponding initial-value problem; see for example [17]. A novelty of our approach is the introduction of suitable tools from abstract harmonic analysis that allow us to give a constructive proof and avoid completely the classical indirect characterizations of time-periodic solutions as fixed points of a Poincaré map, that is, as special solutions to the corresponding initial-value problem. The circumvention of the initial-value problem also enables us to avoid having to assume Agmon’s condition for all ${\lvert\theta\rvert}\geq\frac{\pi}{2}$ and instead carry out our investigation under the the weaker condition that Agmon’s condition is satisfied only for $\theta=\pm\frac{\pi}{2}$.

We shall briefly describe the main ideas behind the proof of Theorem 1.4. We first consider the problem in the whole space $\mathbb{R}\times\mathbb{R}^{n}$ and replace the time axis $\mathbb{R}$ with the torus ${\mathbb{T}}:=\mathbb{R}/T\mathbb{Z}$ in order to reformulate the $T$-time-periodic problem as a partial differential equation on the locally compact abelian group $G:={\mathbb{T}}\times\mathbb{R}^{n}$. Utilizing the Fourier transform $\mathscr{F}_{G}$ associated to $G$, we obtain an explicit representation formula for the time-periodic solution. Since $\mathscr{F}_{G}=\mathscr{F}_{\mathbb{T}}\circ\mathscr{F}_{\mathbb{R}^{n}}$, this formula simply corresponds to a Fourier series expansion in time of the solution and subsequent Fourier transform in space of all its Fourier coefficients. While it is relatively easy to obtain $L^{p}$ estimates (in space) for each Fourier coefficient separately, it is highly non-trivial to deduce from these individual estimates an $L^{p}$ estimate in space and time via the corresponding Fourier series. Instead, we turn to the representation formula given in terms of $\mathscr{F}_{G}$ and show that the corresponding Fourier multiplier defined on the dual group $\widehat{G}$ is an $L^{p}(G)$ multiplier. For this purpose, we use the so-called Transference Principle for Fourier multipliers in a group setting, and obtain the necessary estimate in the whole-space case. In the half-space case, Peetre [21] and Arkeryd [7] utilized the Paley-Wiener Theorem in order to construct a representation formula for solutions to elliptic problems; see also [24, Section 5.3]. We adapt their ideas to our setting and establish $L^{p}$ estimates from the ones already obtained in the whole-space case.

Theorem 1.4 can be reformulated as the assertion that the operator

[TABLE]

is a homeomorphism. By a standard localization and perturbation argument, a purely periodic version of the celebrated theorem of Agmon, Douglis and Nirenberg [3] in the general case of operators with variable coefficients and $\Omega$ being a sufficiently smooth domain follows. In fact, combining the classical result [3] for the elliptic case with Theorem 1.4, we obtain the following time-periodic version of the Agmon-Douglis-Nirenberg Theorem:

Theorem 1.5** (Time-Periodic ADN Theorem).**

Let $p\in(1,\infty)$, $T>0$, $n\geq 2$ and $\Omega$ be a domain with a boundary that is uniformly $C^{2m}$-smooth. Assume $a_{\alpha}$ is bounded and uniformly continuous on $\overline{\Omega}$ for ${\lvert\alpha\rvert}=2m$, and $a_{\alpha}\in L^{\infty}(\Omega)$ for ${\lvert\alpha\rvert}<2m$. Further assume $b_{j,\beta}\in C^{2m-m_{j}}(\partial\Omega)$ with bounded and uniformly continuous derivatives up to the order $2m-m_{j}$. If $A^{H}$ is properly elliptic and $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ satisfies Agmon’s complementing condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$, then the estimate

[TABLE]

holds for all $u\in W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)$, where $\kappa_{j}:=1-\frac{m_{j}}{2m}-\frac{1}{2mp}$ ($j=1,\ldots,m$).

Since time-independent functions are trivially also time-periodic, we have $W^{2m,p}(\Omega)\subset W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)$. If estimate (1.9) is restricted to functions in $W^{2m,p}(\Omega)$, Theorem 1.5 reduces to the classical theorem of Agmon-Douglis-Nirenberg [3], which has played a fundamental role in the analysis of elliptic boundary value problems for more then half a century now. This classical theorem for scalar equations was extended to systems in [4]. We shall only treat scalar equations in the following, but we believe the method developed here could be extended to include systems.

Time-periodic problems of parabolic type have been investigated in numerous articles over the years, and it would be too far-reaching to list them all here. We mention only the article of Liebermann [17], the recent article by Geissert, Hieber and Nguyen [14], as well as the monographs [15, 25], and refer the reader to the references therein. Finally, we mention the article [16] by the present authors in which some of the ideas utilized in the following were introduced in a much simpler setting.

Remark 1.6*.*

The half-space case treated in Theorem 1.4 is also pivotal in the $L^{p}$ theory for parabolic initial-value problems. Denote by $A^{H}_{B}$ the realization of the operator $A^{H}(D)$ in $L^{p}(\mathbb{R}^{n}_{+})$ with domain

[TABLE]

Maximal regularity for parabolic initial-value problems of Agmon-Douglis-Nirenberg type is based on an investigation of the initial-value problem

[TABLE]

Maximal regularity for (1.10) means that for each function $f\in L^{p}(0,T;L^{p}(\mathbb{R}^{n}_{+}))$ there is a unique solution $u\in L^{p}(0,T;D(A^{H}_{B}))\cap W^{1,p}(0,T;L^{p}(\mathbb{R}^{n}_{+}))$ which satisfies the estimate

[TABLE]

We shall briefly sketch how to obtain maximal regularity for (1.10) from Theorem 1.4. For this purpose, it is required that $-A^{H}_{B}$ generates an analytic semi-group $\{\operatorname{e}^{-tA^{H}_{B}}\}_{t>0}$, which follows from resolvent estimates going back to Agmon [2, Theorem 2.1] derived under the assumption that $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ satisfies Agmon’s complementing condition for all rays with ${\lvert\theta\rvert}\geq\frac{\pi}{2}$; see also [23, Theorem 5.5]. We would like to point out that these resolvent estimates can also be established with the arguments in our proof of Theorem 1.4. One can periodically extend any $f\in L^{p}(0,T;L^{p}(\mathbb{R}^{n}_{+}))$ to a $T$-periodic function $f\in L_{\mathrm{per}}^{p}(\mathbb{R}\times\mathbb{R}^{n}_{+})$. With $u$ denoting the solution from Theorem 1.4 corresponding to ${\mathcal{P}}_{\bot}f$, the function

[TABLE]

is the unique solution to (1.10). The desired $L^{p}$ estimates of $u$ follow from Theorem 1.4, while estimates of the two latter terms on the right-hand side in (1.12) follow by standard theory for analytic semi-groups; see for example [19, Theorem 4.3.1]. For more details, see also [20, Theorem 5.1]. The connection between maximal regularity for parabolic initial-value problems and corresponding time-periodic problems was observed for the first time in the work of Arendt and Bu [6, Theorem 5.1].

2 Preliminaries and Notation

2.1 Notation

Unless otherwise indicated, $x$ denotes an element in $\mathbb{R}^{n}$ and $x^{\prime}:=(x_{1},\ldots,x_{n-1})\in\mathbb{R}^{n-1}$. The same notation is employed for $\xi\in\mathbb{R}^{n}$ and $\xi^{\prime}:=(\xi_{1},\ldots,\xi_{n-1})\in\mathbb{R}^{n-1}$.

We denote by $\mathbb{C}_{+}:=\{z\in\mathbb{C}\ \lvert\ \operatorname{Im}(z)>0\}$ and $\mathbb{C}_{-}:=\{z\in\mathbb{C}\ \lvert\ \operatorname{Im}(z)<0\}$ the upper and lower complex plane, respectively.

The notation $\partial_{j}:=\partial_{x_{j}}$ is employed for partial derivatives with respect to spatial variables. Throughout, $\partial_{t}$ shall denote the partial derivative with respect to the time variable. For a multi-index $\alpha\in\mathbb{N}^{n}$, we employ the notation $D^{\alpha}:=i^{|\alpha|}\partial_{x_{1}}^{\alpha_{1}}\ldots\partial_{x_{n}}^{\alpha_{n}}$.

We introduce the parabolic length

[TABLE]

We call a generic function $g$ parabolically $\alpha$-homogeneous if $\lambda^{\alpha}g(\eta,\xi)=g(\lambda^{2m}\eta,\lambda\xi)$ for all $\lambda>0$.

2.2 Paley-Wiener Theorem

Definition 2.1**.**

The Hardy space $\mathscr{H}_{+}^{2}(\mathbb{R})$ consists of all functions $f\in L^{2}(\mathbb{R})$ admitting a holomorphic extension to the lower complex plane $\tilde{f}:\mathbb{C}_{-}\rightarrow\mathbb{C}$ with

[TABLE]

The Hardy space $\mathscr{H}_{-}^{2}(\mathbb{R})$ consists of all functions $f\in L^{2}(\mathbb{R})$ admitting a similar holomorphic extension to the upper complex plane.

Proposition 2.2** (Paley-Wiener Theorem).**

Let $f\in L^{2}(\mathbb{R})$. Then $\operatorname{supp}f\subset\mathbb{R}_{+}$ if and only if $\widehat{f}\in\mathscr{H}_{+}^{2}$. Moreover, $\operatorname{supp}f\subset\mathbb{R}_{-}$ if and only if $\widehat{f}\in\mathscr{H}_{-}^{2}$.

Proof.

See for example [26, Theorems VI.4.1 and VI.4.2]. ∎

2.3 Time-periodic function spaces

Let $\Omega\subset\mathbb{R}^{n}$ be a domain and

[TABLE]

the space of smooth time-period functions with compact support in the spatial variable. Clearly,

[TABLE]

are norms on $C^{\infty}_{0,\mathrm{per}}(\mathbb{R}\times\Omega)$. We define Lebesgue and anisotropic Sobolev spaces of time-periodic functions as completions

[TABLE]

One may identify

[TABLE]

On a similar note, one readily verifies that

[TABLE]

provided $\Omega$ satisfies the segment condition.

We introduce anisotropic fractional order Sobolev spaces (Sobolev-Slobodeckiĭ spaces) by real interpolation:

[TABLE]

For a $C^{2m}$-smooth manifold $\Gamma\subset\mathbb{R}^{n}$, anisotropic Sobolev spaces $W^{s,2ms,p}_{\mathrm{per}}(\mathbb{R}\times\Gamma)$ are defined in a similar manner. We can identify (see also Section 2.4 below) the trace space of $W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)$ as $W^{1-1/2mp,2m-1/p,p}_{\mathrm{per}}(\mathbb{R}\times\partial\Omega)$ in the sense that the trace operator maps the former continuously onto the latter.

2.4 Function Spaces and the Torus Group Setting

We shall further introduce a setting of function spaces in which the time axis $\mathbb{R}$ in the underlying domains is replaced with the torus ${\mathbb{T}}:=\mathbb{R}/T\mathbb{Z}$. In such a setting, all functions are inherently $T$-time-periodic. We shall therefore never have to verify periodicity of functions a posteriori, and it will always be clear in which sense the functions are periodic.

The setting of ${\mathbb{T}}$-defined functions is formalized in terms of the canonical quotient mapping $\pi:\mathbb{R}\times\mathbb{R}^{n}\rightarrow{\mathbb{T}}\times\mathbb{R}^{n},\ \pi(t,x):=([t],x)$. A differentiable structure on ${\mathbb{T}}\times\mathbb{R}^{n}$ is inherited via the quotient mapping form $\mathbb{R}\times\mathbb{R}^{n}$. More specifically, for any domain $\Omega\subset\mathbb{R}^{n}$ we let

[TABLE]

and define for $u\in C^{\infty}({\mathbb{T}}\times\Omega)$ derivatives by \partial^{\alpha}u:=\big{(}\partial^{\alpha}[u\circ\pi]\big{)}\circ\pi_{|[0,T)\times\Omega}^{-1}. We let

[TABLE]

denote the space of compactly supported smooth functions. Introducing the normalized Haar measure on ${\mathbb{T}}$, we define norms $\lVert\cdot\rVert_{p}$ and $\lVert\cdot\rVert_{1,2m,p}$ on $C^{\infty}_{0}({\mathbb{T}}\times\Omega)$ as in (2.1)–(2.2). The quotient mapping trivially respects derivatives and is isometric with respect to $\lVert\cdot\rVert_{p}$ and $\lVert\cdot\rVert_{1,2m,p}$. Letting

[TABLE]

we thus obtain Lebesgue and Sobolev spaces that are isometrically isomorphic to the spaces $L_{\mathrm{per}}^{p}(\mathbb{R}\times\Omega)$ and W^{1,2m,p}_{\mathrm{per}}\big{(}\mathbb{R}\times\Omega\big{)}, respectively. Defining weak derivatives with respect to test functions $C^{\infty}_{0}({\mathbb{T}}\times\Omega)$, one readily verifies that

[TABLE]

provided $\Omega$ satisfies the segment property.

For $s\in(0,1)$, we define fractional ordered Sobolev spaces by real interpolation

[TABLE]

and thereby obtain spaces isometrically isomorphic to $W^{s,2ms,p}_{\mathrm{per}}(\mathbb{R}\times\Omega)$. In the half-space case, we clearly have

[TABLE]

Hence, for $l\in\mathbb{N},l\leq 2m$ the trace operator

[TABLE]

extends to a bounded operator

[TABLE]

that is onto; see for example [24, Theorem 1.8.3]. The existence of a bounded right inverse to $\operatorname{Tr}_{l}$ can be shown by applying [24, Theorem 2.9.1].

We further introduce the operators

[TABLE]

which are clearly complementary projections. Since ${\mathcal{P}}f$ is independent of the time variable $t\in\mathbb{R}$, we may at times treat ${\mathcal{P}}f$ as a function of the space variable $x\in\Omega$ only. Both ${\mathcal{P}}$ and ${\mathcal{P}}_{\bot}$ extend to bounded operators on the Lebesgue space $L^{p}({\mathbb{T}}\times\Omega)$ and Sobolev space W^{1,2m,p}\big{(}{\mathbb{T}}\times\Omega\big{)}. We employ the notation $L^{p}_{\bot}({\mathbb{T}}\times\Omega):={\mathcal{P}}_{\bot}L^{p}({\mathbb{T}}\times\Omega)$ and W^{1,2m,p}_{\bot}\big{(}{\mathbb{T}}\times\Omega\big{)}:={\mathcal{P}}_{\bot}W^{1,2m,p}\big{(}{\mathbb{T}}\times\Omega\big{)} for the subspaces of ${\mathcal{P}}_{\bot}$-invariant functions. This notation is canonical extended to other spaces such interpolation spaces of Lebesgue and Sobolev spaces. We sometimes refer to functions with $f={\mathcal{P}}_{\bot}f$ as purely oscillatory.

Finally, we let

[TABLE]

and put

[TABLE]

2.5 Schwartz-Bruhat Spaces and Distributions

When the spatial domain is the whole-space $\mathbb{R}^{n}$, we employ the notation ${G:={\mathbb{T}}\times\mathbb{R}^{n}}$. Equipped with the quotient topology via $\pi$, $G$ becomes a locally compact abelian group. Clearly, the $L^{p}(G)$ space corresponding to the Haar measure on $G$, appropriately normalized, coincides with the $L^{p}({\mathbb{T}}\times\mathbb{R}^{n})$ space introduced in the previous section.

We identify $G$’s dual group by $\widehat{G}=\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n}$ by associating $(k,\xi)\in\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n}$ with the character $\chi:G\rightarrow\mathbb{C},\ \chi(x,t):=\operatorname{e}^{ix\cdot\xi+ikt}$. By default, $\widehat{G}$ is equipped with the compact-open topology, which in this case coincides with the product of the discrete topology on $\frac{2\pi}{T}\mathbb{Z}$ and the Euclidean topology on $\mathbb{R}^{n}$. The Haar measure on $\widehat{G}$ is simply the product of the Lebesgue measure on $\mathbb{R}^{n}$ and the counting measure on $\frac{2\pi}{T}\mathbb{Z}$.

The Schwartz-Bruhat space $\mathscr{S}(G)$ of generalized Schwartz functions (originally introduced in [9]) can be described in terms of the semi-norms

[TABLE]

as

[TABLE]

The vector space $\mathscr{S}(G)$ is endowed with the semi-norm topology.

The topological dual space $\mathscr{S^{\prime}}(G)$ of $\mathscr{S}(G)$ is referred to as the space of tempered distributions on $G$. Observe that both $\mathscr{S}(G)$ and $\mathscr{S^{\prime}}(G)$ remain closed under multiplication by smooth functions that have at most polynomial growth with respect to the spatial variables. For a tempered distribution $u\in\mathscr{S^{\prime}}(G)$, distributional derivatives $\partial_{t}^{\alpha}\partial_{x}^{\beta}u\in\mathscr{S^{\prime}}(G)$ are defined by duality in the usual manner. Also the support $\operatorname{supp}u$ is defined in the classical way. Moreover, we may restrict the distribution $u$ to a subdomain ${\mathbb{T}}\times\Omega$ by considering it as a functional defined only on the test functions from $\mathscr{S}(G)$ supported in ${\mathbb{T}}\times\Omega$.

A differentiable structure on $\widehat{G}$ is obtained by introduction of the space

[TABLE]

The Schwartz-Bruhat space on the dual group $\widehat{G}$ is defined in terms of the semi-norms

[TABLE]

as

[TABLE]

We also endow $\mathscr{S}(\widehat{G})$ with the corresponding semi-norm topology and denote by $\mathscr{S^{\prime}}(\widehat{G})$ the topological dual space.

2.6 Fourier Transform

As a locally compact abelian group, $G$ has a Fourier transform $\mathscr{F}_{G}$ associated to it. The ability to utilize a Fourier transform that acts simultaneously in time $t\in{\mathbb{T}}$ and space $x\in\mathbb{R}^{n}$ shall play a key role in the following.

The Fourier transform $\mathscr{F}_{G}$ on $G$ is given by

[TABLE]

If no confusion can arise, we simply write $\mathscr{F}$ instead of $\mathscr{F}_{G}$. The inverse Fourier transform is formally defined by

[TABLE]

It is standard to verify that $\mathscr{F}:\mathscr{S}(G)\rightarrow\mathscr{S}(\widehat{G})$ is a homeomorphism with $\mathscr{F}^{-1}$ as the actual inverse, provided the Lebesgue measure ${\mathrm{d}}\xi$ is normalized appropriately. By duality, $\mathscr{F}$ extends to a bijective mapping $\mathscr{F}:\mathscr{S^{\prime}}(G)\rightarrow\mathscr{S^{\prime}}(\widehat{G})$. The Fourier transform provides us with a calculus between the differential operators on $G$ and the polynomials on $\widehat{G}$. As one easily verifies, for $u\in\mathscr{S^{\prime}}(G)$ and $(\alpha,\beta)\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{n}$ we have \mathscr{F}\big{(}\partial_{t}^{\alpha}\partial_{x}^{\beta}u\big{)}=i^{{\lvert\alpha\rvert}+{\lvert\beta\rvert}}\,k^{\alpha}\,\xi^{\beta}\,\mathscr{F}(u) as an identity in $\mathscr{S^{\prime}}(\widehat{G})$.

The projections introduced in (2.5) can be extended trivially to projections on the Schwartz-Bruhat space ${\mathcal{P}},{\mathcal{P}}_{\bot}:\mathscr{S}(G)\rightarrow\mathscr{S}(G)$. Introducing the delta distribution $\delta_{\mathbb{Z}}$ on $\frac{2\pi}{T}\mathbb{Z}$, that is, $\delta_{\mathbb{Z}}(k):=1$ if $k=0$ and $\delta_{\mathbb{Z}}(k):=0$ for $k\neq 0$, we observe that {\mathcal{P}}u=\mathscr{F}^{-1}_{G}\big{[}\delta_{\mathbb{Z}}\mathscr{F}_{G}[u]\big{]} and {\mathcal{P}}_{\bot}u=\mathscr{F}^{-1}_{G}\big{[}(1-\delta_{\mathbb{Z}})\mathscr{F}_{G}[u]\big{]}. Using these representations for ${\mathcal{P}}$ and ${\mathcal{P}}_{\bot}$, we naturally extend the projections to operators ${\mathcal{P}},{\mathcal{P}}_{\bot}:\mathscr{S^{\prime}}(G)\rightarrow\mathscr{S^{\prime}}(G)$. In accordance with the notation introduced above, we put $\mathscr{S^{\prime}_{\bot}}(G):={\mathcal{P}}_{\bot}\mathscr{S^{\prime}}(G)$.

In general, we shall utilize smooth functions $\mathsf{m}\in C^{\infty}(\widehat{G})$ with at most polynomial growth as Fourier multipliers by introducing the corresponding operator

[TABLE]

We call $\mathsf{m}$ an $L^{p}(G)$-multiplier if $\mathsf{op}\,[\mathsf{m}]$ extends to a bounded operator on $L^{p}(G)$ for any $p\in(1,\infty)$. The following lemmas provide us with criteria to determine if $\mathsf{m}$ is an $L^{p}(G)$-multiplier.

Lemma 2.3**.**

Let $\mathsf{m}\in C^{\infty}(\widehat{G})$. If $\mathsf{m}=\mathfrak{m}_{|\widehat{G}}$ for some parabolically [math]-homogeneous $\mathfrak{m}:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{C}$, then $\mathsf{op}\,[\mathsf{m}]$ extends to a bounded operator $\mathsf{op}\,[\mathsf{m}]:L^{p}(G)\rightarrow L^{p}(G)$.

Proof.

The Transference Principle (established originally by de Leeuw [10] and later extended to a general setting of locally compact abelian groups by Edwards and Gaudry [13, Theorem B.2.1]), makes it possible to “transfer” the investigation of Fourier multipliers from one group setting into another. In our case, [13, Theorem B.2.1] yields that $\mathsf{m}$ is an $L^{p}(G)$-multiplier, provided $\mathfrak{m}$ is an $L^{p}(\mathbb{R}\times\mathbb{R}^{n})$-multiplier. To show the latter, we can employ one of the classical multiplier theorems available in the Euclidean setting. Since $\mathfrak{m}$ is parabolically [math]-homogeneous, it is easy to verify that $\mathfrak{m}$ meets for instance the conditions of the Marcinkiewicz’s multiplier theorem ([22, Chapter IV, §6]). Thus, $\mathfrak{m}$ is an $L^{p}(\mathbb{R}\times\mathbb{R}^{n})$-multiplier and by [13, Theorem B.2.1] $\mathsf{m}$ therefore an $L^{p}(G)$-multiplier. ∎

Lemma 2.4**.**

Let $\mathsf{m}\in C^{\infty}(\widehat{G}\setminus\{(0,0)\})$ and $\alpha\leq 0$. If $\mathsf{m}=\mathfrak{m}_{|\widehat{G}}$ for some parabolically $\alpha$-homogeneous function $\mathfrak{m}:\mathbb{R}\times\mathbb{R}^{n}\setminus\{(0,0)\}\rightarrow\mathbb{C}$, then $\mathsf{op}\,[\mathsf{m}]$ extends to a bounded operator $\mathsf{op}\,[\mathsf{m}]:L^{p}_{\bot}(G)\rightarrow L^{p}_{\bot}(G)$.

Proof.

Let $\chi\in C^{\infty}(\mathbb{R})$ be a “cut-off” function with $\chi(\eta)=0$ for ${\lvert\eta\rvert}<\frac{\pi}{T}$ and $\chi(\eta)=1$ for ${\lvert\eta\rvert}\geq\frac{2\pi}{T}$. Put $\mathfrak{M}(\eta,\xi):=\chi(\eta)\mathfrak{m}(\eta,\xi)$. Utilizing that $\mathfrak{m}$ is $\alpha$-homogeneous and $\alpha\leq 0$, one readily verifies that $\mathfrak{M}$ satisfies the conditions of Marcinkiewicz’s multiplier theorem ([22, Chapter IV, §6]). Consequently, $\mathfrak{M}$ is an $L^{p}(\mathbb{R}\times\mathbb{R}^{n})$-multiplier. For $u\in L^{p}_{\bot}(G)$, we have $u={\mathcal{P}}_{\bot}u$ and thus

[TABLE]

Since $\mathsf{m}(1-\delta_{\mathbb{Z}})=\mathfrak{M}_{|\widehat{G}}$, we obtain from [13, Theorem B.2.1] that $\mathsf{m}(1-\delta_{\mathbb{Z}})$ is an $L^{p}(G)$-multiplier. Consequently, $\mathsf{op}\,[\mathsf{m}](u)\leq C\lVert u\rVert_{p}$ for all $u\in L^{p}_{\bot}(G)$. ∎

Corollary 2.5**.**

Let $p\in(1,\infty)$ and $\beta\in[0,1]$. If $M\in C^{\infty}(\frac{2\pi}{T}\mathbb{Z}\setminus\{0\}\times\mathbb{R}^{n})$ is parabolically [math]-homogeneous, then $\mathsf{op}\,[M]$ extends to a bounded operator $\mathsf{op}\,[M]:W^{\beta,2m\beta,p}_{\bot}(G)\rightarrow W^{\beta,2m\beta,p}_{\bot}(G)$.

2.7 Time-Periodic Bessel Potential Spaces

Time-periodic Bessel Potential spaces can be defined via the Fourier transform $\mathscr{F}_{G}$. We shall only introduce Bessel Potential spaces of purely oscillatory distributions:

[TABLE]

Utilizing Lemma 2.4, one readily verifies that $H^{s,p}_{\bot}$ is a Banach space with respect to the norm

[TABLE]

Time-periodic Bessel Potential spaces on the half-space are defined via restriction of distributions in the time-periodic Bessel Potential spaces defined above:

[TABLE]

Identifying $H^{s,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ as a factor space of $H^{s,p}_{\bot}$ in the canonical way, we see that $H^{s,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ is a Banach space in the norm $\lVert\cdot\rVert_{s,p,{\mathbb{T}}\times\mathbb{R}^{n}_{+}}$.

Proposition 2.6**.**

Let $p\in(1,\infty)$. Then $H^{2m,p}_{\bot}(G)=W^{1,2m,p}_{\bot}(G)$ and $H^{2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})=W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ with equivalent norms.

Proof.

It follows from Lemma 2.4 that \mathsf{op}\,\big{[}|\eta,\xi|^{-m}\big{]} extends to a bounded operator on $L^{p}_{\bot}(G)$, which implies that $\lVert f\rVert_{m,p}$ is equivalent to the norm $\lVert f\rVert_{p}+\lVert f\rVert_{m,p}$. From this, we infer that $H^{2m,p}_{\bot}=W^{1,2m,p}_{\bot}(G)$. A standard method (see for example [1, Theorem 4.26]) can be used to construct an extension operator $\operatorname{Ext}:W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})\rightarrow W^{1,2m,p}_{\bot}(G)$. The existence of an extension operator combined with the fact that $H^{2m,p}_{\bot}=W^{1,2m,p}_{\bot}(G)$ implies $H^{2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})=W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$. ∎

Proposition 2.7**.**

Let $s\in\mathbb{R}$. Then

[TABLE]

for all $u\in\mathscr{S^{\prime}}(G)$.

Proof.

The complex function $z\mapsto(iz+|k,\xi^{\prime}|)^{-1}$ is holomorphic in the lower complex plane. Due to Lemma 2.3, we can employ Proposition 2.2 to conclude

[TABLE]

for all $\psi\in\mathscr{S}_{\bot}(G)$. By duality, the same is true for all $\psi\in\mathscr{S^{\prime}_{\bot}}(G)$. We employ Lemma 2.3 to estimate

[TABLE]

It follows from (2.9) that

[TABLE]

We thus conclude

[TABLE]

and thereby (2.6). Furthermore,

[TABLE]

where the last inequality follows by an application of Lemma 2.4. We have thus shown (2.7). One may verify (2.8) in a similar manner. ∎

Lemma 2.8**.**

Let $\beta\in(0,1)$ and \alpha\in\big{(}2m(\beta-1),2m\beta\big{)}. Then \mathsf{op}\,\big{[}|k,\xi|^{\alpha}\big{]} extends to a bounded operator \mathsf{op}\,\big{[}|k,\xi|^{\alpha}\big{]}:W^{\beta,2m\beta,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n})\rightarrow W^{\beta-\frac{\alpha}{2m},2m\beta-\alpha}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}).

Proof.

By interpolation, we directly obtain that \mathsf{op}\,\big{[}|k,\xi|^{\alpha}\big{]} extends to a bounded operator

[TABLE]

for any $\theta\in(0,1)$. Choose $\theta:=\frac{2m\beta-\alpha}{2m-\alpha}$. Using a dyadic decomposition of the Fourier space with respect to the parabolic length, $H^{s,p}_{\bot}$ can be identified as the complex interpolation space $[H^{s_{0},p}_{\bot},H^{s_{1},p}_{\bot}]_{\omega}$ with $\frac{1}{s}=\frac{1-\omega}{s_{0}}+\frac{\omega}{s_{1}}$ by verifying that it is a retract of $L^{p}(l^{s,2})$ as in [8, Theorem 6.4.3]. In fact, relying on the Transference Principle, we have a Mikhlin’s multiplier theorem at our disposal, which is the key ingredient in [8, Theorem 6.4.3]. Hence, by reiteration and Proposition 2.6

[TABLE]

Similarly, \big{(}L^{p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}),H^{2m-\alpha,p}_{\bot}\big{)}_{\theta,p}=W^{\beta-\frac{\alpha}{2m},2m\beta-\alpha}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}). ∎

Finally, we characterize the trace spaces of the time-periodic Bessel Potential spaces.

Lemma 2.9**.**

Let $p\in(1,\infty)$. The trace operator $\operatorname{Tr}_{m}$ defined in (2.3) extends to a bounded operator

[TABLE]

that is onto and has a bounded right inverse. If $u\in H^{m,p}_{\bot}(G)$ with $\operatorname{supp}(u)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$, then \operatorname{Tr}_{m}\big{(}u_{|{\mathbb{T}}\times\mathbb{R}^{n}_{+}}\big{)}=0. If $u\in H^{m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ with \operatorname{Tr}_{m}\big{(}u_{|{\mathbb{T}}\times\mathbb{R}^{n}_{+}}\big{)}=0, then $u$ is the restriction of a function $U\in H^{m,p}_{\bot}(G)$ with $\operatorname{supp}U\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$.

Proof.

For either $I=\mathbb{R}$ or $I=\mathbb{R}_{+}$, put

[TABLE]

We first verify that $H^{m,p}_{\bot}=V(\mathbb{R})$ with equivalent norms. It is straightforward to obtain the embedding $H^{m,p}_{\bot}\hookrightarrow V(\mathbb{R})$. To show the reverse embedding, consider $u\in V(\mathbb{R})$. Then $\lVert\mathsf{op}\,[|k,{\xi^{\prime}}|^{m}]u\rVert_{p}\leq\lVert u\rVert_{V(\mathbb{R})}$ and $\lVert\mathsf{op}\,[\xi_{n}^{m}]u\rVert_{p}\leq\lVert u\rVert_{V(\mathbb{R})}$. By Lemma 2.3,

[TABLE]

is an $L^{p}(G)$ multiplier. It follows that $\lVert\mathsf{op}\,[|k,\xi|^{m}]u\rVert_{p}\leq\lVert u\rVert_{V(\mathbb{R})}$ and thus the embedding $V(\mathbb{R})\hookrightarrow H^{m,p}_{\bot}$. We conclude $H^{m,p}_{\bot}=V(\mathbb{R})$. It is standard to show existence of an extension operator $V(\mathbb{R}_{+})\rightarrow V(\mathbb{R})$; see for example [24, Lemma 2.9.1]. By restriction to ${\mathbb{T}}\times\mathbb{R}^{n}_{+}$, it thus follows that $H^{m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})=V(\mathbb{R}_{+})$. The classical trace method now implies that trace operator extends to a bounded operator

[TABLE]

that is onto; see for example [24, Theorem 1.8.3]. The existence of a bounded right inverse can be shown as in [24, Theorem 2.9.1]. Again by reiteration we identify

[TABLE]

Thus, we conclude (2.10).

Consider now $u\in H^{m,p}_{\bot}(G)$ with $\operatorname{supp}(u)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$. As above we can identify $u$ as an element of $V(\mathbb{R})$, which necessarily satisfies $u(0)=0$. It follows that \operatorname{Tr}_{m}\big{(}u_{|{\mathbb{T}}\times\mathbb{R}^{n}_{+}}\big{)}=0. If on the other hand $u\in H^{m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ with \operatorname{Tr}_{m}\big{(}u_{|{\mathbb{T}}\times\mathbb{R}^{n}_{+}}\big{)}=0, then it is standard to show that $u$ can be approximated by a sequence of functions from $C^{\infty}_{0}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$; see for example [24, Theorem 2.9.1]. Clearly, this sequence also converge in $H^{m,p}_{\bot}(G)$. The limit function $U\in H^{m,p}_{\bot}(G)$ satisfies $\operatorname{supp}U\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$ and $U_{|{\mathbb{T}}\times\mathbb{R}^{n}_{+}}=u$. ∎

3 Constant Coefficients in the Whole- and Half-Space

In this section, we establish the assertion of Theorem 1.4. We first treat the whole-space case, and then show the theorem as stated in the half-space case. Since we consider only the differential operators with constant coefficients in this section, we employ the simplified notation $A(D)$ instead of $A(x,D)$. Replacing the differential operator $D$ with $\xi\in\mathbb{R}^{n}$, we refer to $A(\xi)$ as the symbol of $A(D)$.

3.1 The Whole Space

We consider first the case of the spatial domain being the whole-space $\mathbb{R}^{n}$. The time-space domain then coincides with the locally abelian group $G$, and we can thus employ the Fourier transform $\mathscr{F}_{G}$ and base the proof on an investigation of the corresponding Fourier multipliers.

Lemma 3.1**.**

Assume that $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $p\in(1,\infty)$, $s\in\mathbb{R}$ and

[TABLE]

Then the linear operators

[TABLE]

extend uniquely to bounded operators

[TABLE]

In the setting (3.2), $\mathsf{A}^{-1}$ is the actual inverse of $\mathsf{A}$.

Proof.

Let

[TABLE]

Clearly, $\mathfrak{m}$ is parabolically [math]-homogeneous. By Lemma 2.4, it follows that $\mathsf{m}:=\mathfrak{m}_{|\widehat{G}}$ is an $L^{p}(G)$-multiplier. Since $\mathsf{M}={|k,\xi|^{2m}\mathsf{m}(k,\xi)}$, we conclude that

[TABLE]

Since $A^{H}$ satisfies Agmon’s condition for $\theta=\pm\frac{\pi}{2}$, it follows that $A^{H}(\xi)\notin i\mathbb{R}$ for all $\xi\in\mathbb{R}^{n}\setminus\{0\}$. Consequently, $\mathfrak{m}^{-1}:\mathbb{R}\times\mathbb{R}^{n}\setminus\{(0,0)\}\rightarrow\mathbb{C}$ is well-defined and clearly parabolically [math]-homogeneous. We deduce as above that

[TABLE]

Consequently, $\mathsf{A}$ and $\mathsf{A}^{-1}$ extend uniquely to bounded operators $\mathsf{A}:H^{s,p}_{\bot}\to H^{s-2m,p}_{\bot}$ and $\mathsf{A}^{-1}:H^{s-2m,p}_{\bot}\to H^{s,p}_{\bot}$, respectively. Clearly, $\mathsf{A}^{-1}$ is the actual inverse of $\mathsf{A}$. ∎

Theorem 3.2**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $s\in\mathbb{R}$ and $p\in(1,\infty)$. There is a constant $C>0$ such that

[TABLE]

for all $u\in H^{s,p}_{\bot}(G)$.

Proof.

Since $\mathsf{A}u=\partial_{t}u+A^{H}u$, we employ Lemma 3.1 to estimate

[TABLE]

Since the differential operator $A-A^{H}$ contains derivatives of at most of order $2m-1$, we conclude (3.3) by a similar multiplier argument as in the proof of Lemma 3.1. ∎

3.2 The Half Space with Dirichlet Boundary Condition

In the next step, we consider the case of the spatial domain being the half-space $\mathbb{R}^{n}_{+}$ and boundary operators corresponding to Dirichlet boundary conditions. As in the whole-space case, we shall work with the symbol of $\partial_{t}+A^{H}$. In the following lemma, we collect its key properties.

Lemma 3.3**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let

[TABLE]

1. (1)

For every $(\eta,\xi^{\prime})\in{\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}}$ the complex polynomial $z\mapsto\mathfrak{M}(\eta,\xi^{\prime},z)$ has exactly $m$ roots $\rho_{j}^{+}(\eta,\xi^{\prime})\in\mathbb{C}_{-}$ in the upper complex plane, and $m$ roots $\rho_{j}^{-}(\eta,\xi^{\prime})\in\mathbb{C}_{+}$ in the lower complex plane ($j\in\{1,\ldots,m\})$. 2. (2)

The functions

[TABLE]

are parabolically $m$-homogeneous. 3. (3)

The coefficients of the polynomials $z\mapsto\mathfrak{M}_{\pm}(\eta,\xi^{\prime},z)$, more specifically the functions $c^{\pm}_{\alpha}:\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}\rightarrow\mathbb{C}$ $(\alpha=0,\ldots,m)$ with the property that

[TABLE]

are analytic. Moreover, $c^{\pm}_{\alpha}$ is parabolically $\alpha$-homogeneous.

Proof.

1. (1)

Since $A^{H}$ is properly elliptic, the polynomial $z\mapsto\mathfrak{M}(0,\xi^{\prime},z)$ has exactly m roots in the upper and lower complex plane, respectively. Recall that $A^{H}(x,\xi)\notin i\mathbb{R}$ for all $\xi\in\mathbb{R}^{n}\setminus\{0\}$. Since the roots of a polynomial depend continuously on the polynomial’s coefficients, we deduce part (1) of the lemma. 2. (2)

Since $\mathfrak{M}$ is parabolically $2m$-homogeneous, the roots $\rho_{j}^{\pm}$ are parabolically $1$-homogeneous. It follows that $\mathfrak{M}_{\pm}$ is parabolically $m$-homogeneous. 3. (3)

The analyticity of the coefficients $c^{\pm}_{\alpha}$ follows by a classical argument; see for example [23, Chapter 4.4]. The coefficient $c^{\pm}_{\alpha}$ being parabolically $\alpha$-homogeneous is a direct consequence of $\mathfrak{M}_{\pm}$ being $m$-homogeneous. ∎

Lemma 3.4**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Put $\mathsf{M}_{\pm}:=\mathfrak{M}_{\pm}|_{\widehat{G}}$, where $\mathfrak{M}_{\pm}$ is defined by (3.4). Let $p\in(1,\infty)$ and $s\in\mathbb{R}$. Then the linear operators

[TABLE]

extend uniquely to bounded and mutually inverse operators $\mathsf{A}_{\pm}:H^{s,p}_{\bot}(G)\to H^{s-m,p}_{\bot}(G)$ and $\mathsf{A}_{\pm}^{-1}:H^{s-m,p}_{\bot}(G)\to H^{s,p}_{\bot}(G)$, respectively.

Proof.

The assertion of the lemma follows as in the proof of Lemma 3.1, provided we can show that the restriction to $\widehat{G}$ of the multiplier

[TABLE]

and its inverse are $L^{p}_{\bot}(G)$-multipliers. Although $\mathfrak{m}$ is parabolically [math]-homogeneous, we cannot apply Lemma 2.4 directly since $\mathfrak{m}$ is not defined on all of $\mathbb{R}\times\mathbb{R}^{n}\setminus\{(0,0)\}$. Instead, we recall (3.5) and observe that

[TABLE]

Owing to the $\alpha$-homogeneity of $c^{\pm}_{\alpha}$, Lemma 2.4 yields that both ${\mathfrak{m}_{1}^{\alpha}}_{|\widehat{G}}$ and ${\mathfrak{m}_{2}^{\alpha}}_{|\widehat{G}}$ are $L^{p}(G)$-multipliers. Consequently, also $\mathfrak{m}$ is an $L^{p}(G)$-multiplier, and we thus conclude as in the proof of Lemma 3.1 that $\mathsf{A}_{\pm}$ extends uniquely to a bounded operator $\mathsf{A}_{\pm}:H^{s,p}_{\bot}(G)\to H^{s-m,p}_{\bot}(G)$.

To show the corresponding property for $\mathsf{A}_{\pm}^{-1}$, we introduce a cut-off function $\chi\in C^{\infty}(\mathbb{R})$ with $\chi(\eta)=0$ for ${\lvert\eta\rvert}<\frac{\pi}{T}$ and $\chi(\eta)=1$ for ${\lvert\eta\rvert}\geq\frac{2\pi}{T}$. We claim that

[TABLE]

is an $L^{p}(\mathbb{R}\times\mathbb{R}^{n})$-multiplier. Indeed, utilizing that $\mathfrak{M}_{\pm}$ is $m$-homogeneous, we see that $\mathfrak{M}_{\pm}$ can be bounded below by

[TABLE]

where the infimum above is strictly positive due to the roots in definition (3.4) satisfying $\lim_{(\eta,{\xi^{\prime}})\rightarrow(0,0)}\rho_{j}^{\pm}(\eta,{\xi^{\prime}})=0$. Using only (3.7) and the $\alpha$-homogeneity of the coefficients $c^{\pm}_{\alpha}$ as in (3.6), it is now straightforward to verify that $\mathfrak{\widetilde{m}}$ satisfies the condition of the Marcinkiewicz’s multiplier theorem ([22, Chapter IV, §6]). Thus, $\mathfrak{\widetilde{m}}$ is an $L^{p}(\mathbb{R}\times\mathbb{R}^{n})$-multiplier and by [13, Theorem B.2.1] ${\mathfrak{\widetilde{m}}}_{|{\widehat{G}}}$ therefore an $L^{p}(G)$-multiplier. Since the restriction of the corresponding operator \mathsf{op}\,\big{[}{\mathfrak{\widetilde{m}}}_{|{\widehat{G}}}\big{]}:L^{p}_{\bot}(G)\rightarrow L^{p}_{\bot}(G) to the subspace of purely periodic functions coincides with \mathsf{op}\,\big{[}{\mathfrak{m}^{-1}}_{|{\widehat{G}}}\big{]}:L^{p}_{\bot}(G)\rightarrow L^{p}_{\bot}(G), we deduce as in the proof of Lemma 3.1 that $\mathsf{A}_{\pm}^{-1}$ extends uniquely to a bounded operator $\mathsf{A}_{\pm}^{-1}:H^{s-m,p}_{\bot}(G)\to H^{s,p}_{\bot}(G)$. ∎

The lemma above provides us with at decomposition of the differentiable operators in (3.2), that is, for $\mathsf{A}:H^{s,p}_{\bot}(G)\to H^{s-2m,p}_{\bot}(G)$ and $\mathsf{A}^{-1}:H^{s-2m,p}_{\bot}(G)\to H^{s,p}_{\bot}(G)$ the decompositions $\mathsf{A}=\mathsf{A}_{+}\mathsf{A}_{-}=\mathsf{A}_{-}\mathsf{A}_{+}$ and $\mathsf{A}^{-1}=\mathsf{A}_{+}^{-1}\mathsf{A}_{-}^{-1}=\mathsf{A}_{-}^{-1}\mathsf{A}_{+}^{-1}$ are valid provided $\mathsf{A}$ is normalized accordingly. Employing the Paley-Wiener Theorem, we shall now show that the operators $\mathsf{A}_{\pm}$ and $\mathsf{A}^{-1}_{\pm}$ “respect” the support of a function in the upper (lower) half-space.

Lemma 3.5**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $p\in(1,\infty)$, $s\in\mathbb{R}$ and consider $u\in H^{s,p}_{\bot}(G)$.

1. (i)

If $\operatorname{supp}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$, then $\operatorname{supp}\mathsf{A}_{+}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$ and $\operatorname{supp}\mathsf{A}_{+}^{-1}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$. 2. (ii)

If $\operatorname{supp}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}$, then $\operatorname{supp}\mathsf{A}_{-}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}$ and $\operatorname{supp}\mathsf{A}_{-}^{-1}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}$.

Proof.

We shall prove only part (i), for part (ii) follows analogously. We employ the notation ${H}:={\mathbb{T}}\times\mathbb{R}^{n-1}$ and the canonical decomposition $\mathscr{F}_{G}=\mathscr{F}_{H}\mathscr{F}_{\mathbb{R}}$ of the Fourier transform. In view of Lemma 3.4, it suffices to consider only $u\in\mathscr{S}(G)$ with $\operatorname{supp}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$.

For fixed ${k}\in\frac{2\pi}{T}\mathbb{Z}\setminus\{0\}$ and $\xi^{\prime}\in\mathbb{R}^{n-1}$, we let $\mathsf{D}({k},\xi^{\prime}):=\mathscr{F}_{\mathbb{R}}^{-1}\mathsf{M}_{+}({k},\xi^{\prime},\cdot)\mathscr{F}_{\mathbb{R}}$. Since $\mathsf{M}_{+}$ is a polynomial with respect to the variable $\xi_{n}$, $\mathsf{D}({k},\xi^{\prime})$ is a differential operator in $x_{n}$ and hence $\operatorname{supp}(\mathsf{D}({k},\xi^{\prime})f)\subset\overline{\mathbb{R}_{+}}$ for every $f\in\mathscr{S}^{\prime}(\mathbb{R})$ with $\operatorname{supp}f\subset\overline{\mathbb{R}_{+}}$. Clearly, $\operatorname{supp}([\mathscr{F}_{H}u]({k},\xi^{\prime},\cdot))\subset\overline{\mathbb{R}_{+}}$. Since $\mathscr{F}_{H}[\mathsf{A}_{+}u]({k},\xi^{\prime},\cdot)=[\mathsf{D}({k},\xi^{\prime})\mathscr{F}_{H}u]({k},\xi^{\prime},\cdot)$, we conclude $\operatorname{supp}\mathsf{A}_{+}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$.

To show the same property for $\mathsf{A}_{+}^{-1}u$, we employ the version of the Paley-Wiener Theorem presented in Proposition 2.2. Since $u\in\mathscr{S}(G)\subset L^{2}(G)$, we immediately obtain that for fixed ${k}\in\frac{2\pi}{T}\mathbb{Z}$ and $\xi^{\prime}\in\mathbb{R}^{n-1}$, the Fourier transform $[\mathscr{F}_{G}u]({k},\xi^{\prime},\cdot)$ is in the Hardy space $\mathscr{H}_{+}^{2}(\mathbb{R})$. Let

[TABLE]

denote the extension of $\mathsf{M}_{+}^{-1}({k},\xi^{\prime},\cdot)$ to the lower complex plane. Since all the roots $\rho_{j}^{+}$ lie in the upper complex plane, this extension is holomorphic and bounded. It follows that $[\mathsf{M}_{+}^{-1}\mathscr{F}_{G}u]({k},\xi^{\prime},\cdot)\in\mathscr{H}_{+}^{2}(\mathbb{R})$. Hence, taking the inverse Fourier transform, Proposition 2.2 yields $\operatorname{supp}\mathsf{A}_{+}^{-1}u\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$. ∎

The above properties of $\mathsf{A}_{\pm}$ and $\mathsf{A}^{-1}_{\pm}$ lead to a surprisingly simple representation formula, see (3.8) below, for the solution $u$ to the problem $\partial_{t}u+A^{H}u=f$ in the half-space ${\mathbb{T}}\times\mathbb{R}^{n}_{+}$ with Dirichlet boundary conditions. The problem itself can be formulated elegantly as (3.9).

Lemma 3.6**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $p\in(1,\infty)$ and $f\in H^{-m,p}_{\bot}(G)$. Let $Y_{+}$ denote the characteristic function on ${\mathbb{T}}\times\mathbb{R}^{n}_{+}$. Then

[TABLE]

is the unique solution in $H^{m,p}_{\bot}(G)$ to

[TABLE]

Moreover, there is a constant $c=c(n,p)>0$ such that

[TABLE]

Proof.

By Lemma 3.4, $\mathsf{A}_{-}^{-1}f\in L^{p}_{\bot}(G)$. Clearly then $Y_{+}\mathsf{A}_{-}^{-1}f\in L^{p}_{\bot}(G)$ and trivially $\operatorname{supp}(Y_{+}\mathsf{A}_{-}^{-1}f)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$. Lemma 3.5 now implies $\operatorname{supp}(\mathsf{A}_{+}^{-1}Y_{+}\mathsf{A}_{-}^{-1}f)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$, which concludes the first part of (3.9). Since \operatorname{supp}\big{(}(Y_{+}-\operatorname{\textsf{id}})\mathsf{A}_{-}^{-1}f\big{)}\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}, Lemma 3.5 implies \operatorname{supp}\big{(}\mathsf{A}_{-}(Y_{+}-\operatorname{\textsf{id}})\mathsf{A}_{-}^{-1}f\big{)}\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}. However, $\mathsf{A}u-f=\mathsf{A}_{-}(Y_{+}-\operatorname{\textsf{id}})\mathsf{A}_{-}^{-1}f$, whence the second part of (3.9) follows.

To show uniqueness, let $v\in H^{m,p}_{\bot}$ be a solution to (3.9) with $f=0$. Since $\mathsf{A}_{+}v=\mathsf{A}_{-}^{-1}\mathsf{A}v$, Lemma 3.5 yields $\operatorname{supp}(\mathsf{A}_{+}v)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}$. On the other hand, $\operatorname{supp}v\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$ by assumption, whence $\operatorname{supp}\mathsf{A}_{+}v\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$ by Lemma 3.5. Recalling that $\mathsf{A}_{+}v\in L^{p}_{\bot}(G)$ by Lemma 3.4, we thus deduce $\mathsf{A}_{+}v=0$ and consequently $v=0$. This concludes the assertion of uniqueness.

It remains to show (3.10). For $F\in H^{-m,p}_{\bot}(G)$ with $\operatorname{supp}(F-f)\subset{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{-}}$, we can utilize Lemma 3.4 and Lemma 3.5 as above to conclude that $Y_{+}\mathsf{A}_{-}^{-1}F=Y_{+}\mathsf{A}_{-}^{-1}f$, which in turn implies $u=\mathsf{A}_{+}^{-1}Y_{+}\mathsf{A}_{-}^{-1}F$. Recalling Lemma 3.1, we estimate

[TABLE]

∎

Finally, we can establish the main theorem in the case of the spatial domain being the half-space $\mathbb{R}^{n}_{+}$ and boundary operators corresponding to Dirichlet boundary conditions.

Theorem 3.7**.**

Assume $A^{H}$ is properly elliptic and satisfies Agmon’s condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $p\in(1,\infty)$. For $f\in L^{p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ and $g\in T_{\bot}^{\iota,p}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ there is a unique $u\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ subject to

[TABLE]

Moreover, there is a constant $c=c(n,p)>0$ such that

[TABLE]

Proof.

We first assume $g=0$. Extending $f$ by zero to the whole space ${\mathbb{T}}\times\mathbb{R}^{n}$, we have $f\in L^{p}_{\bot}(G)\subset H^{-m,p}_{\bot}(G)$. Let $u\in H^{m,p}_{\bot}(G)$ be the solution to (3.9) from Lemma 3.6. Lemma 2.9 yields $\operatorname{Tr}_{m}u=0$. Thus, $u$ is a solution to (3.11). We shall establish higher order regularity of $u$ iteratively. For this purpose, we employ Proposition 2.7 to estimate

[TABLE]

Since the symbol of $\mathsf{A}$ reads $\mathsf{M}({k},\xi^{\prime},\xi_{n})=a\xi_{n}^{2m}+i{k}+\sum_{k=0}^{2m-1}\sum_{|\alpha|=2m-k}a_{\alpha,k}(\xi^{\prime})^{\alpha}\xi_{n}^{k}$ with $a\neq 0$, we deduce with the help of Lemma 2.4 that

[TABLE]

Consequently,

[TABLE]

Clearly, \mathsf{op}\,\big{[}|k,{\xi^{\prime}}|\big{]} commutes with $\mathsf{A}_{+}^{-1}Y_{+}\mathsf{A}_{-}^{-1}$, whence

[TABLE]

By Lemma 3.6, we finally obtain

[TABLE]

Iterating this procedure, we obtain after $m$ steps the desired regularity $u\in H^{2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ together with the estimate $\lVert u\rVert_{2m,p,{\mathbb{T}}\times\mathbb{R}^{n}_{+}}\leq C\lVert f\rVert_{p}$. Recalling from Proposition 2.6 that $H^{2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})=W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$, we conclude (3.12) in the case $g=0$.

If $g\neq 0$, we recall the properties (2.4) of the trace operator and choose a function $v\in W^{1,2m,p}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ with $\operatorname{Tr}_{m}v=g$ and $\lVert v\rVert_{2m,p}\leq C\lVert g\rVert_{T_{\bot}^{\iota,p}({\mathbb{T}}\times\mathbb{R}^{n}_{+})}$. With $w:=u-v$, problem (3.11) is then reduced to

[TABLE]

and the assertion readily follows from the homogeneous part already proven.

To show uniqueness, assume $u\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ is a solution the (3.11) with homogeneous data $f=g=0$. By Lemma 2.9 there is an extension $U\in W^{1,2m,p}_{\bot}(G)$ of $u$ with $\operatorname{supp}U\in{\mathbb{T}}\times\overline{\mathbb{R}^{n}_{+}}$. By Lemma 3.6, $U=0$. ∎

3.3 The Half Space with General Boundary Conditions

Let $\Omega:=\mathbb{R}^{n}_{+}$ and $(A,B_{1},\ldots,B_{m})$ be differential operators of the form (1.4) with constant coefficients.

Lemma 3.8**.**

Assume $A^{H}$ is properly elliptic and $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ satisfies Agmon’s complementing condition on the two rays $\operatorname{e}^{i\theta}$ with $\theta=\pm\frac{\pi}{2}$. Let $(k,\xi^{\prime})\in\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}$. Consider as polynomials in $z$ the mappings $z\mapsto\mathsf{M}_{+}(k,\xi^{\prime},z)$ and $z\mapsto B^{H}_{j}(\xi^{\prime},z)$, where $\mathsf{M}_{+}$ is defined as in Lemma 3.4. For $j=1,\ldots,m$ let $F_{(j-1)l}(k,\xi^{\prime})$, $l=0,\ldots,m-1$, denote the coefficients of the polynomial $B^{H}_{j}(\xi^{\prime},z)\mod\mathsf{M}_{+}(k,\xi^{\prime},z)$. The corresponding matrix $F(k,\xi^{\prime})\in\mathbb{C}^{m\times m}$ is called characteristic matrix. The characteristic matrix function $F:\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}\rightarrow\mathbb{C}^{m\times m}$ has an extension $F:\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}\rightarrow\mathbb{C}^{m\times m}$ that satisfies $(j,l=0,\ldots,m-1)$

[TABLE]

Moreover, $F(\eta,{\xi^{\prime}})$ is invertible and the inverse matrix function $F^{-1}(\eta,{\xi^{\prime}})$ satisfies

[TABLE]

Proof.

For $j=1,\ldots,m$ the coefficients $F_{(j-1)l}(\eta,\xi^{\prime})$, $l=0,\ldots,m-1$, of the polynomial z\mapsto\big{[}B^{H}_{j}(\xi^{\prime},z)\mod\mathfrak{M}_{+}(\eta,\xi^{\prime},z)\big{]} is clearly an extension of the $j$’th row of the characteristic matrix $F$ to $(\eta,{\xi^{\prime}})\in\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}$. By definition we have

[TABLE]

for some polynomial $z\mapsto Q_{j}(\eta,\xi^{\prime},z)$. Recalling (3.4) and that the roots $\rho_{j}^{+}$ are parabolically $1$-homogeneous, we deduce for $\lambda>0$ that

[TABLE]

for some polynomial $z\mapsto\widetilde{Q_{j}}(\eta,\xi^{\prime},z)$. Comparing coefficients in the two expressions for the polynomial $z\mapsto B^{H}_{j}(\xi^{\prime},z)$ above yields (3.13). By (3.4) we have $\mathfrak{M}_{+}(k,\xi^{\prime},z)=\sum_{\alpha=0}^{m}c_{\alpha}(\eta,{\xi^{\prime}})z^{m-\alpha}$ with coefficients c_{\alpha}(\eta,{\xi^{\prime}})\in C^{\infty}\big{(}\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}\big{)} and $c_{0}(\eta,{\xi^{\prime}})=1$. Polynomial division of $z\mapsto B^{H}_{j}({\xi^{\prime}},z)$ with $z\mapsto\mathfrak{M}_{+}(k,\xi^{\prime},z)$ thus yields a polynomial with coefficients in C^{\infty}\big{(}\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}\big{)}, which establishes (3.14). Choosing $\lambda:=|\eta,{\xi^{\prime}}|^{-1}$ in (3.13) we obtain ${\lvert F_{jl}(\eta,{\xi^{\prime}})\rvert}\leq{|\eta,{\xi^{\prime}}|}^{m_{j+1}-l}\sup_{|s,\gamma|=1}{\lvert F_{jl}(s,\gamma)\rvert}$ and thus (3.15). As a direct consequence of Definition 1.3, the rows of $F(k,\xi^{\prime})$ are linearly independent and $F(k,{\xi^{\prime}})$ therefore invertible for $(k,{\xi^{\prime}})\in\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}$. The scaling property (3.13) implies that $F(\eta,{\xi^{\prime}})$ is also invertible for $(\eta,{\xi^{\prime}})\in\mathbb{R}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}$. The corresponding scaling property (3.16) for the inverse $F^{-1}(\eta,{\xi^{\prime}})$ follows by multiplying (3.13) with $\lambda^{l}F^{-1}_{l\alpha}(\eta,{\xi^{\prime}})F_{\beta j}(\lambda^{2m}\eta,\lambda{\xi^{\prime}})$ and summing over $j$ and $l$. Since F^{-1}(\eta,{\xi^{\prime}})=\big{(}\det F(\eta,{\xi^{\prime}})\big{)}^{-1}\operatorname{cof}F(\eta,{\xi^{\prime}})^{\top}, (3.17) follows from (3.14). Finally, (3.18) follows from (3.16) in the same way (3.15) was derived from (3.13). ∎

Lemma 3.9**.**

Let $p\in(1,\infty)$. Under the same assumptions as in Lemma 3.8, the operators \mathsf{op}\,\big{[}F\big{]} and \mathsf{op}\,\big{[}F^{-1}\big{]} extend to bounded operators

[TABLE]

Proof.

For $g\in\mathscr{S}_{\bot}({{\mathbb{T}}\times\mathbb{R}^{n-1}})^{m}$ we recall Lemma 2.8 and estimate

[TABLE]

By (3.16), $|k,\xi^{\prime}|^{m_{l}-(j-1)}F^{-1}_{(j-1)(l-1)}(k,{\xi^{\prime}})$ is parabolically [math]-homogeneous. Corollary 2.5 thus implies (3.21). Assertion (3.20) is shown in a similar manner. ∎

Lemma 3.10**.**

Let the assumptions be as in Lemma 3.8. Let $(k,\xi^{\prime})\in\frac{2\pi}{T}\mathbb{Z}\times\mathbb{R}^{n-1}\setminus\{(0,0)\}$ and $\gamma$ be a rectifiable Jordan curve in $\mathbb{C}$ that encircles all the roots $\rho_{j}^{+}(k,\xi^{\prime})$ $(j=0,\ldots,m-1)$ of $z\mapsto\mathsf{M}_{+}(k,\xi^{\prime},z)$. Let $c_{l}^{+}(k,\xi^{\prime})\in\mathbb{C}$, $l=0,\ldots,m$, denote coefficients such that $\mathsf{M}_{+}(k,\xi^{\prime},z)=\sum_{l=0}^{m}c_{l}^{+}(k,\xi^{\prime})z^{m-l}$. Put

[TABLE]

Then for any $g=(g_{0},\ldots,g_{m-1})\in\mathbb{C}^{m}$ the unique solution $u\in W^{2m,2}(\mathbb{R}^{+})$ to

[TABLE]

is given by $u(k,\xi^{\prime},x_{n})=L(k,\xi^{\prime},x_{n})\cdot g$. Moreover,

[TABLE]

Proof.

With $u$ as above, Cauchy’s integral theorem yields

[TABLE]

On the same token

[TABLE]

Since

[TABLE]

which follows by choosing $\gamma$ to be a circle of sufficiently large radius $R$ and letting $R\rightarrow\infty$ (see for example [18, Chapter 2, Proposition 4.1]), (3.23) follows. ∎

Lemma 3.11**.**

Let the assumptions be as in Lemma 3.8. If $u\in W^{1,2m,2}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ satisfies $\mathsf{A}u=0$, then $B^{H}u=\mathsf{op}\,[F]\operatorname{Tr}_{m}u$.

Proof.

Employ the partial Fourier transform $\mathscr{F}_{{{\mathbb{T}}\times\mathbb{R}^{n-1}}}$ to the equation $\mathsf{A}u=0$, which in view of Plancherel’s theorem implies $\mathsf{A}(k,\xi^{\prime},\partial_{n})\mathscr{F}_{{{\mathbb{T}}\times\mathbb{R}^{n-1}}}(u)=0$ for almost every $(k,\xi^{\prime})$. By Lemma 3.10, B^{H}_{j}(\xi^{\prime},\partial_{n})\mathscr{F}_{{{\mathbb{T}}\times\mathbb{R}^{n-1}}}(u)={F_{(j-1)l}(k,\xi^{\prime})\big{(}\operatorname{Tr}_{m}u(0)\big{)}_{l}}. Employing $\mathscr{F}^{-1}_{{{\mathbb{T}}\times\mathbb{R}^{n-1}}}$, we obtain $B^{H}u=\mathsf{op}\,[F]\operatorname{Tr}_{m}u$. ∎

Theorem 3.12**.**

Assume $A^{H}$ is properly elliptic and $(A^{H},B^{H}_{1},\ldots,B^{H}_{m})$ satisfies Agmon’s complementing condition on the two rays $\theta=\pm\frac{\pi}{2}$. Let $p\in(1,\infty)$. For $f\in L^{p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ and $g\in T_{\bot}^{\kappa,p}({\mathbb{T}}\times\partial\mathbb{R}^{n}_{+})$ there is a unique $u\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ subject to

[TABLE]

Moreover,

[TABLE]

where $c=c(n,p)>0$.

Proof.

As in the proof of Theorem 3.7, it suffices to show existence of a solution to (3.24) satisfying (3.25) for $f=0$ and $g\in\mathscr{S}_{\bot}({{\mathbb{T}}\times\mathbb{R}^{n-1}})^{m}$. Since $F^{-1}$ is smooth away from the origin (3.17) and has at most polynomial growth (3.15), it follows that $\mathsf{op}\,[F^{-1}]g\in\mathscr{S}_{\bot}({{\mathbb{T}}\times\mathbb{R}^{n-1}})$. Consequently, Theorem 3.7 yields existence of a solution $u\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})\cap W^{1,2m,2}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ to

[TABLE]

From Lemma 3.11 it follows that $u$ is in fact a solution to (3.24). Additionally, Theorem 3.7 and Lemma 3.9 imply

[TABLE]

It remains to show uniqueness. Assume for this purpose that $u\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ is a solution to (3.24) with homogeneous right-hand side $f=g=0$. Let $\{g_{n}\}_{n=1}^{\infty}\subset\mathscr{S}_{\bot}({{\mathbb{T}}\times\mathbb{R}^{n-1}})^{m}$ be a sequence with $\lim_{n\rightarrow\infty}g_{n}=\operatorname{Tr}_{m}u$ in $T_{\bot}^{\iota,p}({{\mathbb{T}}\times\mathbb{R}^{n-1}})$. By virtue of Theorem 3.7 there is a $u_{n}\in W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})\cap W^{1,2m,2}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ with \big{[}\partial_{t}+A^{H}\big{]}u_{n}=0 and $\operatorname{Tr}_{m}u_{n}=g_{n}$. Theorem 3.7 and Lemma 3.9 imply that $\lim_{n\rightarrow\infty}u_{n}=u$ in $W^{1,2m,p}_{\bot}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ and thus $B^{H}u_{n}\rightarrow B^{H}u=0$ in $T_{\bot}^{\kappa,p}({{\mathbb{T}}\times\mathbb{R}^{n-1}})$. By Lemma 3.11, $B^{H}u_{n}=\mathsf{op}\,[F]g_{n}$. Lemma 3.9 thus yields $\operatorname{Tr}_{m}u=\lim_{n\rightarrow\infty}g_{n}=0$. We conclude $u=0$ by Theorem 3.7. ∎

4 Proof of the Main Theorems

Proof of Theorem 1.4.

As already noted in Section 2.4, the canonical bijection between $C^{\infty}_{0}({\mathbb{T}}\times\mathbb{R}^{n}_{+})$ and $C^{\infty}_{0,\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ implies that W^{1,2m,p}\big{(}{\mathbb{T}}\times\mathbb{R}^{n}_{+}\big{)} and $W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ are isometrically isomorphic. It follows that W^{1,2m,p}_{\bot}\big{(}{\mathbb{T}}\times\mathbb{R}^{n}_{+}\big{)} and ${\mathcal{P}}_{\bot}W^{1,2m,p}_{\mathrm{per}}(\mathbb{R}\times\mathbb{R}^{n}_{+})$ as well as $T_{\bot}^{\kappa,p}({\mathbb{T}}\times\partial\mathbb{R}^{n}_{+})$ and $\Pi_{j=1}^{m}{\mathcal{P}}_{\bot}W^{\kappa_{j},2m\kappa_{j},p}_{\mathrm{per}}(\mathbb{R}\times\partial\mathbb{R}^{n}_{+})$ are isometrically isomorphic. Consequently, Theorem 1.4 follows from Theorem 3.12. ∎

Proof of Theorem 1.5.

Theorem 1.5 follows from Theorem 1.4 by a standard localization and perturbation argument. One can even apply the argument used in the elliptic case [3]; see also [23, Chapter 4.8]. ∎