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

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.


Introduction
We investigate time-periodic solutions to parabolic boundary value problems where A is an elliptic operator of order 2m and B 1 , . . . , B m satisfy an appropriate complementing boundary condition. The domain Ω is either the whole-space, the half-space or a bounded domain, and 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 we decompose (1.1) into an elliptic problem APu = Pf in Ω, and a purely oscillatory problem 3) 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 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 ∈ Ω and all ξ ∈ R n \ {0} it holds A H (x, ξ) = 0, and for all x ∈ Ω and all linearly independent vectors ζ, ξ ∈ R n the polynomial P (τ ) := A H (x, ζ + τ ξ) has m roots in C with positive imaginary part, and m roots in C with negative imaginary part.
Ellipticity, however, does not suffice to establish maximal L p regularity for the timeperiodic problem. We thus recall Agmon's condition, also known as parameter ellipticity. Definition 1.2 (Agmon's Condition). Let θ ∈ [−π, π]. A properly elliptic operator A H is said to satisfy Agmon's condition on the ray e iθ if for all x ∈ Ω and all ξ ∈ R n \ {0} it holds A H (x, ξ) / ∈ {r e iθ | r ≥ 0}.
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 , . . . , 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 , . . . , 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 (1.6) It was shown by Agmon [2] that a necessary and sufficient condition for the resolvent of (A H , B H 1 , . . . , 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 |θ| ≥ π 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 |θ| ≥ π 2 . To apply [12], one has show that (A H , B H 1 , . . . , B H m ) admits bounded imaginary powers. Later, it was shown that maximal regularity is in fact equivalent to 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 initialvalue problem without the use of either bounded imaginary powers or the notion of R-boundedness; see Remark 1.6 below. Under the assumption that (A H , B H 1 , . . . , 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 , . . . , 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 θ = ± π 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 per (R × R n + ) of time-periodic functions on the time-space domain R × R n + . We refer to Section 2 for definitions of the function spaces. ) satisfies Agmon's complementing condition on the two rays e iθ with θ = ± π 2 , then for all functions f ∈ P ⊥ L p per (R × R n + ) and g j ∈ P ⊥ W κ j ,2mκ j ,p per (1.7) Moreover, 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 initialvalue 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 |θ| ≥ π 2 and instead carry out our investigation under the the weaker condition that Agmon's condition is satisfied only for θ = ± π 2 . We shall briefly describe the main ideas behind the proof of Theorem 1.4. We first consider the problem in the whole space R × R n and replace the time axis R with the torus T := R/T Z in order to reformulate the T -time-periodic problem as a partial differential equation on the locally compact abelian group G := T × R n . Utilizing the Fourier transform F G associated to G, we obtain an explicit representation formula for the time-periodic solution. Since F G = F T • F 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 F G and show that the corresponding Fourier multiplier defined on the dual group 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 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 Ω 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: and Ω be a domain with a boundary that is uniformly C 2m -smooth. Assume a α is bounded and uniformly continuous on Ω for |α| = 2m, and a α ∈ L ∞ (Ω) for |α| < 2m. Further assume b j,β ∈ C 2m−m j (∂Ω) 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 , . . . , B H m ) satisfies Agmon's complementing condition on the two rays e iθ with θ = ± π 2 , then the estimate holds for all u ∈ W 1,2m,p Since time-independent functions are trivially also time-periodic, we have W 2m,p (Ω) ⊂ W 1,2m,p per (R × Ω). If estimate (1.9) is restricted to functions in W 2m,p (Ω), 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 Maximal regularity for parabolic initial-value problems of Agmon-Douglis-Nirenberg type is based on an investigation of the initial-value problem (1.10) Maximal regularity for (1.10) means that for each function f ∈ L p (0, T ;  [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 ∈ L p (0, T ; L p (R n + )) to a T -periodic function f ∈ L p per (R × R n + ). With u denoting the solution from Theorem 1.4 corresponding to P ⊥ f , the functionũ 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].
The notation ∂ j := ∂ x j is employed for partial derivatives with respect to spatial variables. Throughout, ∂ t shall denote the partial derivative with respect to the time variable. For a multi-index α ∈ N n , we employ the notation D α := i |α| ∂ α 1 x 1 . . . ∂ αn xn . We introduce the parabolic length We call a generic function g parabolically α-homogeneous if λ α g(η, ξ) = g(λ 2m η, λξ) for all λ > 0.

Paley-Wiener Theorem
consists of all functions f ∈ L 2 (R) admitting a holomorphic extension to the lower complex planef : consists of all functions f ∈ L 2 (R) admitting a similar holomorphic extension to the upper complex plane.

Time-periodic function spaces
Let Ω ⊂ R n be a domain and the space of smooth time-period functions with compact support in the spatial variable. Clearly, are norms on C ∞ 0,per (R × Ω). We define Lebesgue and anisotropic Sobolev spaces of time-periodic functions as completions One may identify On a similar note, one readily verifies that provided Ω satisfies the segment condition. We introduce anisotropic fractional order Sobolev spaces (Sobolev-Slobodeckiȋ spaces) by real interpolation: For a C 2m -smooth manifold Γ ⊂ R n , anisotropic Sobolev spaces W s,2ms,p per (R × Γ) are defined in a similar manner. We can identify (see also Section 2.4 below) the trace space in the sense that the trace operator maps the former continuously onto the latter.

Function Spaces and the Torus Group Setting
We shall further introduce a setting of function spaces in which the time axis R in the underlying domains is replaced with the torus T := R/T 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 T-defined functions is formalized in terms of the canonical quotient mapping π : R × R n → T × R n , π(t, x) := ([t], x). A differentiable structure on T × R n is inherited via the quotient mapping form R × R n . More specifically, for any domain Ω ⊂ R n we let denote the space of compactly supported smooth functions. Introducing the normalized Haar measure on T, we define norms · p and · 1,2m,p on C ∞ 0 (T × Ω) as in (2.1)-(2.2). The quotient mapping trivially respects derivatives and is isometric with respect to · p and · 1,2m,p . Letting we thus obtain Lebesgue and Sobolev spaces that are isometrically isomorphic to the spaces L p per (R × Ω) and W 1,2m,p per R × Ω , respectively. Defining weak derivatives with respect to test functions C ∞ 0 (T × Ω), one readily verifies that provided Ω satisfies the segment property. For s ∈ (0, 1), we define fractional ordered Sobolev spaces by real interpolation and thereby obtain spaces isometrically isomorphic to W s,2ms,p per (R×Ω). In the half-space case, we clearly have Hence, for l ∈ N, l ≤ 2m the trace operator extends to a bounded operator that is onto; see for example [24,Theorem 1.8.3]. The existence of a bounded right inverse to Tr l can be shown by applying [24, Theorem 2.9.1]. We further introduce the operators which are clearly complementary projections. Since Pf is independent of the time variable t ∈ R, we may at times treat Pf as a function of the space variable x ∈ Ω only. Both P and P ⊥ extend to bounded operators on the Lebesgue space L p (T × Ω) and Sobolev space W 1,2m,p T × Ω . We employ the notation L p ⊥ (T × Ω) := P ⊥ L p (T × Ω) and W 1,2m,p ⊥ T × Ω := P ⊥ W 1,2m,p T × Ω for the subspaces of P ⊥ -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 = P ⊥ f as purely oscillatory.
Finally, we let and put

Schwartz-Bruhat Spaces and Distributions
When the spatial domain is the whole-space R n , we employ the notation G := T × R n . Equipped with the quotient topology via π, 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 (T × R n ) space introduced in the previous section. We identify G's dual group by G = 2π T Z×R n by associating (k, ξ) ∈ 2π T Z×R n with the character χ : G → C, χ(x, t) := e ix·ξ+ikt . By default, G is equipped with the compactopen topology, which in this case coincides with the product of the discrete topology on 2π T Z and the Euclidean topology on R n . The Haar measure on G is simply the product of the Lebesgue measure on R n and the counting measure on 2π T Z. The Schwartz-Bruhat space S (G) of generalized Schwartz functions (originally introduced in [9]) can be described in terms of the semi-norms The vector space S (G) is endowed with the semi-norm topology. The topological dual space S (G) of S (G) is referred to as the space of tempered distributions on G. Observe that both S (G) and S (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 ∈ S (G), distributional derivatives ∂ α t ∂ β x u ∈ S (G) are defined by duality in the usual manner. Also the support supp u is defined in the classical way. Moreover, we may restrict the distribution u to a subdomain T × Ω by considering it as a functional defined only on the test functions from S (G) supported in T × Ω.
A differentiable structure on G is obtained by introduction of the space The Schwartz-Bruhat space on the dual group G is defined in terms of the semi-norms We also endow S ( G) with the corresponding semi-norm topology and denote by S ( G) the topological dual space.

Fourier Transform
As a locally compact abelian group, G has a Fourier transform F G associated to it. The ability to utilize a Fourier transform that acts simultaneously in time t ∈ T and space x ∈ R n shall play a key role in the following. The Fourier transform F G on G is given by If no confusion can arise, we simply write F instead of F G . The inverse Fourier transform is formally defined by It is standard to verify that F : S (G) → S ( G) is a homeomorphism with F −1 as the actual inverse, provided the Lebesgue measure dξ is normalized appropriately. By duality, F extends to a bijective mapping F : S (G) → S ( G). The Fourier transform provides us with a calculus between the differential operators on G and the polynomials on G. As one easily verifies, for u ∈ S (G) and (α, β) ∈ N 0 × N n 0 we have F ∂ α t ∂ β x u = i |α|+|β| k α ξ β F (u) as an identity in S ( G).
The projections introduced in (2.5) can be extended trivially to projections on the Schwartz-Bruhat space P, P ⊥ : S (G) → S (G). Introducing the delta distribution δ Z on 2π T Z, that is, δ Z (k) := 1 if k = 0 and δ Z (k) := 0 for k = 0, we observe that . Using these representations for P and P ⊥ , we naturally extend the projections to operators P, P ⊥ : S (G) → S (G). In accordance with the notation introduced above, we put S ⊥ (G) := P ⊥ S (G).
In general, we shall utilize smooth functions m ∈ C ∞ ( G) with at most polynomial growth as Fourier multipliers by introducing the corresponding operator We call m an L p (G)-multiplier if op [m] extends to a bounded operator on L p (G) for any p ∈ (1, ∞). The following lemmas provide us with criteria to determine if m is an L p (G)-multiplier. Let χ ∈ C ∞ (R) be a "cut-off" function with χ(η) = 0 for |η| < π T and χ(η) = 1 for |η| ≥ 2π T . Put M(η, ξ) := χ(η)m(η, ξ). Utilizing that m is α-homogeneous and α ≤ 0, one readily verifies that M satisfies the conditions of Marcinkiewicz's multiplier theorem ([22, Chapter IV, §6]). Consequently, M is an L p (R × R n )-multiplier. For u ∈ L p ⊥ (G), we have u = P ⊥ u and thus

Time-Periodic Bessel Potential Spaces
Time-periodic Bessel Potential spaces can be defined via the Fourier transform F G . We shall only introduce Bessel Potential spaces of purely oscillatory distributions: Utilizing Lemma 2.4, one readily verifies that H s,p ⊥ is a Banach space with respect to the norm Time-periodic Bessel Potential spaces on the half-space are defined via restriction of distributions in the time-periodic Bessel Potential spaces defined above: Identifying H s,p ⊥ (T × R n + ) as a factor space of H s,p ⊥ in the canonical way, we see that H s,p ⊥ (T × R n + ) is a Banach space in the norm · s,p,T×R n + . for all u ∈ S (G).
Proof. The complex function z → (iz + |k, ξ |) −1 is holomorphic in the lower complex plane. Due to Lemma 2.3, we can employ Proposition 2.2 to conclude for all ψ ∈ S ⊥ (G). By duality, the same is true for all ψ ∈ S ⊥ (G). We employ Lemma 2.3 to estimate It follows from (2.9) that We thus conclude u s+1,p,T×R n + ≤ C op iξ n + |k, ξ | u s,p,T×R n + and thereby (2.6). Furthermore, 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 β ∈ (0, 1) and α ∈ 2m(β − 1), 2mβ . Then op |k, ξ| α extends to a bounded operator op |k, ξ| α : W β,2mβ,p Proof. By interpolation, we directly obtain that op |k, ξ| α extends to a bounded operator for any θ ∈ (0, 1). Choose θ := 2mβ−α 2m−α . Using a dyadic decomposition of the Fourier space with respect to the parabolic length, H s,p ⊥ can be identified as the complex interpolation space [H s 0 ,p ⊥ , H s 1 ,p ⊥ ] ω with 1 s = 1−ω s 0 + ω 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 Finally, we characterize the trace spaces of the time-periodic Bessel Potential spaces. Lemma 2.9. Let p ∈ (1, ∞). The trace operator Tr m defined in (2.3) extends to a bounded operator that is onto and has a bounded right inverse. If u ∈ H m,p ⊥ (G) with supp(u) ⊂ T × R n + , then Tr m u |T×R n Proof. For either I = R or I = R + , put We first verify that H m,p ⊥ = V (R) with equivalent norms. It is straightforward to obtain the embedding H m,p ⊥ → V (R). To show the reverse embedding, consider u ∈ V (R).
. It is standard to show existence of an extension operator V (R + ) → V (R); see for example [24, Lemma 2.9.1]. By restriction to T × R n + , it thus follows that H m,p ⊥ (T × R n + ) = V (R + ). The classical trace method now implies that trace operator extends to a bounded operator Thus, we conclude (2.10). Consider now u ∈ H m,p ⊥ (G) with supp(u) ⊂ T × R n + . As above we can identify u as an element of V (R), which necessarily satisfies u(0) = 0. It follows that Tr m u |T×R n + = 0. If on the other hand u ∈ H m,p ⊥ (T × R n + ) with Tr m u |T×R n + = 0, then it is standard to show that u can be approximated by a sequence of functions from C ∞ 0 (T × R n + ); see for example [24,Theorem 2.9.1]. Clearly, this sequence also converge in H m,p ⊥ (G). The limit function U ∈ H m,p ⊥ (G) satisfies supp U ⊂ T × R n + and U |T×R n + = u.

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 ξ ∈ R n , we refer to A(ξ) as the symbol of A(D).

The Whole Space
We consider first the case of the spatial domain being the whole-space R n . The timespace domain then coincides with the locally abelian group G, and we can thus employ the Fourier transform F G and base the proof on an investigation of the corresponding Fourier multipliers.

extend uniquely to bounded operators
In the setting (3.2), A −1 is the actual inverse of A.
Proof. Since Au = ∂ t u + A H u, we employ Lemma 3.1 to estimate 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.

The Half Space with Dirichlet Boundary Condition
In the next step, we consider the case of the spatial domain being the half-space R n + and boundary operators corresponding to Dirichlet boundary conditions. As in the wholespace case, we shall work with the symbol of ∂ t + A H . In the following lemma, we collect its key properties. (1) For every (η, ξ ) ∈ R × R n−1 \ {(0, 0)} the complex polynomial z → M(η, ξ , z) has exactly m roots ρ + j (η, ξ ) ∈ C − in the upper complex plane, and m roots ρ − j (η, ξ ) ∈ C + in the lower complex plane (j ∈ {1, . . . , m}).
is an L p (R × R n )-multiplier. Indeed, utilizing that M ± is m-homogeneous, we see that M ± can be bounded below by where the infimum above is strictly positive due to the roots in definition (3.4) satisfying lim (η,ξ )→(0,0) ρ ± j (η, ξ ) = 0. Using only (3.7) and the α-homogeneity of the coefficients c ± α as in (3.6), it is now straightforward to verify that m satisfies the condition of the Marcinkiewicz's multiplier theorem ([22, Chapter IV, §6]). Thus, m is an L p (R × R n )multiplier and by [13, Theorem B.2.1] m | G therefore an L p (G)-multiplier. Since the restriction of the corresponding operator op m | G : L p ⊥ (G) → L p ⊥ (G) to the subspace of purely periodic functions coincides with op m −1 | G : L p ⊥ (G) → L p ⊥ (G), we deduce as in the proof of Lemma 3.1 that A −1 ± extends uniquely to a bounded operator A −1 ± : The lemma above provides us with at decomposition of the differentiable operators in (3.2), that is, for A : Employing the Paley-Wiener Theorem, we shall now show that the operators A ± and A −1 ± "respect" the support of a function in the upper (lower) half-space.
We shall prove only part (i), for part (ii) follows analogously. We employ the notation H := T × R n−1 and the canonical decomposition F G = F H F R of the Fourier transform. In view of Lemma 3.4, it suffices to consider only u ∈ S (G) with supp u ⊂ T × R n + . For fixed k ∈ 2π T Z \ {0} and ξ ∈ R n−1 , we let D(k, ξ ) := F −1 R M + (k, ξ , ·)F R . Since M + is a polynomial with respect to the variable ξ n , D(k, ξ ) is a differential operator in x n and hence supp(D(k, ξ )f ) ⊂ R + for every f ∈ S (R) with supp f ⊂ R + . Clearly, supp([F H u](k, ξ , ·)) ⊂ R + . Since F H [A + u](k, ξ , ·) = [D(k, ξ )F H u](k, ξ , ·), we conclude supp A + u ⊂ T × R n + . To show the same property for A −1 + u, we employ the version of the Paley-Wiener Theorem presented in Proposition 2.2. Since u ∈ S (G) ⊂ L 2 (G), we immediately obtain that for fixed k ∈ 2π T Z and ξ ∈ R n−1 , the Fourier transform [F G u](k, ξ , ·) is in the Hardy space H 2 + (R). Let denote the extension of M −1 + (k, ξ , ·) to the lower complex plane. Since all the roots ρ + j lie in the upper complex plane, this extension is holomorphic and bounded. It follows that [M −1 + F G u](k, ξ , ·) ∈ H 2 + (R). Hence, taking the inverse Fourier transform, Proposition 2.2 yields supp A −1 + u ⊂ T × R n + .
The above properties of A ± and A −1 ± lead to a surprisingly simple representation formula, see (3.8) below, for the solution u to the problem ∂ t u + A H u = f in the halfspace T × 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 e iθ with θ = ± π 2 . Let p ∈ (1, ∞) and f ∈ H −m,p ⊥ (G). Let Y + denote the characteristic function on T × R n + . Then Moreover, there is a constant c = c(n, p) > 0 such that Finally, we can establish the main theorem in the case of the spatial domain being the half-space 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 e iθ with θ = ± π 2 . Let p ∈ (1, ∞). For f ∈ L p ⊥ (T × R n + ) and g ∈ T ι,p (3.11) Moreover, there is a constant c = c(n, p) > 0 such that Proof. We first assume g = 0. Extending f by zero to the whole space T × R n , we have be the solution to (3.9) from Lemma 3.6. Lemma 2.9 yields 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 Since the symbol of A reads M(k, ξ , ξ n ) = aξ 2m n + ik + 2m−1 k=0 |α|=2m−k a α,k (ξ ) α ξ k n with a = 0, we deduce with the help of Lemma 2.4 that ∂ 2m n u −m+1,p,T×R n + ≤ C Au −m+1,p,T×R n + + op |k, ξ | u m,p,T×R n + .
Clearly, op |k, ξ | commutes with A −1 By Lemma 3.6, we finally obtain Iterating this procedure, we obtain after m steps the desired regularity u ∈ H 2m,p ⊥ (T×R n + ) together with the estimate u 2m,p,T×R n + ≤ C f p . Recalling from Proposition 2.6 that H 2m,p ⊥ (T × R n + ) = W 1,2m,p ⊥ (T × R n + ), we conclude (3.12) in the case g = 0.
If g = 0, we recall the properties (2.4) of the trace operator and choose a function v ∈ W 1,2m,p (T × R n + ) with Tr m v = g and v 2m,p ≤ C g T ι,p ⊥ (T×R n + ) . With w := u − v, problem (3.11) is then reduced to and the assertion readily follows from the homogeneous part already proven.
To show uniqueness, assume u ∈ W 1,2m,p ⊥ (T × R n + ) is a solution the (3.11) with homogeneous data f = g = 0. By Lemma 2.9 there is an extension U ∈ W 1,2m,p ⊥ (G) of u with supp U ∈ T × R n + . By Lemma 3.6, U = 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 ∈ S ⊥ (T × 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 op [F −1 ]g ∈ S ⊥ (T × R n−1 ). Consequently, Theorem 3.7 yields existence of a solution u ∈ W 1,2m,p ⊥ (T × R n + ) ∩ W 1,2m,2 ⊥ (T × R n + ) to ∂ t u + A H u = 0 in T × R n + , Tr m u = op [F −1 ]g on T × ∂R n + . (3.26) 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 It remains to show uniqueness. Assume for this purpose that u ∈ W 1,2m,p ⊥ (T × R n + ) is a solution to (3.24) with homogeneous right-hand side f = g = 0. Let {g n } ∞ n=1 ⊂ S ⊥ (T × R n−1 ) m be a sequence with lim n→∞ g n = Tr m u in T ι,p ⊥ (T × R n−1 ). By virtue of Theorem 3.7 there is a u n ∈ W 1,2m,p ⊥ (T×R n + )∩W 1,2m,2 ⊥ (T×R n + ) with ∂ t +A H u n = 0 and Tr m u n = g n . Theorem 3.7 and Lemma 3.9 imply that lim n→∞ u n = u in W 1,2m,p ⊥ (T×R n + ) and thus B H u n → B H u = 0 in T κ,p ⊥ (T × R n−1 ). By Lemma 3.11, B H u n = op [F ]g n . Lemma 3.9 thus yields Tr m u = lim n→∞ g n = 0. We conclude u = 0 by Theorem 3.7.

Proof of the Main Theorems
Proof of Theorem 1.4. As already noted in Section 2.4, the canonical bijection between C ∞ 0 (T × R n + ) and C ∞ 0,per (R × R n + ) implies that W 1,2m,p T × R n + and W 1,2m,p per (R × R n + ) are isometrically isomorphic. It follows that W 1,2m,p ⊥ T × R n + and P ⊥ W 1,2m,p per (R × R n + ) as well as T κ,p ⊥ (T × ∂R n + ) and Π m j=1 P ⊥ W κ j ,2mκ j ,p per (R × ∂R n + ) are isometrically isomorphic. Consequently, Theorem 1.4 follows from Theorem 3.12.