On time-periodic solutions to parabolic boundary value problems

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document} 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 ∂ t u + Au = f in R × , B j u = g j on R × ∂ , (1.1) where A is an elliptic operator of order 2m and B 1 , . . . , B m satisfy an appropriate complementing boundary condition. The domain is either the whole-space, the half-space or a bounded domain, and R denotes the time-axis. The solutions u(t, x) Communicated by Y. Giga. and a purely oscillatory problem 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 [22] 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) does not satisfy this property. Another 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 with complex coefficients a α : → C and b j,α : ∂ → C. Here, α ∈ N n is a multi-index and D α := (−i) |α| ∂ α 1 x 1 . . . ∂ α n x n . The order of A and B j is 2m and m j ( j = 1, . . . , m), respectively, with no restrictions other than m ∈ N and m j ∈ N 0 . We denote the principle part of the operators by We shall assume that A H is elliptic in the following classical sense: The operator A H is said to be properly elliptic if for all x ∈ and all ξ ∈ R n \ {0} it holds A H (x, ξ) = 0, and for all x ∈ 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 time-periodic problem. We thus recall Agmon's condition, also known as parameter ellipticity.
If A H satisfies Agmon's condition on the ray e iθ , then, since the roots of a polynomial depend continuously on its coefficients, the polynomial Q(τ ) := −r e iθ +A H (x, ζ + τ ξ) has m roots τ + h (r e iθ , x, ζ, ξ) ∈ C with positive imaginary part, and m roots τ − h (r e iθ , x, ζ, ξ) ∈ C with negative imaginary part (h = 1, . . . , m). Consequently, the following assumption on the operator (A H , B H 1 , . . . , B H m ) is meaningful. (i) For all x ∈ ∂ , all pairs ζ, ξ ∈ R n with ζ tangent to ∂ and ξ normal to ∂ at x, and all r ≥ 0, let τ + h (r e iθ , x, ζ, ξ) ∈ C (h = 1, . . . , m) denote the m roots of the polynomial Q(τ ) := −r e iθ +A H (x, ζ + τ ξ) with positive imaginary part. The polynomials P j (τ ) := B H j (x, ζ + τ ξ) ( j = 1, . . . , m) are linearly independent modulo the polynomial m h=1 τ − τ + h (r e iθ , x, ζ, ξ) . 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. We note that it is equivalent to the Lopatinskiȋ-Shapiro condition, see Remark 1.7 below. 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] (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, which leads to the generation of an analytic semi-group, is that Agmon's complementing condition is satisfied for all rays with |θ | ≥ π 2 . The step from analyticity of the semi-group to maximal L p regularity for the parabolic initial-value problem is more complicated though. In the celebrated work of Dore and Venni [12], 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 to 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 initial-value problem without the use of either bounded imaginary powers or the notion of Rboundedness; 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.
The references above to the theory of maximal L p regularity for parabolic initialvalue problems would not be complete without mention of the extensive work of Solonnikov on initial-value problems for parabolic systems; see [23] and the references therein. The investigation of systems requires a more involved definition of parabolicity and complementary condition than Definition 1.1-1.3, but the arguments towards an L p theory follow similar ideas as in the scalar case. As pointed out by Wang [28], the approach of Solonnikov can be reduced to an argument based on Fourier multipliers. This rationale was also proposed by Arkeryd [7] in his study of elliptic boundary value problems and will also be used in our approach in the time-periodic case.
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 (1.7) Moreover, 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 [18]. 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 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 [22] and Arkeryd [7] utilized the Paley-Wiener Theorem in order to construct a representation formula for solutions to elliptic problems; see also [26,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 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: Since time-independent functions are trivially also time-periodic, we have W 2m, p ( ) ⊂ W 1,2m, p per (R× ). If estimate (1.9) is restricted to functions in W 2m, p ( ), 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 will address systems in future works.
We briefly return to the decomposition (1.2)-(1.3). It is well-known in the bounded domain case that ellipticity of (1.2) in the sense of Agmon-Douglis-Nirenberg is equivalent to the corresponding linear operator being Fredholm in the setting of classical Sobolev spaces. From Theorem 1.4 and the similar assertion in the whole-space case, which as mentioned above shall also be provided, one can show that also the operator of the purely oscillatory problem (1.3) is Fredholm in the setting of timeperiodic Sobolev spaces. Indeed, since we show that the operator is a homeomorphism in the whole-and half-space cases, a localization argument (see for example [29, Proof of Theorem 13.1] or [30, Proof of Theorem 9.32]) yields existence of a left and right regularizer in the bounded domain case, which in turn implies the Fredholm property. Since both the elliptic and purely oscillatory problem possess the Fredholm property, so does the full time-periodic problem on bounded domains. Due to the work of Geymonat [15], a comprehensive Fredholm theory is available for the elliptic problem (1.2). Since our proof of Theorem 1.5 successfully demonstrates that time-periodic problems can be approached in much the same way as elliptic problems, it seems likely that a similar comprehensive Fredholm theory can be developed for the purely oscillatory problem (1.3). Although we shall leave this investigation to future works, we note that the Fredholm properties of the operator in (1.3) will in general be different from the Fredholm properties of the elliptic problem (1.2). In fact, the simple example of the Laplace equation with a Neumann boundary condition in a bounded domain shows that the defect numbers of the two problems can be different. This observation further underlines the importance of the decomposition (1.2)-(1.3).
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 [18], the recent article by Geissert, Hieber and Nguyen [14], as well as the monographs [16,27], and refer the reader to the references therein. Finally, we mention the article [17] 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 ;  [25,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 ; With u denoting the solution from Theorem 1.4 corresponding to P ⊥ f , the functioñ 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 [20,Theorem 4.3.1]. For more details, see also [21,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].

Remark 1.7 If A H is a properly elliptic operator that satisfies Agmon's condition on
is said to satisfy the Lopatinskiȋ-Shapiro condition on the ray e iθ , if for all x ∈ ∂ , all pairs ξ, ζ ∈ R n with ζ tangent to ∂ and ξ normal to ∂ at x, all r ≥ 0 and all g = (g 0 , . . . , g m−1 ) ∈ C m the system of ordinary differential equations admits a unique solution u ∈ W 2m,2 (R + ). Often, the Lopatinskiȋ-Shapiro condition is preferred over Agmon's complementing condition specified in Definition 1.3, but the two definitions are fully equivalent, which follows from Lemma 3.10. Although the purely algebraic nature of the complementing condition may seem favorable, in practice it is sometimes easier to verify the Lopatinskiȋ-Shapiro condition.
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.

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 ⊂ 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 × ). 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 satisfies the segment condition. We introduce anisotropic fractional order Sobolev spaces (Sobolev-Slobodeckiȋ spaces) by real interpolation: For a C 2m -smooth manifold ⊂ R n , anisotropic Sobolev spaces W s,2ms, p per (R × ) are defined in a similar manner. We can identify (see also Sect. 2.4 below) the trace 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 : 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 × ) 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 × ) and W 1,2m, p per R × , respectively. Defining weak derivatives with respect to test functions C ∞ 0 (T × ), one readily verifies that provided 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 × ). In the halfspace case, we clearly have Hence, for l ∈ N, l ≤ 2m the trace operator extends to a bounded operator that is onto; see for example [ which are clearly complementary projections. Since P f is independent of the time variable t ∈ R, we may at times treat P f as a function of the space variable x ∈ only. Both P and P ⊥ extend to bounded operators on the Lebesgue space L p (T × ) and Sobolev space W 1,2m, p T× . We employ the notation L p ⊥ (T× ) := P ⊥ L p (T× ) and W 1,2m, p ⊥ T × := P ⊥ W 1,2m, p T × for the subspaces of P ⊥ -invariant functions. This notation is canonically extended to other spaces such as 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 i x·ξ +ikt . By default, G is equipped with the compact-open 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 × by considering it as a functional defined only on the test functions from S (G) supported in T × .
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) . 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. Proof 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 αhomogeneous and α ≤ 0, one readily verifies that M satisfies the conditions of Marcinkiewicz's multiplier theorem ([24, Chapter IV, §6]). Consequently, M is an

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: for all ψ ∈ S ⊥ (G). By duality, the same is true for all ψ ∈ S ⊥ (G). We employ Lemma 2.3 to estimate u s+1, p,T×R n It follows from (2.9) that We thus conclude u s+1, p,T×R n + ≤ op − iξ n + |k, ξ | u s, p,T×R n + and thereby (2.6). Furthermore, ∂ n u s, p,T×R n Proof By interpolation, we directly obtain that op |k, ξ| α extends to a bounded operator for any θ ∈ (0, 1).
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 Proof For either I = R or I = R + , put Thus, we conclude (2.10).

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.

Lemma 3.1 Assume that A H is properly elliptic and satisfies Agmon's condition on
the two rays e iθ with θ = ± π 2 . Let p ∈ (1, ∞), s ∈ R and

extend uniquely to bounded operators
In the setting (3.2), A −1 is the actual inverse of A.

Theorem 3.2 Assume A H is properly elliptic and satisfies Agmon's condition on the
two rays e iθ with θ = ± π 2 . Let s ∈ R and p ∈ (1, ∞). There is a constant > 0 such that u s, p ≤ ∂ t u + Au s−2m, p + u s−1, p . 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 whole-space case, we shall work with the symbol of ∂ t + A H . In the following lemma, we collect its key properties.
Proof (1) Since A H is properly elliptic, the polynomial z → M(0, ξ , z) has exactly m roots in the upper and lower complex plane, respectively. Recall that A H (x, ξ) / ∈ iR for all ξ ∈ R n \ {0}. Since the roots of a polynomial depend continuously on the polynomial's coefficients, we deduce part (1) of the lemma.
(3) The analyticity of the coefficients c ± α follows by a classical argument; see for example [25,Chapter 4.4]. The coefficient c ± α being parabolically α-homogeneous is a direct consequence of M ± being m-homogeneous.

Lemma 3.4 Assume A H is properly elliptic and satisfies Agmon's condition on the
two rays e iθ with θ = ± π 2 . Put M ± := M ± | G , where M ± is defined by (3.4). Let p ∈ (1, ∞) and s ∈ R. Then the linear operators extend uniquely to bounded and mutually inverse operators A ± : H s, p Proof The assertion of the lemma follows as in the proof of Lemma 3.1, provided we can show that the restriction to G of the multiplier Although m is parabolically 0-homogeneous, we cannot apply Lemma 2.4 directly since m is not defined on all of R × R n \ {(0, 0)}. Instead, we recall (3.5) and observe that Owing to the α-homogeneity of c ± α , Lemma 2.4 yields that both m α 1 | G and m α 2 | G are L p (G)-multipliers. Consequently, also m is an L p (G)-multiplier, and we thus conclude as in the proof of Lemma 3.1 that A ± extends uniquely to a bounded operator To show the corresponding property for A −1 ± , we introduce a cut-off function χ ∈ C ∞ (R) with χ(η) = 0 for |η| < π T and χ(η) = 1 for |η| ≥ 2π T . We claim that 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 ([24, Chapter IV, §6]). Thus, m is an The lemma above provides us with at decomposition of the differentiable operators in (3.2), that is, for A : H s, p are valid provided A is normalized accordingly. 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.
Proof 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 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 half-space T × R n + with Dirichlet boundary conditions. The problem itself can be formulated elegantly as (3.9). (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 u m+1, p,T×R n Since the symbol of A reads M(k, ξ , ξ n ) = aξ 2m (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 ≤ 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 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.

21)
where we recall the definitions of the trace spaces T ι, p Proof For g ∈ S ⊥ (T × R n−1 ) m we recall Lemma 2.8 and estimate Proof Employ the partial Fourier transform F T×R n−1 to the equation Au = 0, which in view of Plancherel's theorem implies A(k, ξ , D n )F T×R n−1 (u) = 0 for almost every (k, ξ ). By Lemma 3.10, B H j (ξ , D n )F T×R n−1 (u) = F ( j−1)l (k, ξ ) Tr m u(0) l . 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 ∈ 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 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 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 [25,Chapter 4.8].