REGULARITY OF WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEMS WITH CHARACTERISTIC BOUNDARY

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1 REGULARITY OF WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEMS WITH CHARACTERISTIC BOUNDARY ALESSANDRO MORANDO AND PAOLO SECCHI Abstract. We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be weakly well posed, in the sense that a unique L 2 -solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth. 1. Introduction and main results For n 2, let R n + denote the n dimensional positive half-space R n + := {x = x 1, x, x 1 > 0, x := x 2,..., x n R n 1 }. The boundary of R n + will be sistematically identified with R n 1 x. For T > 0 we set Q T = R n + ]0, T [ and Σ T = R n 1 ]0, T [; we also set Ω T = R n + ], T [ and ω T = R n 1 ], T [. If time t spans the whole real line R, we set Q = R n + R t and Σ = R n 1 R t. We are interested in the following initial-boundary value problem shortly written IBVP Lu = F in Q T, 1 Mu = G on Σ T, 2 u t=0 = f in R n +, 3 where L is a first order linear partial differential operator n L = t + A i x, t i + Bx, t, 4 i=1 t := t and i := x i, i = 1,..., n. The coefficients A i, B, for i = 1,..., n, are real N N matrix-valued functions, defined on Q. The unknown u = ux, t, and the data F = F x, t, G = Gx, t, f = fx are vector-valued functions with N components. M is a given real d N matrix-valued function; M is supposed to have maximal constant rank d. We study the problem 1-3 under the following assumptions. The function spaces involved in D, E and in the statement of Theorems 1, 2 below, as well as the norms appearing in 9 11, 13, will be described in the next Section 2. The square brackets [ ] of a real number denote its integer part. A L is Friedrichs symmetrizable, namely there exists a matrix S 0, definite positive on Q there exists a constant ρ > 0 such that S 0 x, t ρ for every x, t Q, symmetric and such that the matrices S 0 A i, for i = 1,..., n, are also symmetric Mathematics Subject Classification. 35L40, 35L50. Key words and phrases. Symmetrizable systems, symmetric hyperbolic systems, mixed initial-boundary value problem, weak well posedness, loss of derivatives, characteristic boundary, anisotropic Sobolev spaces, tangential regularity. 1

2 2 A. MORANDO AND P. SECCHI B The IBVP is characteristic of constant multiplicity 1 r < N. We assume that the coefficient A 1 of the normal derivative in L displays the block-wise structure A I,I A 1 x = 1 A I,II 1 A II,I 1 A II,II, 5 1 where A I,I 1, AI,II 1, A II,I 1, A II,II 1 are respectively r r, r N r, N r r, N r N r sub-matrices, such that A I,II 1 x 1=0 = 0, AII,I 1 x 1=0 = 0, AII,II 1 x 1=0 = 0, 6 and A I,I 1 is uniformly invertible on the boundary Σ, namely there exists a real positive constant µ such that deta I,I 1 x, t µ, for any x, t Σ. According to the representation above, we split the unknown u as u = u I, u II ; u I R r and u II R N r are said respectively the noncharacteristic and the characteristic components of u. C The matrix M has the form M = I d 0, where I d denotes the identity matrix of order d, 0 is the zero matrix of size d N d, and d r is the constant number of positive eigenvalues of A I,I 1 {x 1=0} the so-called incoming characteristics of problem 1-3. D Existence of the L 2 weak solution. Assume that S 0, A i W 2, Q, for i = 1,..., n. For all T > 0 and matrices B W 1, Ω T, there exist constants γ 0 1 and C 0 > 0 that depend on T, ρ, µ, S 0 W 2, Ω T, A j W 2, Ω T, B W 1, Ω T such that for all γ γ 0 and F Htan,γΩ 1 T, G Hγω 1 T, vanishing for t < 0, the boundary value problem shortly written BVP Lu = F in Ω T, 7 Mu = G on ω T, 8 with B in L, admits a unique solution u L 2 Ω T, vanishing for t < 0, such that u I ω T L 2 ω T. Furthermore u C[0, T ]; L 2 R n +, and it satisfies an a priori estimate of the form 1 γ u γ 2 L 2 Ω + u t γt 2 L 2 R n + + ui γ ω T 2 L 2 ω C t 0 γ 3 F γ 2 Htan,γ 1 Ωt + 1 γ 2 G γ 2 Hγ 1ωt 9 for all γ γ 0 and 0 < t T, where we have set u γ = e γt u, F γ = e γt F, G γ = e γt G. Furthermore, if T = +, for all matrices B 1 W 1, Q and all conormal pseudo-differential operators B 2 with symbol of degree 0 in Γ 0, there exist constants γ 0 1 and C 0 > 0 that depend on ρ, µ, S 0 W 2, Q, A j W 2, Q, B 1 W 1, Q, and on a finite number of seminorms of the symbol of B 2 such that for all F Htan,γQ, 1 G HγΣ, 1 the BVP 7, 8 on Q, with B = B 1 + B 2 in L, admits a unique solution u e γt L 2 Q such that u I Σ eγt L 2 Σ. Furthermore u satisfies the a priori estimate 1 γ u γ 2 L 2 Q + ui γ Σ 2 L 2 Σ C 0 γ 3 F γ 2 Htan,γ 1 Q + 1 γ 2 G γ 2 Hγ 1Σ 10 for all γ γ 0. E Given T > 0, and matrices S 0, A i C T H,γ σ C T H,γ, σ where σ [n + 1/2] + 4, and enjoying properties A - D on Q, let S k 0, Ak i be C matrix-valued functions converging, as k, to S 0, A i in C T H,γ σ C T H,γ σ on [0, T ], and in W 2, Q W 2, Q on R t. Assume also that satisfy properties A, B on Q. Then, for k sufficiently large, property D holds also for the approximating problems with coefficients S k 0, Ak i. S k 0, Ak i When an IBVP admits the solution u enjoying an a priori estimate of type 9 or 10, with F = Lu, G = Mu, the IBVP is weakly L 2 -well posed. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition. More specifically, the loss of derivatives as in 9, 10 occurs when the Kreiss-Lopatinskiĭ determinant has one simple root in the hyperbolic region of the frequency domain, see e.g. [3, 4] for the definitions. In [10], Coulombel and Guès show that the loss of regularity in 9, 10 in such a case is optimal. They also prove that the well posedness result with loss of regularity is

3 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 3 independent of Lipschitzean zero order terms B but is not independent of bounded zero order terms. This is a major difference with the strongly well posed case where there is no loss of derivatives and one can treat lower order terms as source terms in energy estimates. Thus the stability of the problem under lower order perturbations is no longer a trivial consequence of the well posedness itself, and we assume it as an additional hypothesis about the IBVP, see D. Under an a priori estimate of this form, Coulombel [9] has proven the well posedness of the problem, namely the existence of the L 2 solution for all H 1 data. As for E, hyperbolic IBVP that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region as above belong to the WR class defined by Benzoni-Gavage, Rousset, Serre and Zumbrun [3]. This class of problems is stable with respect to small perturbations of the coefficients A j, B, in agreement with E. Examples of problems where the uniform Kreiss-Lopatinskiĭ condition breaks down are provided by elastodynamics with the well-known Rayleigh waves [23, 28], shock waves or contact discontinuities in compressible fluid mechanics, see e.g. [15, 12]. An a priori estimate similar to 9, 10 holds for linearized compressible vortex sheets, see Coulombel and Secchi [11, 12, 13], provided that S 0, A i W 2, Q and B W 1, Q. Under the assumptions A-D it is not hard to obtain the L 2 solvability of the nonhomogeneous IBVP 1-3 on [0, T ], with initial data f 0, that we state in the following theorem. Theorem 1. Assume that problem 1-3 obeys the assumptions A-D. For all T > 0 and matrices B W 1, Q T, there exist constants denoted as above γ 0 1 and C 0 > 0 that depend on T, ρ, µ, S 0 W 2, Q T, A j W 2, Q T and B W 1, Q T such that for all F Htan,γQ 1 T, G HγΣ 1 T, f H 1 tan,γr n + with f 1 := F t=0 n i=1 A i t=0 i f B t=0 f L 2 R n +, and such that Mf = G t=0 on R n 1, the problem 1-3, with data F, G, f, admits a unique solution u L 2 Q T such that u I Σ T L 2 Σ T. Furthermore u C[0, T ]; L 2 R n +, and it satisfies an a priori estimate of the form for all γ γ 0 and 0 < t T. γ u γ 2 L 2 Q + u t γt 2 L 2 R n + + ui γ Σ T 2 L 2 Σ t 1 C 0 γ 3 F γ 2 Htan,γ 1 Qt + 1 γ 2 f 2 1,tan,γ + 1 γ 2 G γ 2 Hγ 1Σt, The proof of Theorem 1 will be given in Appendix A. In order to study the regularity of solutions to the IBVP 1-3, we need to impose some compatibility conditions on the data F, G, f. The compatibility conditions are defined in the usual way, see [22]. Given the equation 1, we recursively define f h by formally taking h 1 time derivatives of Lu = F, solving for h t u and evaluating it at t = 0. For h = 0 we set f 0 := f. The compatibility condition of order m 0 for the IBVP 1-3 reads as 11 Mf h = h t G t=0, on R n 1, h = 0,..., m. 12 The aim of this paper is to prove the following regularity theorem. Theorem 2. Let m N, m 1, and s = max{m + 1, [ n + 1/2 ] + 7}. Given T > 0, assume that S 0, A j C T H,γ, s for j = 1,..., n, and B C T H,γ s 1 or B C T H,γ s if m = s. Assume also that the assumptions A-E are satisfied. Then for all F H,γ m+1 Q T, with tf i t=0 Hγ m i R n + for i = 0,..., m 1, G Hγ m+1 Σ T, f Hγ m+1 R n +, satisfying the compatibility condition 12 of order m, the unique solution u to 1 3, with data F, G, f, belongs to C T H,γ m and u I Σ T Hγ m Σ T. Moreover u satisfies the a priori estimate γ u γ 2 H m,γ Q T + max t [0,T ] u γt 2 m,,γ + u I γ Σ T 2 H m γ Σ T C m 1 γ 2 f 2 m+1,,γ + 1 γ 3 F γ 2 H m+1,γ Q T + 1 γ 2 G γ 2 H m+1 γ Σ T with a constant C m > 0 depending only on A j, B. The function spaces involved in the statement above, as well as the norms appearing in 13, will be described in the next Section

4 4 A. MORANDO AND P. SECCHI In [18], the regularity of weak solutions to the characteristic IBVP 1-3 is studied, under the assumption that the problem is strongly L 2 -well posed, namely that a unique L 2 -solution exists for arbitrarily given L 2 -data, and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that the L 2 -norms of the interior and boundary values of the solution are measured by the L 2 -norms of the corresponding data F, G, f. The statement of Theorem 2 extends the result of [18], to the case where only a weak well posedness property is satisfied by the IBVP 1 3. Here, the L 2 -solvability of 1 3 requires an additional regularity of the data F, G, f, cfr. D. Correspondingly, the regularity of the solution of order m is achieved provided the data have a regularity of order m + 1. To prove the result of [18], the solution u to 1-3 is regularized by a family of tangential mollifiers J ε, 0 < ε < 1, defined by Nishitani and Takayama in [19] as a suitable combination of the operator see Section 3 and the standard Friedrichs mollifiers. The essential point of the analysis performed in [18] is to notice that the mollified solution J ε u solves the same problem 1-3, as the original solution u. The data of the problem for J ε u come from the regularization, by J ε, of the data involved in the original problem for u; furthermore, an additional term [J ε, L]u, where [J ε, L] is the commutator between the differential operator L and the tangential mollifier J ε, appears into the equation satisfied by J ε u. Because the strong L 2 -well posedness is preserved under lower order perturbations, actually this term can be incorporated into the source term of the equation satisfied by J ε u. In the case of Theorem 2, where the L 2 a priori estimate exhibits a finite loss of regularity with respect to the data, this technique seems to be unsuccesful, since [J ε, L]u cannot be absorbed into the right-hand side without losing derivatives on the solution u; on the other hand it seems that the same term cannot be merely reduced to a lower order term involving the smoothed solution J ε u, as well. These observations lead to develop another technique, where the tangential mollifier J ε is replaced by the family of operators 46, involved in the characterization of regularity given by Proposition 4. Instead of studying the problem satisfied by the smoothed solution J ε u, here we consider the problem satisfied by λ m 1,γ Zu. As before, a new term [λ m 1,γ Z, L]u appears which takes account of the commutator between the differential operator L and the conormal operator λ m 1,γ Z. Since we assume the weak well posedness of the IBVP 1-3 to be preserved under lower order terms, the approach is to treat the commutator [λ m 1,γ Z, L]u as a lower order term within the interior equation for λ m 1,γ Zu see 63; this is made possible by taking advantage from the invertibility of the operator λ m 1,γ Z. The paper is organized as follows. In Section 2 we introduce the function spaces and some notations. In Section 3 we give some technical results useful for the proof of the tangential regularity, discussed in Section 4. Sections 5 and 6 contain the proof of the normal regularity for m = 1 and m 2, respectively. The proof of Theorem 1 is given in Appendix A. Some useful properties of the γ-dependent spaces H,γR m n + are proved in Appendix B. 2. Function Spaces The purpose of this Section is to introduce the main function spaces to be used in the following and collect their basic properties Weighted Sobolev spaces. For γ 1 and s R, we set λ s,γ ξ := γ 2 + ξ 2 s/2 14 and, in particular, λ s,1 := λ s. Throughout the paper, for real γ 1, H s γr n will denote the Sobolev space of order s, equipped with the γ depending norm s,γ defined by u 2 s,γ := 2π n R n λ 2s,γ ξ ûξ 2 dξ, 15 û being the Fourier transform of u. The norms defined by 15, with different values of the parameter γ, are equivalent each other. For γ = 1 we set for brevity s := s,1 and, accordingly, the standard Sobolev space H s R n := H s 1R n.

5 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 5 For s N, the norm in 15 turns to be equivalent, uniformly with respect to γ, to the norm H s γ R n defined by u 2 Hγ srn := γ 2s α α u 2 L 2 R n. 16 An useful remark is that for arbitrary s r and γ 1. α s u s,γ γ s r u r,γ, Conormal Sobolev spaces. Let us introduce some classes of function spaces of Sobolev type, defined over the half-space R n +. For j = 1, 2,..., n, we set Z 1 := x 1 1, Z j := j, for j 2. Then, for every multi-index α = α 1,..., α n N n, the conormal derivative Z α is defined by Z α := Z α Z αn n ; we also write α = α αn n for the usual partial derivative corresponding to α. Given an integer m 1 the conormal Sobolev space HtanR m n + is defined as the set of functions u L 2 R n + such that Z α u L 2 R n +, for all multi-indices α with α m. Agreeing with the notations set for the usual Sobolev spaces, for γ 1, Htan,γR m n + will denote the conormal space of order m equipped with the γ depending norm u 2 Htan,γ m Rn + := γ 2m α Z α u 2 L 2 R n + 18 and we have H m tanr n + := H m tan,1r n +. α m 2.3. Anisotropic Sobolev spaces. Keeping the same notations used above, for every positive integer m the anisotropic Sobolev space H m R n + is defined as H m R n + := {w L 2 R n + : Z α k 1 w L 2 R n +, α + 2k m}. For the sake of convenience we also set H 0 R n + = H 0 tanr n + = L 2 R n +. For an extensive study of the anisotropic spaces H m R n + we refer the reader to [18, 27] and references therein. We observe that H m R n + H m R n + H m tanr n + H m loc Rn +, H m R n + H [m/2] R n +, H 1 R n + = H 1 tanr n + except for Hloc m Rn + all imbeddings are continuous. The anisotropic space H,γR m n + is the same space equipped with the γ-depending norm w 2 H,γ m Rn + := γ 2m α 2k Z α 1 k w 2 L 2 R n α +2k m We have H m R n + = H,1R m n +. The spaces Htan,γR m n +, H,γR m n +, endowed with their norms 18, 20 respectively, are Hilbert spaces. In a similar way we define the anisotropic spaces Htan,γQ m T, H,γQ m T, equipped with their natural norms. Given any Banach space X, let C j [0, T ]; X denote the space of all X-valued j-times continuously differentiable functions of t, for t [0, T ]. We denote by W j, 0, T ; X the space of essentially bounded functions, together with the derivatives up to order j on [0, T ], with values in X. We define the spaces m m C T H,γ m := C j [0, T ]; H,γ m j R n +, C T Htan,γ m := C j [0, T ]; Htan,γR m j n +, with norms j=0 u 2 C T H m,γ := j=0 sup ut 2 m,,γ, u 2 C T Htan,γ m := sup ut 2 m,tan,γ, t [0,T ] t [0,T ] 19

6 6 A. MORANDO AND P. SECCHI where ut 2 m,,γ = For the initial data we set m j=0 f 2 m,,γ := j t ut 2 H m j,γ R n +, ut 2 m,tan,γ = m j=0 f j 2 H m j,γ R n +, f 2 m,tan,γ := m j=0 m j=0 j t ut 2 H m j tan,γ Rn +. f j 2 H m j tan,γ Rn +. Some useful properties of the γ-dependent spaces H m,γr n +, that are used in this paper, are proved in Appendix B, where our main concern is to show that the a priori estimates of Section 6 do not explode but are uniformly controlled when γ is taken sufficiently large. 3. Preliminaries and technical tools In this Section, we collect several technical tools that will be used in the subsequent analysis cf. the next Section 4. We start by recalling the definition of two operators and, introduced by Nishitani and Takayama in [19], with the main property of mapping isometrically square integrable resp. essentially bounded functions over the half-space R n + onto square integrable resp. essentially bounded functions over the full space R n. The mappings : L 2 R n + L 2 R n and : L R n + L R n are respectively defined by w x := we x1, x e x1/2, a x = ae x1, x, x = x 1, x R n. They are both norm preserving bijections. It is also useful to notice that the above operators can be extended to the set D R n + of Schwartz distributions in R n +. It is easily seen that both and are topological isomorphisms of the space C 0 R n + of test functions in R n + resp. C R n + onto the space C 0 R n of test functions in R n resp. C R n. Therefore, a standard duality argument yields that and can be defined on D R n + by u, ϕ := u, ϕ 1, u, ϕ := u, ϕ, ϕ C 0 R n, is used to denote the duality pairing between distributions and test functions either in the half-space R n + or the full space R n. In the right-hand sides of 21, 1 is just the inverse operator of, while the operator is defined by 21 ϕ x = 1 x 1 ϕlog x 1, x, x 1 > 0, x R n 1, 22 for functions ϕ C 0 R n. The operators 1 and arise by explicitly calculating the formal adjoints of and respectively. Of course, one has that u, u D R n ; moreover the following relations can be easily verified cf. [19] ψu = ψ u, 23 j u = Z j u, j = 1,..., n, 24 1 u = Z 1 u u, 25 j u = Z j u, j = 2,..., n, 26 whenever u D R n + and ψ C R n + in 23 u L 2 R n + and ψ L R n + are also allowed. From formulas 25, 26 and the L 2 boundedness of, it also follows that : H m tan,γr n + H m γ R n is a topological isomorphism, for each integer m 1 and real γ 1. Following [19] see also [18], in the next Subsection the last property of will be exploited to shift some remarkable properties of the ordinary Sobolev spaces in R n to the functional framework of conormal Sobolev spaces over the half-space R n +. Let us denote by C 0 Rn + the set of restrictions to R n + of functions of C 0 R n. In the end, we observe

7 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 7 that the operator continuously maps the space C 0 Rn + into the space SR n of rapidly decreasing functions in R n note also that the same is no longer true for the image of C 0 Rn + under the operator, which is only included into the space C b Rn of infinitely smooth functions in R n, with bounded derivatives of all orders Parameter depending norms on Sobolev spaces. We start by recalling a classical characterization of ordinary Sobolev spaces in R n, due to Hörmander [14], based upon the uniform boundedness of a suitable family of parameter-depending norms. For given s R, γ 1 and for each ]0, 1] a norm in H s 1 R n is defined by setting u 2 s 1,γ, := 2π n R n λ 2s,γ ξλ 2,γ ξ ûξ 2 dξ. 27 According to Section 2, for γ = 1 and any 0 < 1 we set s 1, := s 1,1, ; the family of weighted norms { s 1, } 0< 1 was deeply studied in [14]; easy arguments relying essentially on a γ rescaling of functions lead to get the same properties for the norms { s 1,γ, } 0< 1 defined in 27 with an arbitrary γ 1. Of course, one has s 1,γ,1 = s 1,γ cf. 15, with s 1 instead of s. It is also clear that, for each fixed ]0, 1[, the norm s 1,γ, is equivalent to s 1,γ in Hγ s 1 R n, uniformly with respect to γ; notice, however, that the constants appearing in the equivalence inequalities will generally depend on see 34. The next characterization of Sobolev spaces readily follows by taking account of the parameter γ into the arguments used in [14, Thm ]. Proposition 3. For every s R and γ 1, u HγR s n if and only if u Hγ s 1 R n, and the set { u s 1,γ, } 0< 1 is bounded. In this case, we have u s 1,γ, u s,γ, as 0. In order to show the regularity result stated in Theorem 13, it is useful providing the conormal Sobolev space Htan,γR m 1 n +, m N, γ 1, with a family of parameter-depending norms satisfying analogous properties to that of norms defined in 27. Such norms were defined by Nishitani and Takayama [19], in the unweighted case γ = 1, just applying the ordinary Sobolev norms m 1, to pull-back of functions on R n +, by the operator; then these norms were used in [18] to characterize the conormal regularity of functions. Following [19], for γ 1, ]0, 1] and all u Htan m 1 R n + we set u 2 R n +,m 1,tan,γ, := u 2 m 1,γ, = 2π n R n λ 2m,γ ξλ 2,γ ξ û ξ 2 dξ. 28 Because is an isomorphism of Htan,γR m 1 n + onto Hγ m 1 R n, the family of norms { R n +,m 1,tan,γ,} 0< 1 keeps all the properties enjoied by the family of norms defined in 27. In particular, the same characterization of ordinary Sobolev spaces on R n, given by Proposition 3, applies also to conormal Sobolev spaces in R n + cf. [19], [18]. Proposition 4. For every positive integer m and γ 1, u H m tan,γr n + if and only if u H m 1 tan,γr n +, and the set { u R n +,m 1,tan,γ,} 0< 1 is bounded. In this case, we have u R n +,m 1,tan,γ, u R n +,m,tan,γ, as A class of conormal operators. The operator, defined at the beginning of Section 3, can be used to allow pseudo-differential operators in R n acting conormally on functions only defined over the positive half-space R n +. Then the standard machinery of pseudo-differential calculus in the parameter depending version well suited to hyperbolic problems, introduced in [1], [7] can be re-arranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to study the conormal regularity of the stationary BVP 52. Let us introduce the pseudo-differential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [8].

8 8 A. MORANDO AND P. SECCHI Definition 5. A parameter-depending pseudo-differential symbol of order m R is a real or complex- valued measurable function ax, ξ, γ on R n R n [1, + [, such that a is C with respect to x and ξ and for all multi-indices α, β N n there exists a positive constant C α,β satisfying: for all x, ξ R n and γ 1. α ξ β x ax, ξ, γ C α,β λ m α,γ ξ, 29 The same definition as above extends to functions ax, ξ, γ taking values in the space R N N resp. C N N of N N real resp. complex-valued matrices, for all integers N > 1 where the module is replaced in 29 by any equivalent norm in R N N resp. C N N. We denote by Γ m the set of γ depending symbols of order m R the same notation being used for both scalar or matrix-valued symbols. Γ m is equipped with the obvious norms a m,k := max sup α + β k x,ξ R n R n, γ 1 λ m+ α,γ ξ α ξ β x ax, ξ, γ, k N, 30 which turn it into a Fréchet space. For all m, m R, with m m, the continuous imbedding Γ m Γ m can be easily proven. For all m R, the function λ m,γ is of course a scalar-valued symbol in Γ m. To perform the analysis of Section 4, it is important to consider the behavior of the weight function λ m,γ λ 1,γ, involved in the definition of the parameter-depending norms in 27, 28, as a γ depending symbol according to Definition 5. In order to simplify the forthcoming statements, henceforth the following short notations will be used λ m 1,γ ξ := λ m,γ ξλ 1,γ ξ λ m+1,γ ξ := λ m 1,γ 1 ξ = λ m,γ ξλ 1,γ ξ, 31 for all real numbers m R, γ 1 and ]0, 1]. One has the obvious identities λ m 1,γ 1 ξ λ m 1,γ ξ, λ m+1,γ 1 ξ λ m+1,γ 1 ξ λ m+1,γ ξ. However, to avoid confusion in the following, we remark that functions λ m+1,γ m+1,γ ξ and λ ξ are no longer the same as soon as becomes strictly smaller than 1; indeed 31 gives λ m+1,γ ξ = λ m+2,γ ξλ 1,γ ξ. A straightforward application of Leibniz s rule leads to the following result. Lemma 6. For every m R and all α N n there exists a positive constant C m,α such that ξ α λ m 1,γ ξ C m,α λ m 1 α,γ ξ, ξ R n, γ 1, ]0, 1]. 32 Because of estimates 32, λ m 1,γ ξ can be regarded as a γ depending symbol, in two different ways. On the one hand, combining estimates 32 with the trivial inequality immediately gives that {λ m 1,γ } 0< 1 is a bounded subset of Γ m. On the other hand, the left inequality in together with 32, also gives λ 1,γ ξ 1 33 λ 1,γ ξ λ 1,γ ξ λ 1,γ ξ, ξ R n, ]0, 1], 34 α ξ λ m 1,γ ξ C m,α 1 λ m 1 α,γ ξ, ξ R n, γ According to Definition 5, 35 means that λ m 1,γ the family {λ m 1,γ For later use, we also need to study the behavior of functions Analogously to Lemma 6, one can prove the following result. actually belongs to Γ m 1 for each fixed ; nevertheless, } 0< 1 is generally unbounded as a subset of Γ m 1. as γ depending symbols. λ m+1,γ Lemma 7. For all m R and α N n there exists C m,α > 0 such that α ξ m+1,γ λ ξ C m,α λ m+1 α,γ ξ, ξ R n, γ 1, ]0, 1]. 36

9 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 9 In particular, Lemma 7 implies that the family { λ m+1,γ } 0< 1 is a bounded subset of Γ m+1 it suffices to combine 36 with the right inequality in 34. Any symbol a = ax, ξ, γ Γ m defines a pseudo-differential operator Op γ a = ax, D, γ on the Schwartz space SR n, by the standard formula u SR n, x R n, Op γ aux = ax, D, γux =: 2π n R n e ix ξ ax, ξ, γûξdξ, 37 where, of course, we denote x ξ := n x j ξ j. Op γ a is called the pseudo-differential operator with symbol j=1 a; m is the order of Op γ a. It comes from the classical theory that Op γ a defines a linear bounded operator Op γ a : SR n SR n ; moreover, the latter extends to a linear bounded operator on the space S R n of tempered distributions in R n. An exhaustive account of the symbolic calculus for pseudo-differential operators with symbols in Γ m can be found in [7]. Here, we just recall the following result, concerning the product and the commutator of two pseudo-differential operators. Proposition 8. Let a Γ m and b Γ l, for l, m R. Then Op γ aop γ b is a pseudo-differential operator with symbol in Γ m+l ; moreover, if we let a#b denote the symbol of the product, one has for every integer N 1 a#b i α ξ α a x α b Γ m+l N. 38 α! α <N Under the same assumptions, the commutator [Op γ a, Op γ b] := Op γ aop γ b Op γ bop γ a is again a pseudo-differential operator with symbol c Γ m+l. If we further assume that one of the two symbols a or b is scalar-valued so that a and b commute in the product, then the symbol c of [Op γ a, Op γ b] has order m + l 1. We point out that when the symbol b Γ l of the preceding statement does not depend on the x variables i.e. b = bξ, γ then the symbol a#b of the product Op γ aop γ b reduces to the point-wise product of symbols a and b; in this case, the asymptotic formula 38 is replaced by the exact formula According to 31, 37, we write: λ m 1,γ D := Op γ λ m 1,γ a#bx, ξ, γ = ax, ξ, γbξ, γ. 39, λ m+1,γ D := Op γ λ m+1,γ. 40 In view of 31 and 39, the operator λ m 1,γ D is invertible, and its two-sided inverse is given by λ m+1,γ D. Starting from the symbolic classes Γ m, m R, we introduce now the class of conormal operators in R n +, to be used in the sequel. Definition 9. Let ax, ξ, γ be a γ depending symbol in Γ m, m R. The conormal operator with symbol a, denoted by Op γ a or equivalently ax, Z, γ is defined by setting u C0 Rn +, Op γ au = Op γ a u. 41 In other words, the operator Op γ a is the composition of mappings Op γ a = 1 Op γ a. 42 As we already noted, u SR n whenever u C 0 Rn +; hence formula 41 makes sense and gives that Op γ au is a C function in R n +. Also Op γ a : C 0 Rn + C R n + is a linear bounded operator that extends to a linear bounded operator from the space of distributions u D R n + satisfying u S R n

10 10 A. MORANDO AND P. SECCHI into D R n + itself 1. Throughout the paper, we continue to denote this extension by Op γ a or ax, Z, γ equivalently. As an immediate consequence of 42, we have that for all symbols a Γ m, b Γ l, with m, l R, there holds u C 0 Rn +, Op γ aopγ bu = Op γ aop γ bu Then, it is clear that a functional calculus of conormal operators can be straightforwardly borrowed from the pseudo-differential calculus in R n ; in particular we find that products and commutators of conormal operators are still operators of the same type, and their symbols are computed according to the rules collected in Proposition 8. Below, let us consider the main examples of conormal operators that will be met in Section 4. As a first example, we quote the multiplication by a matrix-valued function B C 0 Rn +. It is clear that this makes an operator of order zero according to 41; indeed 23 gives for any vector-valued u C 0 Rn + Bu x = B xu x, 44 and B is a C function in R n, with bounded derivatives of any order, hence a symbol in Γ 0. We remark that, when computed for B, the norm of order k N, defined on symbols by 30, just reduces to B 0,k = max α k α B L R n = max α k Zα B L R n +, 45 where the second identity above exploits formulas 24 and that maps isometrically L R n + onto L R n. Now, let L := γi N + n A j xz j be a first order linear differential operator, with matrix-valued coefficients j=1 A j C0 Rn + for j = 1,..., n and γ 1. Since the leading part of L only involves conormal derivatives, applying 23, 25, 26 then gives n γu + A j Z j u = γi 1 n 2 A 1 u + A j ju = Op γ au, j=1 where a = ax, ξ, γ := γi N 12 A 1 +i x n A j xξ j is a symbol in Γ 1. Then L is a conormal operator of order 1, according to 41. In the next Section 4, we will be mainly interested to the family of conormal operators j=1 λ m 1,γ Z := Op γ λm 1,γ, λ m+1,γ j=1 Z := Op γ m+1,γ λ. 46 The operators λ m 1,γ Z are involved in the characterization of conormal regularity provided by Proposition 4 remember that, after Lemma 6, λ m 1,γ Γ m 1. Indeed, from Plancherel s formula and the fact that the operator preserves the L 2 norm, the following identities u R n +,m 1,tan,γ, λ m 1,γ Zu L2 R n + 47 can be straightforwardly established; hence, Proposition 4 can be restated in terms of the boundedness, with respect to, of the L 2 norms of functions λ m 1,γ Zu. This observation is the key point that leads to the analysis performed in Section 4. m+1,γ Another main feature of the conormal operators 46 is that λ Z provides a two-sided inverse of λ m 1,γ Z; this comes at once from the analogous property of the operators in 40 and formulas 41, In principle, Op γ a could be defined by 41 over all functions u C R n +, such that u SR n. Then Op γ a defines a linear bounded operator from the latter function space into itself, provided we equip this space with the topology induced, via, from the Fréchet topology of SR n.

11 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM Sobolev continuity of conormal operators. Proposition 10. If s, m R then for all a Γ m the pseudo-differential operator Op γ a extends as a linear bounded operator from Hγ s+m R n into HγR s n, and the operator norm of such an extension is uniformly bounded with respect to γ. We refer the reader to [7] for a detailed proof of Proposition 10; a sharp calculation shows that the norm of Op γ a, as a linear bounded operator from Hγ s+m R n to HγR s n, actually depends only on a norm of type 30 of the symbol a, besides the Sobolev order s and the symbolic order m cf. [7] for detailed calculations. This observation entails, in particular, that the operator norm is uniformly bounded with respect to γ and other additional parameters from which the symbol of the operator should possibly depend, as a bounded map. Using Proposition 10 and that the operator maps isomorphically conormal Sobolev spaces on R n + to ordinary Sobolev spaces on R n, we easily derive the following result. Proposition 11. If m Z and a Γ m, then the conormal operator Op γ a extends to a linear bounded operator from Htan,γR s+m n + to Htan,γR s n +, for every integer s 0, such that s + m 0; moreover the operator norm of such an extension is uniformly bounded with respect to γ. Remark 12. We point out that, compared to Proposition 10, the statement above only deals with integer orders of symbols and conormal Sobolev spaces. The reason is that, in Section 2, conormal Sobolev spaces were only defined for positive integer orders. In principle, this lack could be removed by extending the definition of conormal spaces H s tanr n + to any real order s: this could be trivially done, just defining H s tanr n + to be the pull-back, by the operator, of functions in H s R n. However, this extension to fractional exponents seems to be useless for the subsequent developments. 4. The homogeneous IBVP. Tangential regularity We introduce the new unknown u γ x, t := e γt ux, t and the new data F γ := e γt F x, t, G γ := e γt Gx, t. Then problem 1-3 becomes equivalent to with L γ u γ = F γ, in Q T, Mu γ = G γ, on Σ T u γ t=0 = f, in R n +, L γ := γ + L. In this section we concentrate on the study of the tangential regularity of the solution to the IBVP 48, where the initial datum f is identically zero and the data F γ and G γ satisfy the compatibility conditions in a more restrictive form than 12. More precisely, we concentrate on the homogeneous IBVP L γ u γ = F γ, in Q T, Mu γ = G γ, on Σ T, u γ t=0 = 0, in R n +. We remark that here and in the following the word homogeneous is referred by convention to the initial datum f. For a given integer m 1, we assume that F γ and G γ satisfy the following conditions h t F γ t=0 = 0, h t G γ t=0 = 0, h = 0,..., m. 50 It is worth to notice that conditions 50 imply the compatibility conditions 12, in the case f = 0. We prove the following theorem for smooth coefficients. The general case with coefficients of finite regularity will be treated later on by a density argument. Theorem 13. Assume that S 0, A i, B, for i = 1,..., n, are in C0 Q, and that problem 49 satisfies assumptions A D; then for all T > 0 and m N there exist constants C m > 0 and γ m, with γ m γ m 1, such that for all γ γ m, F γ Htan,γQ m+1 T and G γ H m+1 Σ T satisfying 50 the unique γ

12 12 A. MORANDO AND P. SECCHI solution u γ to 49 belongs to Htan,γQ m T, the trace of u I γ on Σ T belongs to Hγ m Σ T, and the a priori estimate 1 γ u γ 2 Htan,γ m Q T + ui γ Σ T 2 Hγ mσ T C m γ 3 F γ H m+1 tan,γ Q T γ 2 G γ 2 51 Hγ m+1 Σ T is fulfilled. The first step to prove Theorem 13 is reducing the original problem 49 to a boundary value problem where the time is allowed to span the whole real line and is treated, consequently, as an additional tangential variable. To make this reduction, we extend the data F γ, G γ and the unknown u γ of 49 to all positive and negative times. In the sequel, for the sake of simplicity, we remove the subscript γ from the unknown u γ and the data F γ, G γ. Because of conditions 50, we may extend F, G by setting them equal to zero for all negative times and for t > T by reflection, so that the extended F and G vanish also for all t > T sufficiently large. We get F Htan,γQ m+1 and G Hγ m+1 Σ. As we did for the data, the solution u to 49 is extended to all negative times, by setting it equal to zero. We extend u also for times t > T, by following the argument of [18], where we make use of assumption D. By construction, u solves the BVP L γ u = F, in Q, Mu = G, on Σ. In 52, the time t is involved with the same role of the tangential space variables, as it spans the whole real line R. Therefore, 52 is now a stationary problem posed in Q, with data F Htan,γQ, m+1 G Hγ m+1 Σ. Furthermore, u enjoys the estimate 10, that is 1 γ u 2 L 2 Q + ui Σ 2 L 2 Σ C 0 γ 3 F 2 Htan,γ 1 Q + 1 γ 2 G 2 Hγ 1Σ, 53 for all γ γ 0. The proof of Theorem 13 will be derived as a consequence of the tangential regularity of solutions to the BVP 52. Thus we concentrate from now on this problem. It will be convenient to recover the notations x := t and x := x 1, x, x and denote A = I, Z = t. We argue by induction on the integer order m 1. Let us take arbitrary data F Htan,γQ, m+1 G Hγ m+1 Σ. Because of the inductive hypothesis, we already know that the unique L 2 solution u to 52 actually belongs to Htan,γQ, m 1 and its trace on the boundary u I x 1=0 belongs to Hm 1 γ Σ, provided that γ is taken large enough; moreover the solution u obeys the estimate of order m 1 γ u 2 + H m 1 tan,γ Q ui x 1=0 2 C Hγ m 1 Σ m 1 1 γ F 2 3 Htan,γ m Q γ G 2 2 Hγ mσ, 54 where the positive constant C m 1 only depends on m, µ, and the L norm of a finite number depending on m itself of derivatives of B cf. 45, besides the coefficients A j 1 j n of L The modified conormal Fourier multiplier. In order to increase the conormal regularity of the solution u by order one, we are going to act on the solution u of the BVP 52 by the conormal operator λ m 1,γ Z; then we will consider the problem satisfied by λ m 1,γ Zu. However, due to some technical reasons that will be clarified in the next Section 4.2, we need to slightly modify the conormal operator λ m 1,γ Z. The first step is to decompose the weight function λ m 1,γ as the sum of two contributions. To do so, we proceed as follows 2. Firstly, let us take an arbitrary positive, even function χ C0 R with the following properties 0 χx 1, x R, χx 1, for x 1/2, χx 0, for x > From now on, for all the rest of Section 4, we will use the tools introduced in Section 3 where, due to the use of the time variable t as an additional space variable, the dimension n will be substituted by n + 1.

13 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 13 Then, we set: λ m 1,γ ξ := χdλ m 1,γ ξ = F 1 χ λ m 1,γ ξ, r m, ξ, γ := λ m 1,γ ξ λ m 1,γ ξ = I χdλ m 1,γ ξ. The following result whose proof is given in [17] shows that the function λ m 1,γ λ m 1,γ. 56 essentially behaves like Lemma 14. Let the function χ C0 R satisfy the assumptions in 55. Then λ m 1,γ is a symbol in Γ m 1 ; moreover for every multi-index α N there exists a positive constant C m,α, independent of γ and, such that ξ α λ m 1,γ ξ C m,α λ m 1 α,γ ξ, ξ R. 57 An immediate consequence of Lemma 14 and 56 is that r m, is also a γ depending symbol in Γ m 1. Let us define, with the obvious meaning of the notations: λ m 1,γ λ m 1,γ D := Op γ λ m 1,γ, r m, D, γ := Op γ r m,, Z := Op γ λm 1,γ, r m, Z, γ := Op γ r m,. The second important result is concerned with the conormal operator r m, Z, γ = Op γ r m,, and tells that it essentially behaves as a regularizing operator on conormal Sobolev spaces. Lemma 15. For every k N, the conormal operator r m, Z, γ extends as a linear bounded operator, still denoted by r m, Z, γ, from L 2 Q to H k tan,γq. Moreover there exists a positive constant C m,k, depending only on k and m, such that for all γ 1 and ]0, 1] 58 r m, Z, γu H k tan,γ Q C m,k γ k u L 2 Q, u L 2 Q. 59 The proof of Lemma 15 is given in [17]. According to 56, we decompose λ m 1,γ Z = λ m 1,γ Z + r m, Z, γ. 60 As a consequence of Lemmata 14, 15, the role of the family of conormal operators {λ m 1,γ Z} 0< 1 in the characterization of the conormal regularity provided by Proposition 4 and 47 can also be played by the family of modified operators {λ m 1,γ Z} 0< 1, namely we have the following Corollary 16. For every positive integer m and γ 1, u Htan,γQ m if and only if u H m 1 the set { λ m 1,γ Zu L2 Q} 0< 1 is bounded. tan,γq, and In order to suitably handle the commutator between the differential operator L and the conormal operator λ m 1,γ Z, that comes from deriving the problem satisfied by λ m 1,γ Zu see Subsections 4.2.1, 4.2.2, it is useful to analyze the behavior of the pseudo-differential operators λ m 1,γ D, when interacting with another pseudo-differential operator by composition and commutation. The following Lemma analyzes these situations; actually here the function λ m 1,γ is replaced by a more general symbol a preserving the same kind of decay properties as in 57. Lemma 17. Let {a } 0< 1 be a family of symbols a = a x, ξ, γ Γ r 1, r R, such that for all multi-indices α, β N there exixts a positive constant C r,α,β, independent of γ and, for which: α ξ β x a x, ξ, γ C r,α,β λ r 1 α,γ ξ, x, ξ R. 61 Let b = bx, ξ, γ be another symbol in Γ l, for l R. Then, for every ]0, 1] the product Op γ a Op γ b is a pseudo-differential operator with symbol a #b in Γ l+r 1. Moreover, for all multi-indices α, β N there exists a constant C r,l,α,β, independent of γ and, such that α ξ β x a #bx, ξ, γ C r,l,α,β λ l+r 1 α,γ ξ, x, ξ R. 62

14 14 A. MORANDO AND P. SECCHI Under the same hypotheses, Op γ a Op γ b λ m+1,γ D is a pseudo-differential operators with symbol a #b λ m+1,γ in Γ l+r m ; moreover, {a #b λ m+1,γ } 0< 1 is a bounded subset of Γ l+r m. Eventually, if the symbols a are scalar-valued, [Op γ a, Op γ b] λ m+1,γ D is a pseudo-differential operator with symbol c Γ l+r m 1, and {c } 0< 1 is a bounded subset of Γ l+r m 1. The proof of Lemma 17 is given in [17]. Remark 18. The fact that the symbols of Op γ a Op γ b, Op γ a Op γ b λ m+1,γ D and the symbol of [Op γ a, Op γ b] λ m+1,γ D belong respectively to Γ l+r 1, Γ l+r m and Γ l+r m 1 for scalar-valued a follows at once from the standard rules of symbolic calculus summarized in Proposition 8. The non trivial part of the statement above although deduced from the asymptotic formula 38 with a minor effort is the one asserting that the symbol of Op γ a Op γ b enjoys estimates 62; indeed, it gives the precise dependence on of the decay at infinity of this symbol. Then the remaining assertions in Lemma 17 easily follow from 62 itself. Remark 19. In view of Proposition 11, the results on symbols collected in Lemma 17 can be used to study the conormal Sobolev continuity of the related conormal operators. To be definite, for every nonnegative integer number s, such that s+l+r m is also a nonnegative integer, Proposition 11 and Lemma 17 imply that the conormal operator Op γ a Op γ Z extends as a linear bounded mapping from Htan,γ s+l+r m Q into Htan,γQ; s moreover, its operator norm is uniformly bounded with respect to γ and. If in addition s + l + r m 1 and a are scalar-valued, then [Op γ a, Op γ linear bounded operator from H s+l+r m 1 tan,γ b λ m+1,γ m+1,γ b] λ Z extends as a Q into Htan,γQ, s and again its operator norm is uniformly bounded with respect to γ and. These mapping properties will be usefully applied in the next Sections 4.2, The interior equation. We follow the strategy already explained in the introduction, where now the role of the operator λ m 1,γ Z is replaced by λ m 1,γ Z. Since λ m 1,γ Γ m 1 because of Lemma 14 and u Htan,γQ m 1 from the inductive hypothesis, after Proposition 11 we know that λ m 1,γ Zu L 2 Q. Applying λ m 1,γ Z to 52 we find that λ m 1,γ Zu must solve γ + Lλ γ,m 1 Zu + [λ m 1,γ Z, L]u = λ m 1,γ ZF, in Q. 63 We are going now to show that the commutator term [λ m 1,γ Z, L]u in the above equation can be actually considered as a lower order term with respect to λ m 1,γ Zu. To this end, we may decompose this term as the sum of two contributions corresponding respectively to the tangential and normal components of L. Firstly, in view of 5, 6, we may write the normal coefficient A 1 as A 1 = A A 2 1, A 1 A I,I 1 := hence A = H 1 Z 1, where H 1 x = x 1 1 A2 1x C0 Q. Therefore, we split L as According to this, we have: [λ m 1,γ L = A L tan, L tan := H 1 Z 1 + A j Z j + B., A 2 1 x = 0, 64 1=0 j=2 Z, L]u = [λ m 1,γ Z, A ]u + [λ m 1,γ Z, L tan ]u. 65 Note that L tan is just a conormal operator of order 1, according to the terminology introduced in Section 3.2.

15 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 15 Z, L tan ]u. Us The tangential commutator. Firstly, we study the tangential commutator [λ m 1,γ ing the identity λ m+1,γ be written in terms of λ γ,m 1 Zλ m 1,γ Z = I and the decomposition 60, this tangential commutator can Zu, modulo some infinitely smooth reminder. Indeed we compute: [λ m 1,γ = [λ m 1,γ Z, L tan ]u = [λ m 1,γ Z, L tan ] λ m+1,γ Z, L tan ] λ m+1,γ Zλ m 1,γ Zu + s m, x, Z, γu, Zλ m 1,γ Zu + r m, Z, γu 66 where we have set for short s m, x, Z, γ := [λ m 1,γ Z, L tan ] λ m+1,γ Zr m, Z, γ The normal commutator. We notice that, due to the structure of the matrix A 1 1, the commutator [λ γ,m 1 Z, A ] acts non trivially only on the noncharacteristic component of the vector function u; namely we have: [λ m 1,γ Z, A ]u = [λ m 1,γ Z, A I,I 1 1]u I. 0 Therefore, we focus on the study of the first nontrivial component of the commutator term. Note that the commutator [λ m 1,γ Z, A ] cannot be merely treated by the operator algebra, because of the normal derivative. This subsection is devoted to the study of the normal commutator The following result is of fundamental importance. [λ m 1,γ Z, A I,I 1 1]u I. Proposition 20. For all ]0, 1], γ 1 and m N, there exists a symbol q m, x, ξ, γ Γ m 2 such that [λ m 1,γ Z, A I,I 1 1]w = q m, x, Z, γ 1 w, w C0 Q. 68 Moreover, the symbol q m, x, ξ, γ obeys the following estimates. For all α, β N there exists a positive constant C m,α,β, independent of γ and, such that α ξ β x q m, x, ξ, γ C m,α,β λ m 2 α,γ ξ, x, ξ R. 69 Proof. That q m, x, ξ, γ, satisfying estimates 69, is a symbol in Γ m 2 actually follows arguing from 69 and inequalities 34 as we already did for λ m 1,γ m+1,γ ξ and λ ξ see Section 3.2. ; For given w C0 Q, let us explicitly compute [λ m 1,γ Z, A I,I 1 1]w using the identity 1 w = e x1 Z 1 w and that λ m 1,γ Z and Z 1 commute, we find for every x R : [λ m 1,γ A I,I, Z, A I,I 1 1]w x = λ m 1,γ 1 xe x1 Z 1 λ m 1,γ Zw x D A I,I, 1 e 1 Z 1 w = λ m 1,γ = λ m 1,γ D D A I,I, 1 e 1 Z 1 w x x A I,I, 1 xe x1 λ m 1,γ ZZ 1 w x A I,I, 1 e 1 Z 1 w x A I,I, 1 xe x1 λ m 1,γ DZ 1 w x. 70

16 16 A. MORANDO AND P. SECCHI Observing that λ m 1,γ D acts on the space SR as the convolution by the inverse Fourier transform of λ m 1,γ, the preceding expression can be equivalently restated as follows: [λ m 1,γ Z, A I,I 1 1]w x = F 1 λ m 1,γ A I,I, 1 e 1 Z 1 w x A I,I, 1 xe x1 F 1 λ m 1,γ Z 1 w x = F 1 λ m 1,γ, A I,I, 1 x e x1 1 Z 1 w x A I,I, 1 xe x1 F 1 λ m 1,γ, Z 1 w x 71 = η m 1,γ, χ A I,I, 1 x e x1 1 Z 1 w x η m 1,γ, χ A I,I, 1 xe x1 Z 1 w x = η m 1,γ, χ A I,I, 1 x 1 w x η m 1,γ, χ A I,I, 1 xe 1 1 w x = η m 1,γ, χ [A I,I, 1 x A I,I, 1 xe 1 ] 1 w x, where η m 1,γ := F 1 λ m 1,γ, and the identity F 1 λ m 1,γ = χη m 1,γ following at once from 56 has been used. Just for brevity, let us further set Kx, y := [A I,I, 1 x y A I,I, 1 xe y1 ]χy. 72 Thus the identity above reads as [λ m 1,γ Z, A I,I 1 1]w x =, Kx, 1 w x, 73 η m 1,γ where the kernel Kx, y is a bounded function in C R R, with bounded derivatives of all orders. This regularity of K is due to the presence of the function χ in formula 72; actually the vanishing of χ at infinity prevents the blow-up of the exponential factor e y1, as y 1. We point out that this is just the step of our analysis of the normal commutator, where this function χ is needed. After 72, we also have that Kx, 0 = 0; then, by a Taylor expansion with respect to y, we can represent the kernel Kx, y as follows Kx, y = b k x, yy k, 74 k=1 where b k x, y are given bounded functions in C R R, with bounded derivatives; it comes from 72 that functions b k can be defined so that for some ε > 1 and all x R there holds supp b k x, { y ε}. 75 Inserting 74 in 73 and using standard properties of the Fourier transform we get [λ m 1,γ Z, A I,I 1 n 1]w x = η m 1,γ, b k x, k 1 w x k=1 = k F 1 λ m 1,γ, b k x, 1 w x k=1 = F 1 D k λ m 1,γ, b k x, 1 w x k=1 = D k λ m 1,γ, F 1 b k x, 1 w x k=1 = D R k λ m 1,γ ξf 1 b k x, 1 w x ξdξ k=1 = 2π n 1 D R k λ m 1,γ ξ e iξ y b R k x, y 1 w x ydy dξ k=1 76

17 WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEM 17 where we have set D k := i ξk fo each k = 1,..., n + 1. Note that for w C0 Q and any x R the function b k x, 1 w x belongs to SR ; hence the last expression in 76 makes sense. Henceforth, we replace 1 w by any function v SR. Our next goal is writing the integral operator 2π n 1 D k λ m 1,γ ξ e iξ y b k x, yvx ydy dξ 77 R R as a pseudo-differential operator. Firstly, we make use of the inversion formula for the Fourier transformation and Fubini s theorem to recast 77 as follows: R e iξ y b k x, yvx ydy = 2π n 1 R e iξ y b k x, y R e ix y η vηdη dy = 2π n 1 R e ix η R e iy η ξ b k x, ydy vηdη = 2π n 1 R e ix η b k x, η ξ vηdη ; for every index k, b k x, ζ denotes the partial Fourier transform of b k x, y with respect to y. inserting 78 into 77 we obtain 2π n 1 D R k λ m 1,γ ξ e iξ y b R k x, yvx ydy dξ = 2π 2n 2 D R k λ m 1,γ ξ e ix η b R k x, η ξ vηdη dξ. 78 Then, Recall that for each x R, the function y b k x, y belongs to C0 R and its compact support does not depend on x, see 75; thus, for each x R, b k x, ζ is rapidly decreasing in ζ. Because of the estimates for derivatives of λ m 1,γ and since vη is also rapidly decreasing, Fubini s theorem can be used to change the order of the integrations within 79. So we get 2π 2n 2 D R k λ m 1,γ ξ e ix η b R k x, η ξ vηdη dξ = 2π 2n 2 e ix η b R R k x, η ξd k λ m 1,γ ξdξ vηdη = 2π 80 n 1 e ix η q R k,m, x, η, γ vηdη, where we have set q k,m, x, ξ, γ := 2π n 1 79 R bk x, ηd k λ m 1,γ ξ ηdη. 81 Notice that formula 81 defines q k,m, as the convolution of the functions b k x, and D k λ m 1,γ ; hence q k,m, is a well defined C function in R R. The proof of Proposition 20 will be accomplished, once the following Lemma will be proved. The proof of Lemma 21 is given in [17]. Lemma 21. For every m N, k = 1,..., n + 1 and all α, β N there exists a positive constant C k,m,α,β, independent of γ and, such that α ξ β x q k,m, x, ξ, γ C k,m,α,β λ m 2 α,γ ξ, x, ξ R. 82 It comes from Lemma 21 and the left inequality in 34 that, for each index k, the function q k,m, is a symbol in Γ m 2 ; notice however that the set {q k,m } 0< 1 is bounded in Γ m 1 but not in Γ m 2. End of the proof of Proposition 20. The last row of 80 provides the desired representation of 77 as a pseudo-differential operator; actually it gives the identity 2π n 1 D k λ m 1,γ ξ e iξ y b k x, yvx ydy dξ = Op γ q k,m, vx, R R for every v SR. Inserting the above formula with v = 1 w into 76 finally gives [λ m 1,γ Z, A I,I 1 1]w x = Op γ q m, 1 w x, 83