FACTORIZATION OF SECOND-ORDER STRICTLY HYPERBOLIC OPERATORS WITH LOGARITHMIC SLOW SCALE COEFFICIENTS AND GENERALIZED MICROLOCAL APPROXIMATIONS

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1 Electronic Journal of Differential Equations, Vol , No. 42, pp. 49. ISSN: URL: or FACTORIZATION OF SECOND-ORDER STRICTLY HYPERBOLIC OPERATORS WITH LOGARITHMIC SLOW SCALE COEFFICIENTS AND GENERALIZED MICROLOCAL APPROXIMATIONS MARTINA GLOGOWATZ Communicated by Ludmila Pulkina Abstract. We give a factorization procedure for a strictly hyperbolic partial differential operator of second order with logarithmic slow scale coefficients. From this we can microlocally diagonalize the full wave operator which results in a coupled system of two first-order pseudodifferential equations in a microlocal sense. Under the assumption that the full wave equation is microlocal regular in a fixed domain of the phase space, we can approximate the problem by two one-way wave equations where a dissipative term is added to suppress singularities outside the given domain. We obtain well-posedness of the corresponding Cauchy problem for the approximated one-way wave equation with a dissipative term. Contents. Basic notions 2 2. Pseudodifferential calculus Generalized point values of a generalized symbol Asymptotic Expansion Composition and Adjoint of Pseudodifferential Operators 8 3. Governing equation in the Colombeau setting Derivation of the inhomogeneous wave equation Time extrapolation of the wave field 3.3. Depth evolution processes 4. Ellipticity of pseudodifferential operators 4.. Ellipticity 4.2. Strong ellipticity Construction of a parametrix 9 5. A factorization procedure for the generalized strictly hyperbolic wave operator Technical Preliminaries Factorization procedure 25 2 Mathematics Subject Classification. 35S5, 46F3. Key words and phrases. Hyperbolic equations and systems; algebras of generalized functions. c 28 Texas State University. Submitted January 24, 27. Published February 6, 28.

2 2 M. GLOGOWATZ EJDE-28/42 6. Generalized wave front set Generalized Microlocal Analysis 3 7. Microlocal decomposition of the wave equation Approximated first-order wave equations Requirements on the damping operator 4 9. Well-posedness of the approximated first-order equations 42 Acknowledgments 47 References 47. Basic notions In this section we specify the basic notions that will be needed for our constructions. As the problem is treated within the framework of Colombeau algebras we refer to the literature [6, 24, 23, 5, 9, 8] for a systematic treatment of this field. One of the main objects in our setting are Colombeau generalized functions based on G L 2 which were first introduced in [2]. The elements in this algebra are given by equivalence classes u := [u,] ] of nets of regular functions u in the Sobolev space H = k Z H k satisfying certain asymptotic seminorm estimates. More precisely, we denote by M H the nets of moderate growth whose elements are characterized by the property α N n N N : α u L 2 R n = O N as. Negligible nets will be denoted by N H and are nets in M H whose elements satisfy the following additional condition q N : u L 2 R n = O q as. Then the algebra of generalized functions based on L 2 -norm estimates is defined as the factor space G H = M H /N H. By abuse of notation, we continue to write G L 2R n for G H. For simplicity, we shall also use the notation u instead of u,] throughout the paper. Using [2, Theorem 2.7], we first note that the distributions H = k Z H k are linearly embedded in G L 2R n by convolution with a mollifier ϕ x = n ϕ x where ϕ SR n is a Schwartz function such that ϕx dx =, x α ϕx dx = for all α N n, α.. Further, by the same result, H R n is embedded as a subalgebra of G L 2R n. As different growth types are crucial in regularity theory we introduce the set of logarithmic slow scale nets by { Π := ω R,] : η, ] c >, η] : c ω, η, ] p c p > : ω p c p log },, η]. Further, we call a net ω Π logarithmic slow scale strictly nonzero if there exists r Π and an η, ] such that ω /r for, η]. The different growth type then result by convolution with a mollifier ϕ ω x = ω n ϕω x with the same ϕ SR n as above.

3 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 3 More generally, we introduce Colombeau algebras based on a locally convex vector space E topologized through a family of seminorms {p i } i I as in [3, Section ]. To continue, the elements M E := {u E,] : i I N N : p i u = O N as } M E := {u E,] : N N i I : p i u = O N as } M E := {u E,] : i I ω Π : p i u = Oω as } N E := {u E,] : i I q N : p i u = O q as }.2 are said to be E-moderate, E-regular, E-moderate of logarithmic slow scale type and E-negligible, respectively. Then, N E is an ideal in M E and the Colombeau algebra based on E is defined by the factor space G E = M E /N E and possesses the structure of a C-module. The space of the regular Colombeau generalized functions GE C-module whereas the space of the logarithmic slow scale algebra GE is a C-module. = M E /N E is again a = M E /N E Example.. Setting E = C one gets the ring of complex generalized numbers C = G C with the absolute value as the corresponding seminorm. Furthermore, we denote by R = G R the ring of real generalized numbers. Further, let Ω be an open subset of R n. Then the Colombeau algebra GΩ = E M Ω/N Ω is obtained by taking E = C Ω endowed with the topology induced by the family of seminorms p K,i f = sup{ α fx : x K, α i} with K Ω, i N and f C Ω. Another example is the Colombeau algebra G p,p Ω, p, when E is set to W,p Ω and the topology is determined by the collection of seminorms p i f = sup{ α f p : α i}, f W,p Ω, as i varies over N. We note that G L 2 = G 2,2. For more general Colombeau algebras based on Sobolev spaces we refer to [2, Section 2]. Using the notation there we have the following identities: E M,p Ω = M W,p Ω and N p,p Ω = N W,p Ω. 2. Pseudodifferential calculus In this section we introduce a general calculus for pseudodifferential operators which are standard quantizations of generalized symbols. Since most of the techniques are similar to the usual theory of pseudodifferential operators we also refer to [7, 3]. A detailed discussion on pseudodifferential operators with Colombeau generalized symbols can be found in [9, 8]. As usual, we write D xj = i xj = i xj. We shall initially discuss the main notions of generalized symbols. As already indicated above we will study symbols which satisfy asymptotic growth conditions with respect to ω Π. As usual, we use the notation ξ := + ξ 2 /2. We let ρ, δ [, ] and m R. We denote by Sρ,δ m = Sm ρ,δ Rn R n the set of symbols of order m and type ρ, δ as first introduced by Hörmander in [6, Definition 2.]. Since the symbol class Sρ,δ m satisfies global estimates we remark that our space M S m ρ,δ is different to that in [2, Section.4]. In the following, we will typically encounter subspaces of M S m ρ,δ subjected to logarithmic slow scale asymptotics we can choose in.2 E = Sρ,δ m and obtain symbols of the form:

4 4 M. GLOGOWATZ EJDE-28/42 Definition 2.. Let m R. The set of moderate logarithmic slow scale symbols S m ρ,δ, Rn R n of order m and type ρ, δ consists of all a S m ρ,δ Rn R n,] such that for all α, β N n there exists ω Π : q m α,β a := sup ξ α x β a x, ξ ξ m+ρ α δ β = Oω as. x,ξ R 2n An element of S m ρ,δ, Rn R n is said to be negligible if it fulfills the following condition: α, β N n q N : q m α,β a = O q as. The subset of all negligible elements of S m ρ,δ, Rn R n is denoted by N m ρ,δ Rn R n. Then, logarithmically slow scale symbols of order m are defined as the factor space S m ρ,δ,r n R n := S m ρ,δ,r n R n /N m ρ,δr n R n. and in the following we will assume that δ < ρ. In the case that ρ = and δ = we use the abbreviation S m for S m,,. S Remark 2.2. Moreover, the space of logarithmic slow scale symbols of order consists of equivalence classes a whose representatives a have the property that m R α, β N n ω Π : q m α,β a = Oω as. Since ω is a logarithmic slow scale net, we note that the net of a symbol a in S m can always be estimated as follows α, β N n : q m α,β a = O log as. To give an example, let P x, D x = α m a αxdx α be a partial differential operator with bounded and measurable coefficients. Then the logarithmically slow scale regularization of the symbol is given by p x, ξ := p., ξ ϕ ω x and p is contained in S m Rn R n. Also, we will make use of the following symbol class: Definition 2.3. Let U R n R n be open and conic with respect to the second variable. We say that a generalized symbol a is in S ρ,δ, m U if it has a representative a with a Sρ,δ m U for fixed, ] and for any compact set K pr 2U independent of and V K := {x, ξ U : ξ K} we have: for all α, β N n there exists ω Π exists C > : α ξ β x a x, ξ Cω ξ m ρ α +δ β x, ξ V c K for sufficiently small and where V c K := {x, λξ : x, ξ V K, λ }. Note that Definition 2. is equivalent to Definition 2.3 in the case that U = R n R n. For completeness, we recall the definition for global symbols Sρ,δ m U which is similar to that for local symbols, see [4, Definition 2.3, page 4]: For U being an open subset of R n R n, which is conic in the second variable, we say that a Sρ,δ m U if a C U and for each compact K pr 2 U and V K := {x, ξ U : ξ K} we have α, β N n : sup ξ α x β ax, ξ ξ m+ρ α δ β <. x,ξ VK c

5 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 5 We choose the following convention for defining the Fourier transform Fu of a function u L 2 R n : Fuξ := ûξ := e ixξ ux dx := lim e ixξ σ x ux dx. R n σ + R n Then the Fourier transform is an isomorphism on L 2 and the inverse Fourier transform of u L 2 is given by the formula F ux := e ixξ uξ dξ := lim e ixξ σ ξ uξ dξ. R n σ + R n where we have set dξ := 2π n dξ. More information on the Fourier transform acting on G L 2 can be found in []. As already mentioned above, we will focus on generalized pseudodifferential operators having the following phase-amplitude representation: Definition 2.4. Let a a S ρ,δ, m Rn R n and let u be a representative of u G L 2. We define the corresponding linear operator A : G L 2 G L 2 by Ax, Dux := e ix yξ ax, ξuy dy dξ := A x, Du x + N H R n where A x, D x u x := e ix yξ a x, ξu y dy dξ = e ixξ a x, ξû ξ dξ = F ξ x a x, ξû ξ where the last integral is interpreted as an oscillatory integral. The operator A is called the generalized pseudodifferential operator with generalized symbol a. Later on, we will sometimes write A Ψ m ρ,δ, Rn to denote that A is a generalized pseudodifferential operator with symbol in S ρ,δ, m Rn R n. ρ, δ =,, we will write A Ψ m Rn for short. In the case that 2.. Generalized point values of a generalized symbol. As above, let U Γ R n R n be open and conic with respect to the second variable and a a generalized symbol in Sρ,δ, m U Γ. Definition 2.5. Let Γ R n be an open cone. On Γ M := {ζ Γ,] : N N : ζ = O N as } we introduce the equivalence relation ζ ζ 2 m N : ζ ζ 2 = O m as and denote by Γ := Γ M / the generalized cone with respect to Γ. Lemma 2.6. Let a S m ρ,δ, U Γ and ζ Γ. Then the generalized point value of a at z, ζ = [z, ζ ], is a well-defined element of C. az, ζ := [a z, ζ ]

6 6 M. GLOGOWATZ EJDE-28/42 Proof. The proof is similar to that of [5, Proposition.2.45]. We let z, ζ U Γ M and a Sρ,δ, m U Γ. For any index i N C i denotes a positive constant, ω i, a logarithmic slow scale net and N i a natural number. Then a z, ζ C ω, + ζ m C ω, + C 2 N2 maxm, C 3 N3 so a z, ζ R M = M R. We now show that ζ ζ 2 implies az, ζ az, ζ 2. So let ζ ζ 2. Then a z, ζ a z, ζ 2 ζ ζ 2 D ζ a z, ζ + σζ 2 ζ dσ C 4 ω 4, + C 5 N5 maxm, ζ ζ 2 C 6 N6 p for all p and small enough. Hence a z, ζ a z, ζ 2. Finally, if a N S mu Γ we have a z, ζ C p + ζ m C p + C 7 N7 maxm, for any p as. So a z, ζ. In the case of generalized logarithmic slow scale coneswe obtain a similar result. Definition 2.7. Let Γ R n be an open cone. On Γ M, := {ζ Γ,] : ω Π : ζ = Oω as } we introduce the equivalence relation ζ ζ 2 m N : ζ ζ 2 = O m as and denote by Γ := Γ M, / the generalized logarithmic slow scale cone of Γ. Lemma 2.8. Let a S m ρ,δ, U Γ and ζ Γ. Then the generalized point value of a at z, ζ = [z, ζ ], is a well-defined element of C. az, ζ := [a z, ζ ] 2.2. Asymptotic Expansion. To prepare the factorization theorem of section 5 we will have to consider products of pseudodifferential operators. In the sequel we will construct a complete symbolic calculus for generalized pseudodifferential operators. We therefore start with some general observations concerning the notion of asymptotic expansion of a generalized symbol in S ρ,δ, m. The presentation of the results are inspired by the results given in [8, Section 2.5] and [9]. There, the techniques are described by means of generalized symbols as well as for slow scale symbols. The modification to logarithmic slow scale symbols is evident. The definition of the asymptotic expansion is now the following: Definition 2.9. Let {m j } j be a strictly decreasing sequence of real numbers such that m j as j, m = m.

7 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 7 i Further, let {a j, } j be a sequence with a j, M m S j. We say that the ρ,δ formal series j= a j, is the asymptotic expansion for a C R n R n,], denoted by a j a j,, if and only if N a a j, M m S N ρ,δ j= N. ii Further, let {a j, } j be a sequence with a j, S mj ρ,δ,. We say that the formal series j= a j, is the asymptotic expansion for a C R n R n,], denoted by a j a j,, if and only if N a a j, S m N ρ,δ, N. j= In both cases, a, is said to be the principal symbol of a. Also, we introduce the special case M S m ρ,δ,cl R n R n M S m ρ,δ of classical generalized symbols. Definition 2.. We say that a generalized symbol a M S m ρ,δ is classical, denoted by a M S m ρ,δ,cl, if there exists a sequence {a j, } j with symbols a j, in M S m j R n R n \ homogeneous of degree m j in ξ, j N, ρ,δ such that for any cut-off function ϕ C R n equal to near the origin we have N a x, ξ ϕξa j, x, ξ j= M S m N ρ,δ quad N. 2. As above, we will write a j a j, if 2. holds and we call a, the principal symbol of a. Replacing M S m ρ,δ by M S one obtains classical symbols of logarithmic slow ρ,δ, m scale type. As a result, we obtain that any infinite sum of symbols of strictly decreasing orders can be summed as follows. Theorem 2.. i Let a j, M m S j with m j as j, m = m as in i of Definition 2.9. Then there exists a M S m ρ,δ ρ,δ such that a j a j,. Moreover, if a j a j,, then a a M S. ii Let a j, S mj ρ,δ, with m j as j, m = m as in ii of Definition 2.9. Then there exists a Sρ,δ, m such that a j a j,. Moreover, if a j a j,, then a a S. Remark 2.2. The proof of this theorem can be found in [8, Theorem 2.2]. The same proof remains valid for classical symbol classes. We recall the construction of the symbol a of Theorem 2.: Let ψ C R n, ψξ such that ψξ = for ξ and ψξ = for ξ 2. As in [8, Theorem 2.2] one can define a x, ξ := j N ψλ j, ξa j, x, ξ

8 8 M. GLOGOWATZ EJDE-28/42 where λ j, are appropriate positive constants with λ j+, < λ j, <, λ j, as j. In case i of Theorem 2., λ j, are taken to be the inverse of an appropriate strictly positive net. In Theorem 2. ii, it turns out that λ j, can be chosen to be an inverse of a logarithmic slow scale net. We also remark that Theorem 2. can be carried over to classical symbols in a corresponding way. Noting that in Theorem 2. i one can replace the moderateness by negligibility we introduce as in [8, Definition 2.6 ii]: Definition 2.3. Let {m j } j be as in Definition 2.9. Further let {a j } j be a sequence mj in S ρ,δ,. We say that the formal series j= a j is the asymptotic expansion of a S ρ,δ, m, denoted by a j a j, if and only if there exists a representative a of a and, for every j representatives a j, of a j, such that a j a j,. Similarly, we introduce the following definition for classical symbols. S m j Definition 2.4. Let {a j } j N be a sequence with a j ρ,δ, Rn R n \ homogeneous of degree m j. We say that the formal series j= a j is the asymptotic expansion of a S ρ,δ,cl, m, denoted by a j a j, if and only if there exists a representative a of a and, for every j representatives a j, of a j, such that a j a j, Composition and Adjoint of Pseudodifferential Operators. In this subsection we briefly recall the composition law of two pseudodifferential operators and adjoint operators. A detailed discussion for slow scale regular generalized symbols can be found in [9, Section 5]. Again, the adaptation to logarithmic slow scale symbols is evident. Therefore, for logarithmic slow scale symbols,we obtain the following result. Theorem 2.5. Let Ax, D x and Bx, D x be two pseudodifferential operators m m2 with generalized symbols a S ρ,δ, and b S ρ,δ, respectively. Then the product AB is well-defined and maps G L 2 into itself. Moreover AB is a pseudodifferential operator with generalized symbol a#b in having the representation a#bx, ξ α S m+m2 ρ,δ, α! Dα ξ ax, ξ α x bx, ξ. For the proof we refer to [9, Theorem 5.5]. We note that Theorem 2.5 can also m m2 be stated for classical symbols. In detail, if a S ρ,δ,cl, and b S ρ,δ,cl,, then m+m2 a#b in S ρ,δ,cl,. Also useful is the following theorem. Theorem 2.6. Let Ax, D x be a pseudodifferential operators with generalized symbol a S m ρ,δ,. Then the adjoint operator A is in Ψ m ρ,δ, and its symbol a is given by the asymptotic expansion a x, ξ α For the proof see [9, Theorem 5.2]. i α α! Dα x D α ξ ax, ξ.

9 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 9 3. Governing equation in the Colombeau setting The present survey is devoted to wave propagation in inhomogeneous acoustic media in the case of non-smooth background data. The operator structure is motivated from wave propagation phenomena occuring in a number of physical applications, such as underwater acoustics or seismology. Hence, the evolution direction is the depth coordinate. 3.. Derivation of the inhomogeneous wave equation. The acoustic wave equation characterizes fluid motions and can be derived from the conservation laws of mass and momentum together with the equation of state of thermodynamical equilibrium. This system of acoustic equations can be linearized and therefore describes small perturbations from a state of rest of the pressure Ut, x R, the density ρ t, x R and the velocity v t, x R n that moves in a fluid with given wave speed cx and density ρx, x R n, t R. To explain how these waves are generated or added to the fluid one can introduce so-called sources by adding source terms in the equation of mass, momentum and energy. Assuming an isentropic process the linearized acoustic system can be written as cf. [2, 2, 25, 3] ρ t = divρ v + m mass conservation v t = ρ U + f momentum conservation U t = c2 ρ t + v ρ equation of state where the mass source term m and the force source term f are supposed to vanish in the undisturbed state. Here m represents a volume injection of a source such as, for example, bodies whose volume is oscillating. An example for a body force f of a source is an oscillating rigid body of constant volume. Note that these equations also hold for the fluid at rest. Substituting the state equation into the equation of mass and using the conservation of momentum to eliminate the velocity term gives 2 div ρ U c 2 ρ t 2 U = F where F := m/ρ t + div f/ρ denotes the source function. We remark that in the absence of sources one derives the homogeneous wave equation. In the following we will study the Colombeau generalized partial differential equation of the form LU := n z ρ z + xj ρ x j ρ j= W, /N W, c 2 2 t U = F 3. with the pressure wave field U G L 2R n+ and F G L 2R n+ a source term. For the space coordinates we will allocate the vertical direction z, which we call the depth, the lateral directions are denoted by x. We assume that the coefficients ρ = ρ x, z, c = c x, z are Colombeau generalized functions in G, := M and meet the following requirements: there are representatives ρ ρ, c 2 c 2 such that

10 M. GLOGOWATZ EJDE-28/42 i there exist Hölder continuous functions ρ, c, such that for all x, z R n we have ρ = ρ C,µ R n for some µ ϕ ω, c = c ϕ ω where ϕ is as in., ω Π and the convolution is taken with respect to x and z ii η, ] constants c, c, ρ and ρ such that < c c x, z c < and < ρ ρ x, z ρ < for all x, z R n and, η]. The lower bound assumption is sometimes referred to as strong positivity cf. [, Section ] or [7, Section 2]. The upper bound means uniform boundedness of the -th derivative of the representatives. For more details on the assumption i we refer to the remark given below. Given a regularized operator as above, we carry out all transformations within algebras of generalized functions from now on. More explicitly, we will study the action of the linear operator L from G L 2 into itself in the following sense: on the level of representatives L acts as u n z z + xj xj ρ ρ ρ c 2 2 t u j= This explains our governing equation in 3. where U = [u ]. u M H. Remark 3.. To realize the logarithmic slow scale type conditions on the coefficients of a partial differential operator, one usually uses a rescaling in the mollification. In our case, the regularization is obtained by convolution with the logarithmically scaled mollifier ϕ ω. := ωn ϕω. with ϕ SR n as in. and ω Π. For completeness, we give the following implications for our type of coefficients: Let u C,µ R n be a real-valued Hölder continuous function with exponent µ, and such that infu c for some positive constant c. Further ϕ is a mollifier as above and ω Π. One may think of ω = loglog as an explicit example. Then c > : ω c for all, ] and for all p, c p > : ω p c p log for all, ]. So given some ω Π the logarithmic slow scale regularization of u is given by u x = u ϕ ω x and satisfies the following estimates see [8, Theorem 7], or [28, Section 5]: { α O α = u L = O ω α µ α > and u > η, ] : u x u x R n,, η]. Here, the latter inequality is the strong positivity and the first estimate guarantees that the net u is in M W,. Moreover, note that u C,µ implies that for every x R n we have u x ux ux ω y ux ϕy dy = Oω µ. Again, with a slight abuse of notation we write u for the equivalence class of the Colombeau regularization of u C,µ, i.e: u := u + N W, G,.

11 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 3.2. Time extrapolation of the wave field. In a paper of Garetto and Oberguggenberger, see [3], the authors studied well-posedness of Cauchy problems with respect to the time variable for strictly hyperbolic systems and higher order equations in the Colombeau setting using symmetrisers and the theory of generalized pseudodifferential operators. There they imposed conditions concerning the asymptotic scale of the regularization parameter of the operator. Concretely, the authors proved existence, uniqueness and regularity of generalized solutions in the case that the regularization parameter is chosen logarithmic slow scale. More details can be found in [3, Theorem 4.2] Depth evolution processes. The concept of one-way wave equations, also known as paraxial wave equations, was first introduced by [5] and has become a standard tool in depth migration processes due to ill-posedness of the full wave equation. In fact, they are expressions for the first depth derivative of a wave field and thus of the form z + ib ± x, z, D t, D x. 3.2 where B ± are microlocal pseudodifferential operators. Our derivation of one-way wave equations starts with a factorization of the operator L in 3. into terms of the form 3.2. Remark 3.2. As already noted in subsection 3.2, one obtains well-posed problems in the model of time migration for strictly hyperbolic operators. We therefore give the following link. Note that the operator L in 3. is strictly hyperbolic in the following sense: Let θ, π/2 be a fixed angle and { } I θ := x, z, τ, ξ R n R n : τ, c x, zτ ξ < sin θ. For completeness we recall the following definition. Definition 3.3. The operator L in 3. is called generalized strictly hyperbolic in I θ if there exists a choice of representatives of the coefficients in G, R n such that the corresponding principal symbol of L ζ; x, z, τ, ξ has 2 distinct real-valued roots ζ j,, j =, 2 such that ζ, x, z, τ, ξ ζ 2, x, z, τ, ξ ζ τ, ξ 3.3 holds for some strictly nonzero net ζ, for all x, z, τ, ξ I θ and sufficiently small. 4. Ellipticity of pseudodifferential operators In this section, we introduce the notion of elliptic operators which enables us to construct a parametrix for such operators. Furthermore such operators will play an important role in the characterization of the generalized wave front set cf. section Ellipticity. In [, Definition.2] and [8, Proposition 2.7] the authors introduced the concept of slow scale ellipticity which can be carried over to logarithmic slow scale ellipticity easily. With this in mind, we define the property of logarithmic slow scale ellipticity for our class of operators as follows.

12 2 M. GLOGOWATZ EJDE-28/42 Definition 4.. We say that a generalized symbol a S m is logarithmic slow scale micro-elliptic at x, ξ T R n \ if it has a representative a such that the following is satisfied: There exists a relatively compact open neighborhood U of x, exists a conic neighborhood Γ of ξ, exist nets r Π, s Π and a constant η, ] such that a x, ξ s ξ m x, ξ U Γ, ξ r,, η]. 4. In the sequel, we will sometimes use the abbreviation for logarithmic slow scale. We denote by Ell a the set of points x, ξ T R n \ where a is logarithmic slow scale micro-elliptic. If there exists a representative a a such that 4. holds for all x, ξ T R n \ then the symbol a is called logarithmic slow scale elliptic. Definition 4.2. More generally, we say that a generalized symbol a S m sc is slow scale micro-elliptic at x, ξ T R n \ if 4. holds for some nets r, s Π sc. Note that a logarithmic slow scale elliptic symbol is also slow scale elliptic but not vice versa. Remark 4.3. We first observe that condition 4. is independent of the choice of representative. Indeed, let a a as in Definition 4. and a another representative of a. Then for some arbitrary but fixed p > there c >, η, ] such that on U Γ we have a x, ξ a x, ξ a a x, ξ s ξ m c s 2p ξ r,, η ]. Since there exist η 2, ] and c 2 > such that s c 2 p for, η 2 ],we obtain c s 2p c c 2 p /2, minη 2, 2c c 2 /p]. Redefining η := minη, η 2, 2c c 2 /p we obtain a x, ξ 2s ξ m x, ξ U Γ, ξ r,, η]. Moreover, we notice that the notion of logarithmic slow scale ellipticity is stable under lower order logarithmic slow scale perturbations. For a proof one reworks essentially the lines in [, Proposition.3]. For completeness we give the proof of the following proposition. Proposition 4.4. Let a S m Rn R n be logarithmic slow scale micro-elliptic at the point x, ξ. Then i for all α, β N n there exists λ Π η, ]: α ξ β x a x, ξ λ a x, ξ ξ α U Γ, ξ r,, η] ii if b S m Rn R n with m < m, then 4. is valid for the net a + b. Proof. We first show i. Since a S m is logarithmic slow scale elliptic we obtain: for all α, β N n there exist ω Π, c >, and η, ] such that α ξ β x a x, ξ cω ξ m α cω s ξ α a x, ξ

13 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 3 on U Γ for ξ r,, η]. Setting λ := ω s then λ Π as desired. To show the second part let b S m. Then a x, ξ + b x, ξ s ξ m cω ξ m = s ξ m cs ω ξ m m 2s ξ m on U Γ, ξ r := maxr, 2cs ω /m m and r Π. We note that the previous proposition can also be carried over for symbols that are slow scale micro-elliptic. Remark 4.5. In case of smooth symbols the notion of -ellipticity reduces to the classical one and is therefore equivalent to the complement of the characteristic set. Indeed, given a S m R n R n and a representative a of an -elliptic symbol a S m Rn R n then there exist nets s, r Π and constants c >, η, ] such that for every q N we have: ax, ξ a x, ξ ax, ξ a x, ξ ξ m c q for ξ r,, η]. s As we are allowed to fix an small enough the last expression is bounded away from. In particular Ell a c is equal to the characteristic set Charax, D. Note that this result remains valid if one replaces by sc cf. [, Remark.4]. Before we proceed, we give an equivalent characterization of the notion of ellipticity concerning the principal symbol. In [9, Proposition 3.3] it is proved that a generalized partial differential operator with regular coefficients in G = GC is W-elliptic weak elliptic if its principal symbol is S-elliptic strong elliptic. For the precise meaning of weak and strong ellipticity consult [9]. Referring to this we want to give the following proposition. Proposition 4.6. Given a generalized symbol a Scl, m then the following properties are equivalent: i a is -elliptic ii the principal symbol a m, satisfies the following condition: there exists s Π, η, ] and r > such that a m, x, ξ s ξ m x, ξ T R n with ξ r and, η]. Proof. The proof follows similar arguments to those used in [9, Proposition 3.3]. We first assume that condition ii holds. Then there exist nets s Π, ω Π and constants c, r > such that a x, ξ = a m, x, ξ + j a m j, x, ξ s ξ m cω ξ m s ξ m cω s ξ for ξ r and small enough. Setting r = maxr, 2cω s then r Π and we obtain a x, ξ 2s ξ m ξ r,, η] for some η, ], showing i.

14 4 M. GLOGOWATZ EJDE-28/42 To prove that i implies ii, suppose that a satisfies 4. on R n R n. Let ζ R n, ζ = and choose ξ R n such that ξ r and ζ = ξ/ ξ with r Π as in 4.. Then s Π, r Π, η, ] so that a m j, x, ζ ξ j = a x, ξ ξ m j s ξ m ξ m = s ξ m min2m/2, s =: for ξ r := max, r,, η]. Now fix N N, N. Since a S m we obtain t Π, j N, ω j, Π, η, ] such that s a m j, x, ζ ξ j ω j, ξ j t for ξ max j N r, t ω j, /j and, η]. At this point we are free to choose t Π such that N t = 4s. On the other hand, we can use Theorem 2. for classical symbols to show that ω Π, η, ] such that j N+ a m j, x, ζ ξ j ξ N+ j N+ 2m N ω ξ N+ 4s a m j, x, ζ ξ N+ ω ζ m N for any ξ max, r, 2 m N ω s N+ and, η]. At this point we reset r := max j N, r, 2 m N ω s N+, t ω j, /j. Then r Π and we obtain: η, ] such that a m, x, ζ a m j, x, ζ ξ j a m j, x, ζ ξ j j= j= N s a m j, x, ζ ξ j + s j= 2s = 2s j=n+ a m j, x, ζ ξ j for ξ r,, η] by the above. Thus, since a m, is homogeneous of degree m in the second variable for sufficiently small. a m, x, ξ 2s ξ m 2s ξ m for all ξ R n, ξ Remark 4.7. Again, Proposition 4.6 remains valid if we replace logarithmic slow scale by slow scale.

15 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS Strong ellipticity. As already announced in section 3 we are interested in the depth extrapolation of the wave field U in 3.. We will therefore introduce a notion of logarithmic slow scale ellipticity which in addition has a certain behavior with respect to ζ, the dual variable of the depth. We first introduce some notation. We denote by τ, ξ, ζ R R n R the dual variables corresponding to t, x, z R R n R. Thus, we will work on a 2n + -dimensional phase space. We already mentioned in subsection 2. that given a symbol a S m Rn+ R n+ and a generalized point t, x, z, τ, ξ, ζ in R n+ R n R then the generalized point value of the symbol a at t, x, z, τ, ξ, ζ is well-defined in C. Moreover let Γ s R n+ be a classical conic neighborhood of τ, ξ, ζ of the form Γ s = { λτ, ξ; ζ R n+ τ, ξ ; ζ \ : τ, ξ; ζ B s S, λ > } τ, ξ ; ζ for some s > small enough, where B s η denotes the open ball with radius s around η R n+ and S := {η R n+ : η = }. We set Γ s,m := {λτ, ξ; ζ Γ,] s N N : ζ = O N as, λ > } and introduce an equivalence relation on Γ s by ζ ζ 2 m > : ζ ζ 2 = O m. We then denote by Γ s := Γ s,m / the generalized cone with respect to ζ generated from Γ s. So Γ s is a generalized conic neighborhood of τ, ξ, ζ. Moreover we recall that Γ s Γ s. The canonical embedding of Γ s into Γ s is given by τ, ξ, ζ τ, ξ, ζ + N R. Also note that in the definition of Γ s,m, τ, ξ; ζ Γ,] s τ, ξ; ζ B s τ, ξ ; ζ τ, ξ ; ζ means that S for every fixed, ]. With this notation we now give the definition of strong -micro-ellipticity. Definition 4.8. Let a be a generalized symbol in S m Rn+ R n+. We say that a is strong logarithmic slow scale micro-elliptic at a point t, x, z, τ, ξ ; ζ T R n+ \, and we write t, x, z, τ, ξ ; ζ Ell a, if there exist a relatively compact open neighborhood U of t, x, z, a classical conic neighborhood Γ of τ, ξ, ζ such that Γ Γ where Γ = Γ M / is the generalized conic neighborhood of τ, ξ, ζ generated from Γ as above, nets r, s Π and an η, ] such that a t, x, z, τ, ξ; ζ s τ, ξ; ζ m U Γ M, τ, ξ; ζ r,, η]. 4.2 Figure below shall illustrate the relation of the sets Γ s and Γ s,m in the following special situation. We fix a point x, z R n, let ζ 2 := c x 2, z τ 2 ξ 2 and let Γ s be a conic neighborhood around τ, ξ, ζ. Then for any s >, however small, we can define { ζ, s2 c x := 2, z τ 2 ξ 2 C s, ] c x 2, z τ 2 ξ 2, C s ]

16 6 M. GLOGOWATZ EJDE-28/42 for some positive constant C s depending on s > and which will be specified below. Then ζ 2 ζ, s 2 = c 2 c 2 x, z τ 2 Cτ ω µ, ]. Now let rs be a function depending on s such that rs as s. Then, since Cτ ω µ rs/2 for all small enough, i.e. for less or equal a C s > with C s as s we obtain By the definition of ζ s2, we even get ζ 2 ζ s2, rs/2, C s ]. ζ 2 ζ s2, rs/2, ]. So in particular we have τ, ξ, ζ, s2 Γ s for all, ] and ζ, s2 ζ 2 as. Therefore, if we are working in an open ball around τ, ξ, ζ with radius rs > we can find a generalized point τ, ξ, ζ, s such that τ, ξ, ζ, s is contained in B rs τ, ξ, ζ for all, ]. ζ Γ s ζ, s ζ rs S τ, ξ τ, ξ Figure. Here the shaded area corresponds to the set Γ s. The dashed line through τ, ξ, ζ s, and the origin move to to the line through τ, ξ, ζ and the origin as respectively. Remark 4.9. In particular, since Γ Γ condition 4.2 implies a t, x, z, τ, ξ; ζ s τ, ξ; ζ m U Γ, τ, ξ; ζ r,, η]. 4.3 so the symbol a is logarithmic slow scale micro-elliptic at t, x, z, τ, ξ ; ζ. Remark 4.. The definition of strong slow scale micro-ellipticity is straightforward. Again, one simply replaces by sc in Definition 4.8. Lemma 4.. Let a S m Rn+ R n+. Then a is strong logarithmic slow scale micro-elliptic at t, x, z, τ, ξ ; ζ if and only if it is logarithmic slow scale micro-elliptic there. So if a satisfies 4.3, then it also satisfies 4.2 and vice versa. Proof. The first direction is already shown by Remark 4.9. On the other hand suppose that the symbol a is logarithmic slow scale microelliptic at t, x, z, τ, ξ ; ζ T R n+ \. Then there exists a representative a of a such that the following is satisfied: relatively compact open neighborhood U

17 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 7 of t, x, z, classical conic neighborhood Γ of τ, ξ ; ζ, nets r, s Π, η, ] such that a t, x, z, τ, ξ; ζ s τ, ξ; ζ m on U Γ, τ, ξ; ζ r,, η]. In particular, since Γ is conic a neighborhood of τ, ξ ; ζ, it is of the form Γ = {λτ, ξ; ζ R n+ τ, ξ ; ζ \ τ, ξ; ζ B s S, λ > } τ, ξ ; ζ for some s > small enough. We define Γ := Γ M / the generalized cone with respect to ζ constructed from Γ as above. Take τ, ξ; ζ Γ M. Then τ, ξ; ζ Γ for any fixed, ] and N N such that ζ = O N as. Hence η, ]: a t, x, z, τ, ξ; ζ s τ, ξ; ζ m on U Γ M, τ, ξ; ζ r,, η]. As a conclusion we obtain that t, x, z, τ, ξ ; ζ Ell a, i.e., the symbol a is strong logarithmic slow scale micro-elliptic at the point t, x, z, τ, ξ ; ζ. We have therefore shown that logarithmic slow scale micro-ellipticity at a particular point is equivalent to strong logarithmic slow scale micro-ellipticity at that point. An inspection of the proof of Lemma 4. shows the following result. Corollary 4.2. Given a S m sc then a is strong slow scale elliptic at a phase space point if and only if a is slow scale elliptic at that point. Lemma 4.3. Let L be the generalized differential operator given in 3.. Then Ell sc L is a subset of the complement of Σ, where Σ := {t, x, z, τ, ξ, ζ T R n+ \ : ζ 2 c 2 x, zτ 2 + ξ 2 = } and c C,µ R n is as in the beginning of section 3. Note that Σ is a conic set 2 and independent of the regularization parameter, ]. Moreover we have the following inclusion relations: Σ Ell sc L c Ell L c. Proof. Let t, x, z, τ, ξ, ζ T R n+ \ be an arbitrary but fixed point of Σ. Hence, ζ 2 = c 2 x, z τ 2 ξ 2. Assume that the operator L with generalized symbol in S 2 Rn+ R n+ is slow scale micro-elliptic at the point t, x, z, τ, ξ, ζ T R n+ \. Then by Corollary 4.2 there is a relatively compact open neighborhood U of t, x, z, a conic neighborhood Γ of τ, ξ, ζ such that for some r, s Π sc and some η, ] we have L x, z, τ, ξ, ζ s τ, ξ, ζ 2 for all t, x, z, τ, ξ, ζ U Γ M, τ, ξ, ζ r,, η] and Γ = Γ M / is the generalized conic neighborhood of τ, ξ, ζ from above such that Γ Γ. For small enough we now define ζ 2, := c 2 x, z τ 2 ξ 2 which tends to ζ 2 for fixed τ R as tends to.

18 8 M. GLOGOWATZ EJDE-28/42 Then one can choose τ, ξ, ζ, Γ M, i.e. there is a positive constant s such that τ, ξ ; ζ τ, ξ ; ζ, B s S for every fixed, ] τ, ξ ; ζ since τ, ξ ; ζ, τ, ξ ; ζ as. On the other hand the principal symbol of L vanishes on x, z, τ, ξ, ζ,, i.e. σl x, z, τ, ξ, ζ, = for all small enough. So, by Proposition 4.6 and Remark 4.7, the symbol L is not strong slow scale micro-elliptic and therefore not slow scale micro-elliptic at t, x, z, τ, ξ, ζ - a contradiction. Lemma 4.4. Let L and Σ be as in Lemma 4.3. Then Σ = Ell c L. Proof. By Lemma 4.3 it suffices to show the inclusion Σ c Ell L. To do this, we introduce the continuous function fx, z, τ, ξ, ζ := ζ 2 c 2 x, zτ 2 + ξ 2, which coincides with the limit of the principal symbol of ρ L as. So, Σ c = {t, x, z, τ, ξ, ζ T R n+ \ : fx, z, τ, ξ, ζ > } is an open subset of the phase space. Now given a point t, x, z, τ, ξ, ζ Σ c, there exist a relatively compact open neighborhood U of t, x, z, a conic neighborhood Γ of τ, ξ, ζ and a small δ > such that fx, z, τ, ξ, ζ = ζ 2 c 2 x, zτ 2 + ξ 2 δ τ, ξ, ζ 2 U Γ S. Since f is homogeneous of degree 2 in τ, ξ, ζ it follows that fx, z, τ, ξ, ζ = ζ 2 c 2 x, zτ 2 + ξ 2 δ τ, ξ, ζ 2 U Γ, τ, ξ, ζ. Hence, there exists a constant C δ, ], depending on δ >, such that σl t, x, z, τ, ξ, ζ ζ 2 c 2 x, zτ 2 + ξ 2 c 2 c 2 x, zτ 2 δ τ, ξ, ζ 2 Cω µ τ, ξ, ζ 2 δ 2 τ, ξ, ζ 2 U Γ, τ, ξ, ζ,, C δ ]. Therefore, L is -elliptic at t, x, z, τ, ξ, ζ, which is the desired conclusion. Remark 4.5. We define the set { Λ M := t, x, z, τ, ξ; ζ R n+ R n R,] \ : ζ 2 = } c 2 x, zτ 2 ξ 2, N N : ζ = O N as. If a is a symbol on Σ, then the evaluation on Λ M / is in general not defined. If a is a symbol on Γ, then the generalized point value of a at a point of Γ M / is a well-defined element on C.

19 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 9 A fixed generalized point t, x, z, τ, ξ, ζ, Λ M satisfies Since τ R is fixed we obtain ζ 2, = c 2 x, z τ 2 ξ 2. ζ, 2 = c 2 x, z τ 2 ξ 2 ζ 2 = c 2 x, z τ 2 ξ 2 as and where t, x, z, τ, ξ, ζ Σ. Moreover, ζ, 2 ζ 2 = c 2 c 2 x, z τ 2 Cω µ τ 2. In the next subsection we will construct an approximate inverse for logarithmic slow scale elliptic pseudodifferential operators Construction of a parametrix. In this subsection we will discuss uniqueness and existence results concerning the invertibility of elliptic pseudodifferential operators. Recall that the asymptotic expansion is unique modulo operators of order. As a consequence we will show that one obtains uniqueness of a parametrix modulo operators of infinite order. Note that given a a symbol of class S the operator defined by u a x, Du, u M H has a regular kernel representation in the sense that moderate nets are mapped into regular nets. In the following theorem we shall work on the level of representatives. Theorem 4.6. Let a S m be logarithmic slow scale elliptic, i.e. r Π, s Π, η, ]: Then there exists p S m a x, ξ s ξ m ξ r,, η]. such that p #a = modulo S. Proof. Step : As in [8, Proposition 2.8] we show that there exists p m, S m such that p m, a S. 4.4 For the proof we let ψ C R n, ψ, ψξ = for ξ and ψξ = for ξ 2. Then p m, x, ξ := a x, ξψξ/r with r Π as in the theorem does the job. Indeed one can show that ξ α x β a x, ξ ω α, β ξ α a x, ξ ξ r for some ω α, β Π cf. [9, Lemma 6.3]. Then, by the properties of the function ψ,we obtain p m, S m for ξ 2r and ξ r. On the set r ξ 2r we observe that ξ α x β p m, x, ξ = α α ξ α α α α α α α β x a x, ξ α α ψξ/r ξ ω α, β a x, ξ ξ α r α α χ [re,2r ]ξ sup α α ψξ/r

20 2 M. GLOGOWATZ EJDE-28/42 α α α α ω α, βr α α s 2r α α ξ m α sup α α ψξ and therefore, p m, is in S m. To show 4.4 we write p m, a x, ξ = a a x, ξ + a a x, ξψξ/r = + ψξ/r. Since for any l > and α N n \, sup ξ l ψξ/r cψ 2r l r ξ 2r sup ξ l ξ α ψξ/r = sup ξ l r α α ψξ/r r ξ 2r r ξ 2r γ +j=k j<k cψr α 2r l we have shown 4.4. Step 2: We recursively define for k : { i γ } p m k, x, ξ := γ γ! xa x, ξ γ ξ p m j,x, ξ a x, ξψξ/r. By the same arguments as in [9, Propositions 6.5 and 6.6], one can show that each p m k, is in S m k and by Theorem 2. there exists a net p S m with p j p m j,. Step 3: The aim is to show that p #a S. Therefore, let σ be the generalized symbol with the asymptotic expansion σ x, ξ γ ξ γ! p Dxa γ x, ξ. γ and we will show that σ in S. We write σ γ! Dγ xa γ ξ p γ <N = σ γ <N γ! Dγ xa γ ξ p m l, + l<n γ <N γ! Dγ xa γ ξ p m l,. Since the left-hand side and the last sum on the right-hand side are in S N we obtain σ γ! Dγ xa γ ξ p m l, S N. We now write γ <N γ <N γ! Dγ xa γ ξ p m l, = p m, a + + γ +l N γ <N,l<N l<n N k= l<n {p m k, a + γ! γ ξ p m l,d γ xa, γ +l=k l<k l N } γ! γ ξ p m l,dxa γ

21 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 2 where the last expression is in S N. The second sum on the right-hand side vanishes for ξ 2r and ξ r by construction. So it remains to estimate this expression on the set r ξ 2r. We observe for any l > and α N n that sup ξ l ξ α ψξ/r = r ξ 2r sup ξ l r α α ψξ/r cψr α 2r l r ξ 2r and we conclude that this term is contained in S N. Using Step, i.e. p m, a = mod S, we obtain for every N σ γ! γ ξ p Dxa γ = σ mod S N γ <N and the proof is complete. Recall that a S m is called -elliptic if one of its representatives a is elliptic. Then, the operator px, D has the symbol p + N S m S m and p is as in Theorem 4.6. Furthermore, by Step 3 of the same theorem we obtain px, D ax, Dux = e ix yξ u y dy dξ + N H R n. Therefore, we obtain the following result. Theorem 4.7. Let a S m be a logarithmic slow scale elliptic symbol of order m. Then there exists a logarithmic slow scale elliptic pseudodifferential operator of m order m with symbol p S such that for all u G L 2 ax, D px, Du = u + rx, Du px, D ax, Du = u + sx, Du where rx, D and sx, D are regularizing operators, that is they map G L 2 into G L 2 see section A factorization procedure for the generalized strictly hyperbolic wave operator Concerning products of logarithmic slow scale pseudodifferential operators that approximate the generalized operator L from 3., we will follow the ideas in [22, Appendix II], [7, Chapter 23]. First we recall from 3., n Lx, z, D t, D x, D z = z ρ z + xj ρ x j ρ j= =: z ρ z + A ρ x, z, D t, D x c 2 2 t = 5. where the coefficients /ρ and /c 2 are in G, R n in the variables x, z. This implies that L is an operator of class Ψ 2 Rn R n+ and a representative l of the symbol of L is in S 2 Rn R n+. Furthermore, the symbol of the operator A ρ is in S 2 Rn R n and is given by A ρ x, z, D t, D x = ρ n c 2 2 t x 2 j + j= n j= xj ρ xj.

22 22 M. GLOGOWATZ EJDE-28/42 Note that this operator is only a pseudodifferential operator in x, t depending smoothly on the parameter z on the level of representatives for any fixed, ]. We denote by a ρ S 2 Rn R n the principal symbol of A ρ. Moreover, we set Ax, z, D t, D x := n c 2 x, z 2 t + x 2 j. As already mentioned in section 3, the operator L is not globally strictly hyperbolic but we can restrict the analysis to an appropriate space on which the operator becomes strictly hyperbolic. In particular, we follow Stolk in [29] and introduce the following set of points on which we will then establish the factorization. Therefore we recall from section 3: { } I θ := x, z, τ, ξ R n R n : τ, c x, zτ ξ sin θ 5.2 for some fixed θ, π/2 and c x, z = lim c x, z in C,µ R n as in section 3. Then I θ is a subset of Rn R n \, independent of the regularization parameter, ] and conic with respect to τ, ξ. Fix θ, π/2 and denote by a the principal symbol of A, i.e. a x, z, τ, ξ := τ 2 j= c 2 x, z ξ 2, ]. Then a is logarithmic slow scale elliptic on I θ and the generalized roots [ζ j, ] of the principal symbol of L are given by ζ j, := i a x, z, τ, ξ which satisfy 3.3 on I θ, i.e. ζ, x, z, τ, ξ ζ 2, x, z, τ, ξ C τ, ξ on I θ, for τ, ξ M for some constants C >, M > and all sufficiently small. Indeed, on I θ we have τ 2 a x, z, τ, ξ = c 2 x, z ξ 2 τ 2 c 2 x, z ξ 2 c 2 x, z c 2 τ 2 x, z τ 2 c 2 x, z sin2 θ Cω µ τ 2 Cτ 2 + ξ 2 for some generic constant C > and small enough. So, in particular a ρ is logarithmic slow scale elliptic, since the generalized function /ρ is strongly positive see assumption ii in section 3. Remark 5.. In the classical theory of pseudodifferential operators the characteristic set plays an important role in microlocal analysis. We recall that the characteristic set is defined as the set of points where the homogeneous principal symbol of the operator vanishes. Concerning vanishing properties of the homogeneous principal symbol in the generalized setting, we make the following remarks. We denote by l the principal symbol of L. First note that l ζ; x, z, τ, ξ = ρ x, z ζ2 + a ρ, x, z, τ, ξ = iζ i a x, z, τ, ξ ρ x, z iζ + i a x, z, τ, ξ =

23 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 23 on I θ, τ, ξ M for some constant M > is not classically, i.e. independent of solvable in T R n+ as. But for any x, z, τ, ξ I θ there exists two distinct ζ x, z, τ, ξ such that l ζ ; x, z, τ, ξ = ρ x, z ζ2 + a ρ, x, z, τ, ξ = iζ i a x, z, τ, ξ ρ x, z iζ + i a x, z, τ, ξ = for all τ, ξ M > and, ]. In particular, this is satisfied if we set for fixed x, z, τ, ξ I θ τ ζ x, z, τ, ξ := ± 2 c 2 x, z τ ξ 2 ζx, z, τ, ξ := ± 2 c 2 x, z ξ 2 as. So, the equation has a generalized solution for fixed x, z, τ, ξ I θ. Note that the generalized principal symbol can always be factorized, i.e. l ζ; x, z, τ, ξ = ρ x, z ζ2 + a ρ, x, z, τ, ξ = iζ + i a x, z, τ, ξ ρ x, z iζ i a x, z, τ, ξ on I θ, τ, ξ M regardless of whether the symbol vanishes or not. 2 The roots ζ, x, z, τ, ξ, ζ 2, x, z, τ, ξ of the equation ρ x, z ζ2 + a ρ, x, z, τ, ξ = are called the generalized characteristic roots. Before we state the main theorem of this section, we give a few more details about notation. In the following we will study operators of the form S = 2 j= S j x, z, D t, D x 2 j z on R n+ where the coefficients are operators with symbols S j S kj for some real numbers k j, j =,, 2. Further, we write S = 2 j= S j x, z, D t, D x 2 j z on I θ when the symbols of the coefficients S j are restricted to the set I θ. We will now establish the factorization procedure in order to write the operator L in terms of two first-order pseudodifferential operators of the form L j = z + A j on I θ where A j are pseudodifferential operators with generalized symbols in S, j =, 2. We are now in the position to show the following result. Theorem 5.2. Let L = z ρ z + A ρ x, z, D t, D x and I θ be as in 5. and 5.2 respectively. Then, on the set I θ there are generalized roots {ζ,, ζ 2, } of the principal symbol of L that fulfill the separation property ζ, x, z, τ, ξ ζ 2, x, z, τ, ξ C τ, ξ on I θ, for τ, ξ M

24 24 M. GLOGOWATZ EJDE-28/42 for some constants C >, M >, η, ] and all, η]. Furthermore, the operator L can be factorized into L = L ρ L 2 + R on I θ 5.3 where L j is written as L j = z + A j and A j = A j x, z, D t, D x is a parameterdependent logarithmic slow scale classical generalized pseudodifferential operator of order in t, x. The principal symbol of A j can be chosen either equal to iζ j, = i a, j =, 2 or equal to iζ j, = i a, j =, 2 on I θ. Furthermore, the remainder is given by R = Γ + Γ 2 z for some pseudodifferential operators Γ j = Γ j x, z, D t, D x with parameter z and generalized symbol γ j in on I θ, j =, 2. S Remark 5.3. Note that the product L ρ L 2 in 5.3 is not a pseudodifferential operator on R n+. But one can overcome this by introducing a generalized cut-off ψt, x, z, D t, D x, D z such that the difference ψl ρ L 2 L ρ L 2 is insignificant on some adequate subdomain of the phase space T R n+ under a microlocal point of view. This will be specified in the next section where we introduce a generalized version of microlocal analysis. 5.. Technical Preliminaries. As already mentioned above, the symbol a is logarithmic slow scale elliptic on I θ, θ, π/2, i.e. for any x, z, τ, ξ I θ there exist a relatively compact open neighborhood U of x, z and a conic neighborhood Γ of τ, ξ such that for some r Π, s Π and a constant η, ] we have a x, z, τ, ξ s τ, ξ 2 for x, z, τ, ξ U Γ, τ, ξ r,, η]. As demonstrated in Proposition 4.4, we have stability under lower order logarithmic slow scale perturbations, and therefore the total symbol of the operator A ρ, itself is logarithmic slow scale elliptic on I θ. To describe a factorization for the operator L, we have to give a meaning to the square root of the symbol of A on the set I θ. Note that the square root of a is prescribed on the set I θ. Outside of I θ it is in general not defined but we choose it without the singularity of the square root. We remark that such an extension is equal to a when restricted to I θ. To cut off singularities of the square root of a, we define a generalized symbol χ x, z, τ, ξ S Rn R \ R n homogeneous of degree for τ, ξ and such that { on I θ χ x, z, τ, ξ = outside I θ 2 for angles θ, θ 2, π/2, θ < θ 2. Then, χ a S Rn R \ R n and its restriction to I θ is equal to a. Moreover, χ a vanishes outside I θ 2. Remark 5.4. For the construction of χ we define for every fixed, ] the function f x, z, τ, ξ := c x, zξτ on R n R \ R n.

25 EJDE-28/42 FACTORIZATION OF HYPERBOLIC OPERATORS 25 Then, for every fixed, ] we let χ be the smooth function defined on R n R \ R n, χ, which is given by, f sin γ 2 χ x, z, τ, ξ :=, f sin γ 5.4, sin γ < f < sin γ 2 +e f sin γ sin γ 2 f for some fixed angles γ, γ 2 with < θ < γ < γ 2 < θ 2 < π/2. We therefore get χ = on I θ since there exists η, ] such that I θ {x, z, τ, ξ : τ, c x, zξ/τ sin γ }, η]. To show the inclusion, let x, z, τ, ξ I θ and fx, z, τ, ξ := lim f x, z, τ, ξ. Then there exists a constant C > such that f x, z, τ, ξ f fx, z, τ, ξ + fx, z, τ, ξ c x, z c x, zξ/τ + c x, zξ/τ sin θ Cω µ Since ω µ tends to as and θ < γ, there exists η, ] such that the right-hand side of the last equation is bounded from above by sin γ. Similarly, one can show that there exists η, ] such that for all, η]: {x, z, τ, ξ : τ, c x, zξ/τ sin θ 2 } {x, z, τ, ξ : τ, c x, zξ/τ sin γ 2 } and hence χ = outside I θ 2 note χ is defined only for τ. Indeed f x, z, τ, ξ fx, z, τ, ξ f fx, z, τ, ξ c x, zξ/τ c x, z c x, z c x, z sin θ 2 Cω µ sin γ 2 as. Now, since c G 2, and for every, ] the function χ is smooth on R n R\ R n and homogeneous of order for τ, ξ, we can conclude that χ S Rn R \ R n [4, Example.2] Factorization procedure. The aim is to decompose the operator L as announced in 5.3 in Theorem 5.2. Therefore, we give a construction scheme for the generalized symbols a j, of the operators A j, j =, 2 by means of their asymptotic expansions in the sense of Definition 2.4, that is a j x, z, τ, ξ b µ j x, z, τ, ξ in S cl,i θ 5.5 µ= where the sequence {b µ j } µ N consists of appropriate elements b µ j S µ I θ, j =, 2 and will be constructed recursively. Recall that 5.5 means that there are representatives a j, of a j and b µ j, of b µ j such that for any cut-off ϕ equal to near the origin we have N a j, ϕb µ j, µ= S N I θ. 5.6