J. Pseudo-Differ. Oper. Appl. (2014) 5:455–479 DOI 10.1007/s11868-014-0100-x

Iterative properties of pseudo-differential operators on edge spaces W. Rungrottheera · B.-W. Schulze · M. W. Wong

Received: 27 July 2014 / Accepted: 10 August 2014 / Published online: 20 August 2014 © Springer Basel 2014

Abstract Pseudo-differential operators with twisted symbolic estimates play a large role in the calculus on manifolds with edge singularities. We study here aspects of the underlying abstract concept and establish a new result on iteration of quantizations. Keywords Pseudo-differential operators · Twisted symbolic estimates · Quantizations Mathematics Subject Classification

35J70 · 47G30 · 58J40

1 Introduction The cone, edge and corner pseudo-differential theories of [6,20,25,27], organized as algebras of operators with symbolic structures, suggest an iterative approach. This

This research has been supported by Research Fund SFR-PRG-2557-04 of Faculty of Science, Silpakorn University and the Natural Sciences and Engineering Research Council of Canada. W. Rungrottheera Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand e-mail: [email protected] B.-W. Schulze Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany e-mail: [email protected] M. W. Wong (B) Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada e-mail: [email protected]

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paper is devoted to new elements of this program. In Sects. 2 and 3 we give the abstract edge spaces, twisted symbolic estimates and associated operators. Edge spaces modelled on Hilbert and more general spaces with group action have been introduced in [24]. The edge pseudo-differential calculus in such spaces based on operators with operator-valued symbols arises in the analysis of boundary value problems. See, in particular, [1] for operators with transmission property on the boundary, and [21] for the case without the transmission property. The corresponding boundary symbolic calculus in L 2 spaces on the half-axis and relations to Mellin operators have been studied in Eskin’s book [7]. This is one of the sources of the pseudo-differential calculus on manifolds with conical singularities in the sense of [20]. Other results can be traced back to Kondratyev [14], Rabinovich [19], Kondratyev and Oleynik [15]. Operator-valued symbolic structures have also been used in Vishik and Grushin [33] in the context of boundary value problems for certain degenerate operators. The paper [18] of Luke contains an index theorem for elliptic operators of order zero based on operator-valued symbols on L 2 spaces. The lack of the higher order case is due to the lack of spaces with group actions, which are given later in [24]. The development of the singular analysis up to iterative concepts for higher singularities has been outlined in Chapter 10 of [11]. See also [2,10,22,23,27,28,30,31]. Many specific contributions and applications are given in [3–5,8,9,29,32]. Let us finally mention the useful paper [12] where interpolation properties of edge Sobolev spaces have been studied by a more abstract integral transform than the Fourier transform. In Sect. 4 we establish a new theorem for iterated pseudo-differential operators on edge spaces. This is a result in the larger program of completing the calculus of k-fold iterated corner pseudodifferential operators for k ≥ 2. 2 Abstract edge spaces Abstract edge spaces, to be defined in Definition 2.1, play a large role in the following investigations. Spaces of that kind have been introduced and widely investigated in a first version of the edge algebra in [24]. In a more “concrete” form they have already appeared in [21]. In a paper of Hirschmann [12] these spaces are investigated in connection with interpolations and other useful functional analytic properties. Vector-valued spaces without group action in the reference spaces H are employed in Luke [18] in connection with the index theory of elliptic operators with operator-valued symbols. In the paper [33] of Vishik and Grushin, degenerate operators in terms of operator-valued symbols are studied. Certain versions of edge spaces have been applied by Dreher and Witt [5] to hyperbolic problems. In Flad and Harutyunyan [8] the edge algebra machinery, including edge spaces, has been applied to models of particle physics. Let us first recall some notation and definitions on pseudo-differential operators with operator-valued symbols. The starting point is a Hilbert space H with a group of isomorphisms κ = {κδ }δ∈R+ , κδ : H → H, such that δ → κδ h defines an element of C(R+ , H ) for every h ∈ H . In that case we say that H is endowed with a group action. There are then constants c, g > 0 such that

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κδ L(H ) ≤ c(max{δ, δ −1 })g .

(2.1)

More generally, if E is a Fr´echet space, written as a projective limit E = lim

←− j∈N

Ej

of Hilbert spaces E j , continuously embedded in E 0 for all j, a group action on E is defined by a group action on E 0 such that the restriction κδ | E j is a group action on E j for every j. Definition 2.1 (i) Let H be a Hilbert space with group action κ. Then for s ∈ R, W s (Rq , H ) is defined to be the completion of S(Rq , H ) with respect to the norm  W s (Rq ,H ) given by  uW s (Rq ,H ) =

η

2s

Rq

−1 2 − κη u(η) ˆ Hd η

1/2 ,

(2.2)

where u(η) ˆ = (Fy→η u)(η) is the Fourier transform, d− η := (2π )−q dη. (ii) For a Fr´echet space E with group action in the above-mentioned sense, we define W s (Rq , E) by W s (Rq , E) = lim W s (Rq , E j ). ← − j∈N

If necessary, in order to indicate the dependence of the spaces on κ we also write W s (Rq , H )κ and W s (Rq , E)κ , respectively. Clearly the case id consisting of κδ = id for all δ ∈ R+ is admitted. Then we have W s (Rq , H )id = H s (Rq , H ). The spaces in Definition 2.1 are also referred to as abstract edge Sobolev spaces. Recall that the operator K given by K u = F −1 κη Fu induces an isomorphism K : W s (Rq , H )id → W s (Rq , H )κ for every s ∈ R. For any positive function w(η) ∈ C(Rq ) such that there exist positive constants c1 and c2 for which c1 w(η) ≤ η ≤ c2 w(η)

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for all η ∈ Rq , the integral  w(η)

Rq

2s

−1 2 − κη u(η) ˆ Hd η

1/2

gives an equivalent norm to (2.2). The case when w(η) = σ η , where σ ∈ R+ is fixed, is of particular interest to us. Moreover, there are different choices of group actions κ = {κδ }δ∈R+ and ϑ = {ϑδ }δ∈R+ on H such that (2.2) is equivalent to  Rq

−1 2 − η 2s ϑη u(η) ˆ Hd η

1/2 .

This is the case, for instance, for ϑδ u = κσ δ u for any fixed σ ∈ R+ . More generally, κ is equivalent to ϑ if supδ∈R+ κδ ϑδ−1 L(H ) < ∞. Moreover,  Rq

−1 2 − σ η 2s κσ ˆ Hd η η u(η)

1/2 (2.3)

is equivalent to (2.2) for any fixed σ ∈ R+ . For a fixed u ∈ W s (Rq , H ) we can rewrite (2.3) as  Rq

−1 η 2s κη uˆ σ (η)2H d− η

1/2

for a continuous family R+ → W s (Rq , H ), σ → u σ . In fact, we may set uˆ σ (η) =

σ η s −1 κ u(η) ˆ η s σ η /η

(2.4)

for all η ∈ Rq . Theorem 2.2 Let H be a Hilbert space with group action κ. Then W s (Rq , H )κ is a Hilbert space with group action χ = {χδ }δ∈R+ defined by (χδ u)(y) = δ q/2 (κδ u)(δy), δ ∈ R+ ,

(2.5)

where κδ acts pointwise on the values of u in the space H , and for every p ∈ N we have W s (R p , W s (Rq , H )κ )χ = W s (R p+q , H )κ , for all s ∈ R. A similar result holds for a Frechet ´ space E with group action κ.

(2.6)

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Proof Let us verify that (2.5) defines a group action on W s (Rq , H )κ . We employ Definition 2.1, assume that u ∈ W s (Rq , H )κ , and compute the W s (Rq , H )κ -norm of χδ u, δ ∈ R+ . Indeed, 

η

2s

Rq

−1 κη (χδ u)∧ (η)2H d− η =



−1 −q/2 η 2s κη δ u(δ ˆ −1 η)2H d− η   −1 −q/2 −1 2 q− 2 − δ η δη 2s κδη δ u( ˆ η) ˜ δ d η ˜ = u(η) ˆ ˜ 2s κδ = H Hd η η ˜ Rq Rq  δη 2s −1 2 − ) η 2s κδη ( u(η) ˆ (2.7) = H d η, q η R

χδ u2W s (Rq ,H ) =

Rq

using the relation Fy→η (ϑδ−1 h)(η) = ϑδ (Fy→η h)(η), δ ∈ R+ ,

(2.8)

on a function h(y), y ∈ Rq , where (ϑδ−1 h)(y) = δ q/2 h(δ −1 y). The right hand side of (2.7) can be written as 

−1 η 2s κη uˆ δ (η)2H d− η

Rq

−1 u(η) ˆ in view of the formula (2.4). As noted before the for uˆ δ (η) = (δη / η )s κδη /η −1 correspondence δ → Fη→y uˆ δ (η) for fixed u ∈ W s (Rq , H )κ represents a continuous function on δ ∈ R+ with values in W s (Rq , H )κ . For convenience, norms will be identified when they are equivalent. First we write

  f W s (R p+q ,H ) =



2

Rq

Rp

−1 (ξ 2 + |η|2 )s κ(ξ fˆ(ξ, η)2H dξ dη, 2 +|η|2 )1/2

where the “hat” indicates the Fourier transform F(x,y)→(ξ,η) . Moreover, employing the expression (2.2) where “hat” has the meaning of Fy→η , we obtain   f W s (R p+q ,H ) =





2

=

ξ 

2s

Rp



Rq

ξ

2s

Rp

|η|2 1+ ξ 2

η

2s

Rq

s

−1 κ(ξ fˆ(ξ, η)2H dηdξ 2 +|η|2 )1/2

−1 κ(ξ 2 +ξ 2 |η|2 )1/2

 2 q ˆ f (ξ, ξ η) H ξ dη dξ (2.9)

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and   f W s (R p ,W s (Rq ,H )) = 2



−1 2 ξ 2s χξ (Fx→ξ f )(ξ, y)W s (Rq ,H ) dξ

Rp

−1 −q/2 ξ 2s κξ (Fx→ξ f )(ξ, ξ −1 y)2W s (Rq ,H ) dξ ξ    −1 −1 q/2 ˆ 2 ξ 2s η 2s κξ ξ ξ κ dη dξ. f (ξ, η) = H η

=

Rp Rp

Rq

(2.10) In the identification of the expressions in {· · · } occurring in (2.9) and (2.10) we have −1 −1 −1 

employed the relation (2.8) and the fact that κξ κη = κξ η . The Eq. (2.6) is an extension of Lemma 1 in Subsection 3.1.1 of [25], i.e., for the case H = C and κδ = idC , δ ∈ R+ , i.e., W s (R p , H s (Rq )) = H s (R p+q ) when H s (Rq ) is endowed with the group action (κδ u)(y) = δ q/2 u(δy), δ ∈ R+ . The Eq. (2.6) is formula (24) in Subsection 3.1.2 of [25]. Details are given in Proposition 1.3.44 of [26], however, under the some extra assumptions as in [25]. The assumptions are, in fact, redundant. We have presented the general proof here since (2.6) belongs to the iterative concept of corner pseudo-differential operators.

3 Symbols with twisted estimates  with group actions κ and κ, Given two Hilbert spaces H and H ˜ respectively, a function q )) is called twisted homogeneous in η ∈ Rq \{0} of order f ∈ C ∞ (Rη \{0}, L(H, H μ ∈ R if f (δη) = δ μ κ˜ δ f (η)κδ−1 for all δ ∈ R+ . Let ) S (μ) (Rq \{0}; H, H

(3.1)

denote the space of those functions f . Then, if χ (η) is an excision function on Rq , i.e., χ ∈ C ∞ (Rq ), χ (η) = 0 for |η| < ε0 , χ (η) = 1 for |η| > ε1 for some 0 < ε0 < ε1 , the function a(η) := χ (η) f (η) is an example of an operator-valued symbol in the following sense. The space ), S μ ( × Rq ; H, H

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)) such  ⊆ R p open, is defined to be the set of all a(y, η) ∈ C ∞ ( × Rq , L(H, H that −1 {D αy Dηβ a(y, η)}κη L(H, H) ≤ c η μ−|β| κ˜ η

(3.2)

for all (y, η) ∈ K × Rq , K  , α ∈ N p , β ∈ Nq , for some constant μ ) of classical symbols is c = c(α, β, K ) > 0. The subspace Scl ( × Rq ; H, H μ q  defined as the set of all a(y, η) ∈ S ( × R ; H, H ) such that there are functions ), j ∈ N, with a(μ− j) (y, η) ∈ S (μ− j) ( × (Rq \{0}); H, H r N +1 (y, η) := a(y, η)−

N 

) (3.3) χ (η)a(μ− j) (y, η) ∈ S μ−(N +1) ( × Rq ; H, H

j=0

for every N ∈ N. )) and Example 3.1 Let a(y, η) ∈ C ∞ ( × Rq , L(H, H a(y, δη) = δ μ κ˜ δ a(y, η)κδ−1 μ

). for all δ ≥ 1, |η| ≥ c for some c > 0. Then a(y, η) ∈ Scl ( × Rq ; H, H ) is a Fr´echet space where the From the definition it follows that S μ ( × Rq ; H, H μ ) semi-norms are the best constants c in the estimates (3.2). Also Scl ( × Rq ; H, H μ q  is Fr´echet in the projective limit topology of the mappings Scl ( × R ; H, H ) → ), a(y, η) → a(μ− j) (y, η), j ∈ N, and S μ ( × S (μ− j) ( × (Rq \{0}); H, H cl q μ−(N +1) ) → S ), a(y, η) → r N +1 (y, η), cf. (3.3). ( × Rq ; H, H R ; H, H If a consideration is valid in the classical or the general case we write subscript “(cl)”. By μ ) S(cl) (Rq ; H, H

(3.4)

we denote the subspaces of symbols with constant coefficients, i.e., which are indeμ ). Then pendent of y. The space (3.4) is closed in Scl ( × Rq ; H, H μ μ ) = C ∞ (, S μ (Rq ; H, H )) = C ∞ ()⊗ ), ˆ π S(cl) (Rq ; H, H S(cl) ( × Rq ; H, H (cl) (3.5)

ˆ π denotes the (completed) projective tensor product between the involved where ⊗ spaces. Clearly our spaces of symbols depend on the choice of κ, κ. ˜ If necessary, then we write μ

)κ,κ˜ S(cl) ( × Rq ; H, H μ ). instead of S(cl) ( × Rq ; H, H

(3.6)

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 be Hilbert spaces with group actions κ and κ, Lemma 3.2 Let H and H ˜ respectively, : H  →  and let T : H → L and T L be isomorphisms to Hilbert spaces L and  L, r espectively. Then the spaces L ,  L are endowed with group actions λδ , λ˜ δ given by λδ = T κδ T −1 and κ˜ δ T −1 , λ˜ δ = T  induce an isomorphism and T, T μ

μ

)κ,κ˜ → S ( × Rq ; L ,  Q : S(cl) ( × Rq ; H, H L)λ,λ˜ , (cl) where a(y, η) → b(y, η) and  ◦ a(y, η) ◦ T −1 , μ ∈ R. b(y, η) = T Proof First note that  ◦ a(y, η) ◦ T −1 ∈ C ∞ ( × Rq , L(L ,  b(y, η) = T L)).

(3.7)

Moreover, we have D αy Dηβ a(y, η)T −1 D αy Dηβ b(y, η) = T for arbitrary α ∈ N p , β ∈ Nq . Let us now check that b(y, η) ∈ S μ ( × Rq ; L ,  L)λ,λ˜ . To this end we have to verify the symbolic estimates to the effect that α β μ−|β| λ˜ −1 L) ≤ c η η {D y Dη b(y, η)}λη L(L ,

for all (y, η) ∈ K × Rq , K  , and α ∈ N p , β ∈ Nq , c = c(α, β, K ) > 0. From (3.2) and (3.7) we obtain −1 λ˜ −1 T  α β −1 b(y, η)T }T −1 λη T L(H, H) T η {D y Dη T −1 λ˜ −1 {D αy Dηβ b(y, η)}λη T L(H, H) = T η −1 = κ˜ η {D αy Dηβ a(y, η)}κη L(H, H) ≤ c η μ−|β| ,

using the fact that the operator )) Q : C ∞ ( × Rq , L(L ,  L)) → C ∞ ( × Rq , L(H, H

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defined by −1 b(y, η)T Q : b(y, η) → T has the property that D αy Dηβ (Qb)(y, η) = Q(D αy Dηβ b)(y, η), or, equivalently, −1 b(y, η)T }T −1 = D αy Dηβ b(y, η). {D αy Dηβ T T  ◦ a(μ− j) ◦ T −1 The assertion for classical symbols is a consequence of the fact that T induces isomorphisms )κ,κ˜ → S (μ− j) ( × (Rq \{0}); L ,  S (μ− j) ( × (Rq \{0}); H, H L)λ,λ˜ for every j; here subscripts κ, κ; ˜ λ, λ˜ have a similar meaning as in (3.6).



Let  be an open subset of Rq . Then for (y, y  ) ∈  × , and a(y, y  , η) ∈ ), we define Op y (a)u(y) by H, H

μ S(cl) ( ×  × Rq ;



 Op y (a)u(y) =

Rq

Rq



ei(y−y )η a(y, y  , η)u(y  )dy  d− η, μ

first for u ∈ C0∞ (, H ), and define L (cl) (; H, H˜ ) by μ

μ

) = {Op y (a) : a(y, y  , η) ∈ S ( ×  × Rq ; H, H )}, L (cl) (; H, H (cl) and define L −∞ (; H, H˜ ) by ) = L −∞ (; H, H



). L μ (; H, H

μ∈R

 = C and the trivial group action id both on Recall from [25,26] that for H = H μ  H and H , we recover the spaces S(cl) ( ×  × Rq ) of scalar symbols and pseudo

μ differential operators L (cl) (). In particular, L −∞ () = μ∈R L μ () coincides with the space of smoothing operators, i.e., operators of the form C0∞ ()

  u →

for some c(y, y  ) ∈ C ∞ ( × ).



c(y, y  )u(y  )dy 

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There are also variants with parameter λ ∈ Rl , namely, q+l μ ) and L μ (; H, H ; Rlλ ). S(cl) ( ×  × Rη,λ ; H, H (cl)

In this case we define L −∞ (; H. H˜ ; Rl ) by ; Rl ) = S(Rl , L −∞ (; H, H )). L −∞ (; H, H

(3.8)

) induces a continuous operator An A ∈ L μ (; H, H A : C0∞ (, H ) → C ∞ (, H ).

(3.9)

). Then Op(a) Proposition 3.3 [26, Proposition 1.3.24] Let a(η) ∈ S μ (Rη ; H, H extends to a continuous operator q

) Op(a) : W s (Rq , H ) → W s−μ (Rq , H for every s ∈ R. The proof in this case is straightforward. In fact, by virtue of Op(a) = F −1 a F for the Fourier transform F in y ∈ Rq we have  Op(a)uW s−μ (Rq , H) = 2

≤

−1 2 − η 2(s−μ) κ˜ η a(η)u(η) ˆ d η H

Rq



−1 −1 2 − η 2(s−μ) κ˜ η a(η)κη 2L(H, H) κη u(η) ˆ Hd η

Rq

≤ c2 u2W s (Rq ,H ) −1 where c = supη∈Rq η −μ κ˜ η a(η)κη L(H, H) , which is finite because of (3.2). In particular, it follows that a → Op(a) induces a continuous operator

) → L(W s (Rq , H ), W s−μ (Rq , H )) S μ (Rq ; H, H

(3.10)

for every s ∈ R. ) we have for all s ∈ R, continuous More generally, for a(y, η) ∈ S μ (×Rq ; H, H operators s ) Op(a) : Wcomp (, H ) → Wloc (, H s−μ

(3.11)

between comp/loc-versions of the abstract edge Sobolev spaces over an open set  ⊆ Rq . There are different ways of proving such a continuity. One relatively “unspecific” way is the tensor product argument. This refers to the fact that when E and F are ˆ π F can Fr´echet spaces, every element g in the completed projective tensor product E ⊗

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 ∞ be written as a convergent sum ∞ j=0 ν j e j ⊗ f j for suitable ν j ∈ C, j=0 |ν j | < ∞, and e j ∈ E, f j ∈ F, j ∈ N, tending to zero in the respective spaces, as j → ∞. In order to show the continuity of (3.11), we employ the fact that the operator Mϕ of multiplication by ϕ ∈ C0∞ () induces a continuous operator Mϕ : W s (Rq , H ) → W s (Rq , H ), where Mϕ L(W s (Rq ,H ),W s (Rq ,H )) → 0 as ϕ → 0 in C0∞ (),

(3.12)

see [26, Proposition 1.3.34]. Because of (3.5), we can represent a(y, η) as a convergent sum a(y, η) =

∞ 

ν j Mϕ j a j (η)

j=0 μ q ) tending to zero in the respective for sequences ϕ j ∈ C ∞ (), a  j ∈ S (R ; H, H ∞ spaces as j → ∞, and ν j ∈ C, j=0 |ν j | < ∞. Then (3.11) follows from (3.10) and (3.12), using the fact that

Op(a) =

∞ 

ν j Mϕ j Op(a j )

j=0 s−μ )) for every ψ ∈ C ∞ (). converges in L([ψ]W s (Rq , H ), Wloc (, H 0 Here if E is a Fr´echet space that is a module over an algebra, then for any ψ in that algebra we define

[ψ]E = completion of {ψe : e ∈ E} in E.

(3.13)

Similar arguments of proving the continuity of Op(a) between edge spaces can be found in [25, Subsection 3.2.1, Theorem 6], however, under an extra assumption that turns out to be superfluous. Other methods are applied in [26, Theorem 1.3.59] or in [32]. ) and assume that Remark 3.4 Let a0 ∈ L(H, H )). κ˜ δ a0 κδ−1 ∈ C ∞ (R+ , L(H, H Then a(η) defined by −1 a(η) = η μ κ˜ η a0 κη μ ). is in Scl (Rq ; H, H

(3.14)

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It also makes sense to consider g(η) given by −1 g(η) = [η]μ κ˜ [η] a0 κ[η]

for any fixed strictly positive function [·] in C ∞ (Rη ) such that [η] = |η| for |η| ≥ c for some c > 0. Clearly g(η) is C ∞ in η if and only if a(η) is C ∞ in η. For g, we have q

−1 −1 −1 μ μ μ −1 g(σ η) = [σ η]μ κ˜ [σ η] a0 κ[σ η] = σ [η] κ˜ σ κ˜ [η] a0 κ[η] κσ = σ κ˜ σ g(η)κσ μ

). See Example 3.1. for σ ≥ 1 and |η| ≥ c. We also have g(η) ∈ Scl (Rq ; H, H Remark 3.5 It can happen that the property (3.14) is violated. For instance, Let H =  = L 2 (Rn ) and let a0 = M f be the operator of multiplication by a function f such H that f ≡ 1 for |x| ≥ 1, f ≡ 21 for |x| < 1. Then for κδ defined by κδ u(x) = u(δx), δ ∈ R+ , we have κδ M f κδ−1 u = M fδ u for f δ (x) = f (δx). However, because of the discontinuity of f we cannot differentiate f δ with respect to δ. ) such that κδ cκ −1 does not belong Note that if we consider an operator c ∈ L(H, H δ ∞ )), we can generate smoothness by a mollifying process [17,25]. to C (R+ , L(H, H Indeed, let c(δ) = κδ cκδ−1 and let a() be defined by 

∞

a() = 0



−1

 −δ c(δ)dδ ϕ  

a function ϕ ∈ C0∞ (R) such that supp ϕ ⊂ [−ε, ε] for some 0 < ε < 1/2 and for ∞ ∞  −∞ ϕ(δ)dδ = 1. Then we have a() ∈ C (R+ , L(H, H )). Moreover, for h ∈ H , we have      ∞  ∞ −δ −δ −1 −1 −1 −1 κ˜ σ a()κσ h =  ϕ  ϕ κ˜ σ c(δ)κσ hdδ = c(σ δ)hdδ   0 0     ∞  ∞ ˜   − σδ σ  − δ˜ −1 ˜ ˜ ˜ = −1 ϕ dδ = σ −1 −1 ϕ c(δ)hσ c(δ)hd δ˜  σ 0 0 = a(σ ).

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In other words, we obtain the homogeneity a(σ ) = κσ a()κσ−1 , σ ∈ R+ . −1 )) and a(η) given by ∈ C ∞ (Rq , L(H, H We conclude that κ˜ η a1 κη −1 a(η) = η μ κ˜ η a1 κη μ

). is in Scl (Rq ; H, H Concrete cases in connection with Remark 3.4 are as follows. Let μ

b(x, x  , ξ ) ∈ S(Rnx × Rnx  , S(cl) (Rnξ )) and let a0 u = Op(b)u, where 

 Op(b)u =

Rn



Rn

ei(x−x )ξ b(x, x  , ξ )u(x  )d x  d− ξ.

Then a0 : H s (Rn ) → H s−μ (Rn ) is continuous for every s ∈ R. Let κδ be given by (κδ u)(x) = u(δx), δ ∈ R+ , and compute κδ a0 κδ−1 . By some obvious computations, we obtain (κδ Op(b)κδ−1 u)(x) = =







R R n

n

Rn

Rn

ei(δx−x )ξ b(δx, x  , ξ )u(δ −1 x  )d x  d− ξ ˜ ξ˜ ˜ xd ˜ − ξ˜ . ei(x−x) b(δx, δ x, ˜ δ −1 ξ˜ )u(x)d

Thus κδ Op(b)κδ−1 = Op(bδ )

(3.15)

for bδ (x, x  , ξ ) = b(δx, δx  , δ −1 ξ ). It follows in this case that Dδk (κδ Op(b)κδ−1 ) = Op(Dδk bδ ) for every k ∈ N. Therefore κδ Op(b)κδ−1 ∈ C ∞ (R+ , L(H s (Rn ), H s−μ (Rn ))). μ

q

q

Applying the construction to symbols b(x, y, ξ, η) ∈ S(Rnx , S(cl) (R y × Rnξ × Rη )), it follows that

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where bdec (x, y, ξ, η) = b(η −1 x, y, η ξ, η).

(3.16)

The symbol bdec (x, y, ξ, η) is often useful as a “decoupled” symbol. The decoupling b → bdec is known [29, Proposition 2.2.1] to generate a continuous operator μ

μ

q

q

μ

S(cl) (Rnx × R y × Rnξ × Rqη ) → S(cl) (R y × Rqη , S(cl) (Rnx × Rnξ )), Let us also recall that in boundary value problems (when n = 1) or in the edge pseudo-differential calculus (for arbitrary n), it is useful to interpret Op y (b0 )(y, η) for b0 (y, ξ, η) = b(0, y, ξ, η) as an element of μ

q

Op(b0 )(y, η) ∈ Scl (R y × Rqη ; H s (Rn ), H s−μ (Rn )), n

s ∈ R, where the symbol space on the right refers to κδ : u(x) → δ 2 u(δx), both on H s (Rn ) and H s−μ (Rn ). 4 Operators in iterated representation  be Hilbert spaces with group actions κ and κ, Theorem 4.1 Let H and H ˜ respectively. p+q μ )κ,κ˜ and p(η) defined by Then for a(ξ, η) ∈ S (Rξ,η ; H, H p(η) = Opx (a)(η), we have )κ˜ )χ ,χ˜ p(η) ∈ S μ (Rq ; W s (R p , H )κ , W s−μ (R p , H for every s ∈ R, where (χδ f )(x) = δ p/2 (κδ f )(δx) and (χ˜ δ f˜)(x) = δ p/2 (κ˜ δ f˜)(δx), δ ∈ R+ . Proof Indeed, −1 α κ˜ ξ,η {Dξ,η a(ξ, η)}κξ,η L(H, H) ≤ c ξ, η μ−|α|

(4.1)

for every α ∈ N p+q , (ξ, η) ∈ R p+q , for some c = c(α) > 0. We first note that η ξ, η 2 = 1 + η 2 |ξ |2 + |η|2 = η 2 + η 2 |ξ |2 = ξ 2 η 2 .

(4.2)

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469

In (4.1) we first assume α = 0 and obtain −1 κ˜ η ξ,η a(η ξ, η)κη ξ,η L(H, H) ≤ c η ξ, η μ = c ξ μ η μ .

(4.3)

−1 p(η)χη u(x), we have For p(η) = Opx (a)(η) and vη (x) = χ˜ η −1 vη (x) = χ˜ η −1 = κ˜ η

=

−1 κ˜ η

−1 = κ˜ η

 

 Rp



Rp



Rp





Rp



Rp



Rp

Rp

ei(x−x )ξ a(ξ, η)χη u(x  )d x  d− ξ ei(η

−1 x−x  )ξ



ei(x−x )η

−1 ξ

a(ξ, η)κη u(η x  )d x  d− ξ a(ξ, η)κη u(x  ) η − p d x  d− ξ



Rp

ei(x−x )ξ a(η ξ, η)κη u(x  )d x  d− ξ

−1 −1 = κ˜ η Fξ →x a(η ξ, η)κη u(ξ ˆ ).

Thus −1 −1 (χ˜ η p(η)χη u)∧ (ξ ) = κ˜ η a(η ξ, η)κη u(ξ ˆ ).

(4.4)

In view of (4.2), the estimate (4.3) gives −1 −1 κ˜ η κ˜ ξ a(η ξ, η)κξ κη L(H, H) ≤ c ξ μ η μ .

(4.5)

Let us now show that −1 χ˜ η p(η)χη L(W s (R p ,H ),W s˜ (R p , H)) ≤ c η μ .

(4.6)

p

For u(x) ∈ W s (Rx , H ) and s˜ = s − μ, we have  vη W s˜ (R p , H) = 2

=

Rp



Rp

−1 −1 ∧ 2 ξ 2˜s κ˜ ξ  dξ (χ˜ η p(η)χη u) (ξ ) H −1 −1 ξ 2˜s κ˜ ξ ˆ )2Hdξ. κ˜ η a(η ξ, η)κη u(ξ

Here the relation (4.4) is used. The right hand side of (4.7) is equal to 

−1 −1 −1 ξ 2(s−μ) κ˜ ξ ˆ )2Hdξ κ˜ η a(η ξ, η)κη κξ κξ u(ξ  −1 −1 −1 2 2s ξ −2μ κ˜ ξ ≤ ˆ )2H dξ ) ξ κξ u(ξ κ˜ η a(η ξ, η)κη κξ L(H, H Rp  −1 2 2μ η ξ 2s κξ ≤c ˆ )2H dξ, u(ξ

Rp

Rp

(4.7)

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where c=

sup

(ξ,η)∈R p+q

−1 −1 ξ −μ η −μ κ˜ ξ ) κ˜ η a(η ξ, η)κη κξ L(H, H

comes from the estimate (4.5). Summing up, we have proved the symbolic estimate (4.6) for every s ∈ R. Finally, using Dηβ p(η) = Opx (Dηβ a)(η) and p+q ) Dηβ a(ξ, η) ∈ S μ−|β| (Rξ,η ; H, H

from (4.6), we immediately obtain −1 β χ˜ η Dη p(η)χη L(W s (R p ,H ),W s−μ+|β| (R p , H)) ≤ c η μ−|β|

(4.8)

) on the left hand for all β ∈ Nq . This then implies a similar estimate for W s−μ (R p , H side of (4.8). So, the proof is complete. 

p+q μ  ; H, H )κ,κ˜ and p(η) = Opx (a)(η), then Proposition 4.2 If a(ξ, η) ∈ S (R cl

ξ,η

μ )κ˜ )χ ,χ˜ p(η) ∈ Scl (Rqη ; W s (R p , H )κ , W s−μ (R p , H

for all s ∈ R. Proof We first look at a(μ) (ξ, η), the twisted homogeneous principal component of a(ξ, η). Then for p(μ) (η) = Opx (a(μ) )(η) with η = 0, we have p(μ) (δη) = δ μ χ˜ δ p(μ) (η)χδ−1 for every δ ∈ R+ . In fact,    p(μ) (δη)u(x) = ei(x−x )ξ a(μ) (ξ, δη)u(x  )d x  d− ξ p p R R  = ei(x−x )ξ a(μ) (δ −1 δξ, δη)u(x  )d x  d− ξ p p R R  = ei(x−x )ξ δ μ κ˜ δ a(μ) (δ −1 ξ, η)κδ−1 u(x  )d x  d− ξ p p R R     μ =δ ei(x−x )δξ κ˜ δ a(μ) (ξ  , η)κδ−1 u(x  )δ p d x  d− ξ  p p R R   μ =δ ei(δx−x )ξ κ˜ δ a(μ) (ξ  , η)κδ−1 u(δ −1 x  )d x  d− ξ  Rp Rp    μ = δ χ˜ δ ei(x−x )ξ a(μ) (ξ, η)χδ−1 u(x  )d x  d− ξ. Rp

Rp

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For every N ∈ N we can write a(ξ, η) =

N 

χ (ξ, η)a(μ− j) (ξ, η) + r N +1 (ξ, η)

j=0

for any excision function χ (ξ, η) in R p+q and some ), r N +1 (ξ, η) ∈ S μ−(N +1) (Rξ,η ; H, H p+q

(see the formula (3.3)). Then the above computation for μ − j rather than μ, using that χ (δξ, δη)a(μ− j) (δξ, δη) = δ μ− j κ˜ δ χ (ξ, η)a(μ− j) (ξ, η)κδ−1 for δ ≥ 1 and |η| large enough, gives us for pμ− j (η) = Opx (χa(μ− j) )(η) the relation pμ− j (δη) = δ μ− j χ˜ δ pμ− j (η)χδ−1 for δ ≥ 1 and |η| large. (See also Example 3.1). Thus for every N ∈ N we obtain p(η) =

N 

pμ− j (η) + p N +1 (η)

j=0

for )κ˜ )χ ,χ˜ . p N +1 (η) = Opx (r N +1 )(η) ∈ S μ−(N +1) (Rq ; W s (R p , H )κ , W s−μ (R p , H (See Theorem 4.1). Thus p(η) is a classical symbol.



Proposition 4.3 The correspondence a(ξ, η) → p(η) induces a continuous operator μ

μ

)κ,κ˜ → S (Rqη ; W s (R p , H )κ , W s−μ (R p , H )κ˜ )χ ,χ˜ , S(cl) (Rξ,η ; H, H (cl) p+q

for all s ∈ R. Proof The continuity for general symbols is an immediate consequence of the proof of Theorem 4.1. The assertion in the classical case is straightforward for the terms that are homogeneous for large absolute values of covariables. The arguments for the remainder terms are as in the first part of the proof. 

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p q p+q )κ,κ˜ ) Remark 4.4 (i) More generally, let a(x, y, ξ, η) ∈ S(Rx , S μ (R y ×Rξ,η ; H, H and let p(y, η) = Opx (a)(y, η). Then we have

)κ˜ )χ ,χ˜ p(y, η) ∈ S μ (R y × Rqη ; W s (R p , H )κ , W s−μ (R p , H q

for every s ∈ R, and the map a → p induces a continuous operator p q p+q )κ,κ˜ ) → S μ (Rqy × Rqη ; W s (R p , H )κ , S(Rx , S μ (R y × Rξ,η ; H, H

)κ˜ )χ ,χ˜ . W s−μ (R p , H

q p+q μ )κ,κ˜ and p(y, η) = Opx (a)(y, η), then we (ii) If a(y, ξ, η) ∈ S(cl) (R y × Rξ,η ; H, H have μ

)κ˜ )χ ,χ˜ , p(y, η) ∈ S(cl) (Rq × Rq ; W s (R p , H )κ , W s−μ (R p , H and a → p defines a continuous operator between the respective symbol spaces. Let H1 be a Hilbert space with group action κ1 . Then H2 = W s (Rq1 , H1 ) for q1 ∈ N, admits the group action κ2 in accordance with (κ2,δ u 2 )(y 1 ) = δ q1 /2 (κ1,δ u 2 )(δy 1 ),

y 1 ∈ Rq1 , δ ∈ R+ .

(See Theorem 2.2). If we let Hk = W s (Rqk−1 , Hk−1 )

(4.9)

for k ≥ 2, qk−1 ∈ N, then we have the group action κk on Hk given in (4.9) defined by (κk,δ u k )(y k−1 ) = δ qk−1 /2 (κk−1,δ u k )(δy k−1 ),

y k−1 ∈ Rqk−1 .

As a corollary of Theorem 4.1, we obtain the following iteration. p μ 0 )χ ,χ˜ , Corollary 4.5 Let a(ξ, η1 , η2 , . . . , ηk ) ∈ S(cl) (Rξ × Rq1 × . . . × Rqk ; H0 , H 0 0 0 and χ0 , χ˜ 0 are as in Theorem 4.1. For s ∈ R, let where H0 , H

Iterative properties of pseudo-differential operators

473

p 1 = W s−μ (Rxp , H 0 )χ˜ , H1 = W s (Rx , H0 )χ0 , H 0

s ∈ R, with group actions χ1,δ f 1 (x) = δ p/2 (χ0,δ f 1 )(δx), χ˜ 1,δ f˜1 (x) = δ p/2 (χ˜ 0,δ f˜1 )(δx), δ ∈ R+ . Then by Theorem 4.1, we have μ

a1 (η1 , η2 , . . . , ηk ) = Opx (a)(η1 , η2 , . . . , ηk ) ∈ S(cl) 1 )χ ,χ˜ . ×(Rq1 × Rq2 × . . . × Rqk ; H1 , H 1 1 For 2 ≤ l ≤ k − 1 and l = W s−μ (Rql−1 , H l−1 )χ˜ Hl = W s (Rql−1 , Hl−1 )χl−1 , H l−1 endowed with group actions χl , χ˜l defined by χl,δ fl (y l−1 ) = δ ql−1 /2 (χl−1,δ fl )(δy l−1 ), χ˜l,δ f˜l (y l−1 ) = δ ql−1 /2 (χ˜l−1,δ f˜l )(δy l−1 ), we have μ l )χ ,χ˜ . al (ηl , . . . , ηk ) = Op yl−1 (al−1 )(ηl , . . . , ηk ) ∈ S(cl) (Rql × . . . × Rqk ; Hl , H l l

The correspondence a(ξ, η1 , . . . , ηk ) → ak (ηk ) induces a continuous operator μ

μ

)κ,κ˜ → S (Rqk ; Hk , H k )χ ,χ˜ S(cl) (Rξ × Rq1 × . . . × Rqk ; H, H k k (cl) p

for every k ∈ N, s ∈ R. The following proposition is related with the shape of decoupled symbols (3.16). p

Proposition 4.6 Let ϕ ∈ S(Rx ) and Mϕ the operator of multiplication by ϕ. Then for every s ∈ R, we have Mϕ ∈ S 0 (Rq ; W s (R p , H )κ , W s (R p , H )κ )χ ,χ , and ϕ → Mϕ gives rise to a continuous injective operator S(R p ) → S 0 (Rq ; W s (R p , H )κ , W s (R p , H )κ )χ ,χ .

(4.10)

Proof We have to verify the symbolic estimates −1 χη Mϕ χη L(W s (R p ,H )) ≤ c

(4.11)

p

for a c = cϕ > 0 and cϕ → 0 as ϕ → 0 in S(Rx ). We have −1 Mϕ χη = Mϕη χη

for ϕη (x) = ϕ(η −1 x). We now employ the fact that for the scalar symbol ξ s , the operator Op(ξ s ) for Op = Opx induces an isomorphism

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Op(ξ s ) : W s (R p , H ) → W 0 (R p , H ) with the inverse Op(ξ −s ). We have Op(Mϕη ) = Op(ξ −s )Op(ξ s )Op(Mϕη )Op(ξ −s )Op(ξ s ) = Op(ξ −s )Op( f η )Op(ξ s ),

(4.12)

where f η (x, ξ ) = ξ s #(Mϕη ξ −s ) and # is the Leibniz product between symbols in (x, ξ ). Now, we show that Op( f η )L(W 0 (R p ,H )) ≤ c p

for some c = cϕ > 0 independent of η ∈ Rq and cϕ → 0 as ϕ → 0 in S(Rx ). Writing for the moment a(ξ ) = ξ s , b(x, ξ ) = ϕη (x)ξ −s , we have f η (x, ξ ) = (a#b)(x, ξ ) after Kumano-go’s formalism [16] as an oscillatory integral, namely, f η (x, ξ ) = ξ −s



 Rp

Rp

e−i zζ ξ + ζ s ϕ(η −1 (x + z))dzd− ζ.

Applying a regularization argument, we obtain ξ s f η (x, ξ )   = e−i zζ ζ −2M (1−ζ ) N ξ + ζ s z −2N (1−z ) M ϕ(η −1 (x + z))dzd− ζ Rp

Rp

for any M, N ∈ N. We choose M, N in such a way that |s| − 2M < − p, −2N < − p. Then it follows that  −s | f η (x, ξ )| ≤ ξ ζ −2M (1 − ζ ) N ξ + ζ s d− ζ Rp  |z −2N (1 − z ) M ϕ(η −1 (x + z))|dz. (4.13) Rp

Let us first consider the expression −s

F(ξ ) = ξ

 Rp

ζ −2M (1 − ζ ) N ξ + ζ s d− ζ.

We have (1 − ζ ) N ξ + ζ s =

2N  ξ + ζ s− j B j (ξ, ζ ) j=0

Iterative properties of pseudo-differential operators

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for functions B j (ξ, ζ ) satisfying the estimates sup |B j (ξ, ζ )| ≤ constant

ξ,ζ ∈R p

for different positive constants. This implies that |F(ξ )| ≤ cξ −s

2N  

ζ −2M ξ + ζ s− j d− ζ.

p j=0 R

Using Peetre’s inequality to the effect that ξ + ζ t ≤ Cξ t ζ |t| , we get −s



ζ −2M ξ + ζ s− j d− ζ ≤ Cζ −2M+|s− j| d− ζ ≤ C 

ξ

for some C  > 0. Thus, |F(ξ )| ≤ c for some c > 0. Moreover, z −2N (1 − z ) M ϕ(η −1 (x + z)) is a finite linear combination of terms of the form z −2N Dzα ϕ(η −1 (x + z)) = η −|α| z −2N (Dzα ϕ)(η −1 (x + z)), |α| ≤ 2M. Thus, the second factor on the right hand side of (4.13) is a finite linear combination of integrals −|α|



η

z −2N |(Dzα ϕ)(η −1 (x + z))|dz, |α| ≤ 2M.

(4.14)

For ψα (η −1 (x + z)) = (Dzα ϕ)(η −1 (x + z)) we obtain by substituting v = c−1 (x + z) for c = η , 

−2N

Rp

z

−1

|ψα (η



cv − x −2N |ψα (v)|c p dv  ≤ sup |ψα (v)| cv − x −2N c p dv.

(x + z))|dz =

Rp

Rp

v∈R p

Let w = cv. Then we obtain 

−2N p

Rp

cv − x



c dv =

Rp

−2N

w − x

 dw =

Rp

w −2N dw = constant.

It follows that | f η (x, ξ )| ≤ c



sup |Dxα ϕ(x)|.

p |α|≤2M x∈R

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In a similar manner, we see that for any fixed α, β ∈ N p there is a semi-norm ν(·) on the space S(R p ) such that β

sup{|Dxα Dξ f η (x, ξ )| : (x, ξ ) ∈ R2 p , α ≤ α, β ≤ β} ≤ cν(ϕ) for a positive constant c, which is independent of η ∈ Rq . Let us now recall a version of Calder´on-Vaillancourt’s theorem, proved by Seiler [32]. (See also Hwang’s paper  be Hilbert spaces with group actions κ and κ, [13] for the scalar case). Let H and H ˜ ∞ )) be a function such that respectively, and let f (x, ξ ) ∈ C (R2 p , L(H, H β

−1 α 2p π( f ) = sup{κ˜ ξ ) : (x, ξ ) ∈ R , α ≤ α, β ≤ β} {D x Dξ f η (x, ξ )}κη L(H, H

is finite for α = (gκ˜ + 1, . . . , gκ˜ + 1), β = (1, . . . , 1), where gκ˜ is the constant in (2.1) corresponding to κ. ˜ Then ) Op( f ) : W 0 (R p , H ) → W 0 (R p , H is continuous, and we have Op( f )L(W 0 (R p ,H ),W 0 (R p , H)) ≤ cπ( f ) for a positive constant c > 0 independent of f . We apply this theorem to the case , κ = κ, H =H ˜ and to f (x, ξ ) = f η (x, ξ ) · id H . Then for α = β = (1, . . . , 1), we have β

π( f η ) = sup{|Dxα Dξ f η (x, ξ )| : (x, ξ ) ∈ R2 p , α ≤ α, β ≤ β} ≤ cν(ϕ). It follows that Op( f η )L(W 0 (R p ,H )) ≤ cν(ϕ). Applying (4.12), we can return to (4.11) for an arbitrary s ∈ R and obtain −1 Mϕ χη L(W s (R p ,H )) ≤ cν(ϕ). sup χη

η∈Rq

This is the only semi-norm in S 0 (Rq ; W s (R p , H )κ , W s (R p , H )κ )χ ,χ that we have to control. Summing up, we have proved that Mϕ is a symbol as claimed and the continuity of (4.10). 

The following theorem extends Theorem 4.1 to symbols with variable coefficients.  and κ, κ˜ be as in Theorem 4.1. Then for Theorem 4.7 Let H, H p q p+q )κ,κ˜ ) a(x, y, ξ, η) ∈ S(Rx , S μ (R y × Rξ,η ; H, H

Iterative properties of pseudo-differential operators

477

and p(y, η) = Opx (a)(y, η), we have q )κ˜ )χ ,χ˜ p(y, η) ∈ S μ (R y × Rqη ; W s (R p , H )κ , W s−μ (R p , H

(4.15)

for every s ∈ R. Proof Let us first observe that the extension of Theorem 4.1 to symbols q p+q )κ,κ˜ a(y, ξ, η) ∈ S μ (R y × Rξ,η ; H, H

is straightforward because the additional variable y as an action from the left does not influence the proof, and we obtain (4.15) in this case. Therefore, without loss of generality, we omit y and only consider the case ˆπE a(x, ξ, η) ∈ S(Rx )⊗ p

)κ,κ˜ . On the right hand side, we use the fact that for E = S μ (Rξ,η ; H, H p+q

ˆπE S(R p , E) = S(R p )⊗ for the Fr´echet space E. As in the proof of Proposition 3.3 we employ a tensor product argument. We write a(x, ξ, η) as a convergent sum a(x, ξ, η) =

∞ 

λ j Mϕ j a j (ξ, η)

j=0

for sequences ϕ j ∈ S(R p ), a j ∈ E, tending to zero in the respective spaces as j → ∞, and λ j ∈ C, ∞ j=0 |λ j | < ∞. From Proposition 4.3 we conclude that Opx (a j )(η) → 0 )κ˜ )χ ,χ˜ and from Proposition 4.6 that in S μ (Rq ; W s (R p , H )κ , W s−μ (R p , H Opx (Mϕ j ) → 0 )κ˜ , W s−μ (R p , H )κ˜ )χ, in S 0 (Rq ; W s−μ (R p , H ˜ χ˜ as j → ∞. The operator Opx (Mϕ j ) representing a symbol that is independent of η can be still identified with the operator of multiplication Mϕ j . Therefore, it is justified to interpret Opx (a)(η) as a convergent sum Opx (a)(η) =

∞  j=0

λ j Opx (Mϕ j )Opx (a j )(η)

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p )κ˜ )χ ,χ˜ continuously embedded in ˆ π S μ (Rq ; W s (R p , H )κ , W s−μ (R p , H in S(Rx )⊗ μ q s p s−μ p  (R , H )κ˜ )χ ,χ˜ . Thus Opx (a)(η) is an element in the latter S (R ; W (R , H )κ , W symbol space. 

Remark 4.8 As in Corollary 4.5 we can iterate Theorem 4.7 for symbols p 0 )χ ,χ˜ ) a(x, y 1 , . . . , y k , ξ, η1 , η2 , . . . , ηk ) ∈ S(Rx × Rq1 +...+qk−1 , S μ (Rqk ; H0 , H 0 0

(4.16) 0 and χ0 , χ˜ 0 as in Theorem 4.1. Then we have the same iterative process to for H0 , H successively build up symbols al (y l , . . . , y k , ηl , . . . , ηk ) = Op yl−1 (al−1 )(y l , . . . , y k , ηl , . . . , ηk ) l )χ ,χ˜ ) ∈ S(Rql +...+qk−1 , S μ (Rqk × Rql +...+qk ; Hl , H l l for every 1 ≤ l ≤ k, starting with a0 = a defined by (4.16), then a1 = Opx (a0 ), and so on. References 1. Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971) 2. Chang, D.-C., Habal, N., Schulze, B.-W.: Quantisation on a manifold with singular edge. J. PseudoDiffer. Oper. Appl. 4(3), 317–343 (2013) 3. Dines, N.: Ellipticity of a class of corner operators. In: Pseudo-differential Operators: PDE and TimeFrequency Analysis. Fields Institute Communication, vol. 52, pp. 131–169. American Mathematical Society, Providence, RI (2007) 4. Dorschfeldt, Ch.: Algebras of pseudo-differential operators near edge and corner singularities. In: Math. Res., vol. 102. Wiley-VCH, Berlin, Weinheim (1998) 5. Dreher, M., Witt, I.: Edge Sobolev spaces and weakly hyperbolic operators. Ann. Mat. Pura Appl. 180, 451–482 (2002) 6. Egorov, Ju.V., Schulze, B.-W.: Pseudo-differential operators, singularities, applications. In: Operator Theory: Advances and Applications, vol. 93. Birkh¨auser Verlag, Basel (1997) 7. Eskin, G.I.: Boundary value problems for elliptic pseudo-differential equations. Transl. Nauka, Moskva, 1973. In: Mathematical Monographs, vol. 24. American Mathematical Society (1980) 8. Flad, H.-J., Harutyunyan, G.: Ellipticity of quantum mechanical Hamiltonians in the edge algebra. In: Proceedings of the AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden (2010) 9. Gil, J.B., Schulze, B.-W., Seiler, J.: Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37, 219–258 (2000) 10. Habal, N., Schulze, B.-W.: Mellin quantisation in corner operators. In: Karlovich, Y.I. et al. (eds) Operator Theory: Advances and Applications, vol. 228. Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. The Vladimir Rabinovich Anniversary Volume, Birkhäuser, Basel, 2013, pp. 151–172 11. Harutyunyan, G., Schulze, B.-W.: Elliptic mixed. In: Transmission and Singular Crack Problems. European Mathematical Society, Zürich (2008) 12. Hirschmann, T.: Functional analysis in cone and edge Sobolev spaces. Ann. Global Anal. Geom. 8, 167–192 (1990) 13. Hwang, I.L.: The L 2 -boundedness of pseudodifferential operators. Trans. Amer. Math. Soc. 302, 55–76 (1987) 14. Kondratyev, V.A.: Boundary value problems for elliptic equations in domains with conical points. Trudy Mosk. Mat. Obshch. 16, 209–292 (1967)

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