Boundary symbols associated with interior symbols

316 D. Kapanadze – B.-W. Schulze T HEOREM 2. a ∈ D µ, d + ; N , N + and b ∈ D ν, e + ; N − , N implies ab ∈ D µ+ν, h + ; N − , N + for h = maxν + d, e. T HEOREM

3. Let a ∈ D

µ, d + ; N − , N + where aτ 6= 0 for all τ ∈ , and assume that a defines an invertible operator 26 for some s ∈ , s maxµ, d − 1 2 . Then 26 is invertible for all s ∈ , s maxµ, d − 1 2 . In addition a : ✁ + ⊕ N − −→ ✁ + ⊕ N + is invertible, and we have a −1 ∈ D −µ,d−µ + + ; N + , N − ; here ν + = maxν, 0.

3.2. Boundary symbols associated with interior symbols

In this section we introduce a special symbol class on n that gives rise to operator-valued symbols in the sense of Section 2.4. Let S µ;δ cl ξ ; x n × n ≍ defined to be the subspace of all ax, ξ ∈ S µ cl ξ ; x n × n vanishing on the set T R := x = y, t ∈ n : |x| ≥ R, |t | R|y| 28 for some constant R = Ra. In an analogous manner we define the more general space S µ;δ n × n ≍ . Set S µ;δ cl ξ ; x n × n tr,≍ = S µ;δ cl ξ ; x n × n ≍ ∩ S µ;δ cl n × n tr and S µ;δ cl ξ ; x n + × n tr,≍ = n a = ˜a| n + × n : ˜ax, ξ ∈ S µ;δ cl ξ ; x n × n tr,≍ o , n + = n−1 × + . Similarly, we can define the spaces S µ;δ cl ξ n × n ≍ , S µ;δ cl ξ n × n tr,≍ , S µ;δ cl ξ n + × n tr,≍ , where cl ξ means symbols that are only classical in ξ . For a ∈ S µ;δ cl ξ n × n ≍ we form opay, ηut = RR e it −t ′ τ ay, t, η, τ ut ′ dt ′ d¯τ and set op + ay, η = r + opay, ηe + , where r + and e + are of analogous meaning on as the corresponding operators r + and e + in Section 2.2. We also form op + ay, η for symbols ay, t, η, τ ∈ S µ;δ cl ξ n + × n tr,≍ ; the extension e + includes an extension of a to a corresponding ˜ a, though op + ay, η does not depend on the choice of ˜ a. P ROPOSITION 1. ax, ξ ∈ S µ;δ n × n ≍ implies op ay, η ∈ S µ;δ n−1 × n−1 ; H s , H s−µ for every s ∈ . The simple proof is left to the reader. P ROPOSITION 2. ax, ξ ∈ S µ;δ cl ξ n + × n tr,≍ implies op + ay, η ∈ S µ;δ n−1 × n−1 ; H s + , H s−µ + Boundary value problems 317 for every s − 1 2 and op + ay, η ∈ S µ;δ n−1 × n−1 ; ✁ + , ✁ + . The proof of this result can be given similarly to Theorem 2.2.11 in [20]. Given a symbol ax, ξ ∈ S µ;δ cl ξ n + × n tr we call the operator family op + a| t =0 y, η : H s + −→ H s−µ + , s − 1 2 or op + a| t =0 y, η : ✁ + −→ ✁ + the boundary symbol associated with ax, ξ . R EMARK 10. For ax, ξ ∈ S µ;δ cl ξ ; x n + × n tr we have op + a| t =0 y, η ∈ S µ;δ cl η; y n−1 × n−1 ; H s + , H s−µ + , s − 1 2 , and op + a| t =0 y, η ∈ S µ;δ cl η; y n−1 × n−1 ; ✁ + , ✁ + . For ax, ξ ∈ S µ;δ cl ξ ; x n + × n we form σ a = σ ψ

a, σ

e

a, σ

ψ, e a; σ ∂

a, σ

e ′

a, σ

∂, e ′ a , for σ ψ a = σ ψ ˜ a| n + × n \0 , σ e a = σ e ˜ a| n + \0 × n , σ ψ, e a = σ ψ, e ˜ a| n + \0 × n \0 with an ˜ a ∈ S µ;δ cl ξ n × n tr where a = ˜ a| n + × n , and σ ∂ ay, η := σ ∂ op + a| t =0 y, η , y, η ∈ n−1 × n−1 \ 0 , σ e ′ ay, η := σ e ′ op + a| t =0 y, η , y, η ∈ n−1 \ 0 × n−1 , σ ∂, e ′ ay, η := σ ∂, e ′ op + a| t =0 y, η , y, η ∈ n−1 \ 0 × n−1 \ 0 , where the right hand sides are understood in the sense of 18. Here, e ′ is used for the exit symbol components along y ∈ n−1 , while e indicates exit symbol components of interior symbols with respect to x ∈ n . It is useful to decompose symbols in S µ;δ cl ξ ; x n + × n into a ≍-part and an interior part by a suitable partition of unity. D EFINITION 3. A function χ ≍ ∈ C ∞ n + is called a global admissible cut-off function in n + if i 0 ≤ χ ≍ x ≤ 1 for all x ∈ n + , ii there is an R 0 such that χ ≍ λ x = χ ≍ x for all λ ≥ 1, |x| R, iii g χ ≍ x = 1 for 0 ≤ t ε for some ε 0, χ ≍ x = 0 for |x| ≥ R, t e R|y| and χ ≍ x = 0 for |x| ≤ R, t ˜ε for some ˜ε ε and non-negative reals R and e R. A function χ ≍ ∈ C ∞ n + is called a local admissible cut-off function in n + if it has the properties i , ii and 318 D. Kapanadze – B.-W. Schulze iii l χ ≍ x = νx1 − ωx for ω = ˜ ω| n + for some ˜ ω ∈ C ∞ n , 0 ≤ ˜ ω x ≤ 1 for all x ∈ n and ˜ ω x = 1 in a neighbourhood of x = 0 and ν = ̹| n + for some ̹ ∈ C ∞ n \ 0 with ̹λx = ̹x for all λ ∈ + , x ∈ n \ 0, such that for some y ∈ n−1 with |y| = 1, and certain 0 ε ˜ε 1 2 we have ̹x = 1 for all x ∈ S n−1 ∩ n + with |x − y| ε and ̹x = 0 for all x ∈ S n−1 ∩ n + with |x − y| ˜ε. For ax, ξ ∈ S µ;δ cl ξ ; x n + × n and any local or global admissible cut-off function χ ≍ we then get a decomposition ax, ξ = χ ≍ xax, ξ + 1 − χ ≍ xax, ξ where a ≍ x, ξ := χ ≍ xax, ξ ∈ S µ;δ cl ξ ; x n + × n ≍ and 1 − χ ≍ xax, ξ ∈ S µ;δ cl ξ ; x n + × n . R EMARK 11. The operator of multiplication M χ ≍ by any χ ≍ ∈ C ∞ n with χ ≍ λ x = χ ≍ x for all |x| R for some R 0 and λ ≥ 1, can be regarded as an element in L 0;0 cl n . In other words, we have M χ ≍ A, A M χ ≍ ∈ L µ;δ cl n for every A ∈ L µ;δ cl n . If χ ≍ and ˜ χ ≍ are two such functions with supp χ ≍ ∩ supp ˜ χ ≍ = ∅ we have χ ≍ A ˜ χ ≍ ∈ L −∞;−∞ n for arbitrary A ∈ L µ;δ cl n . A similar observation is true in the operator-valued case.

3.3. Green symbols