Green symbols Let a ∈ D

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

Pseudo-differential boundary value problems are described by a symbol structure that reflects an analogue of Green’s function and generates boundary and potential conditions of elliptic boundary value problems. This is summarized by the following definition. D EFINITION 4. The space µ, 0;δ G n−1 × n−1 ; N − , N + of Green symbols of order µ ∈ , type 0 and weight δ ∈ is defined to be space of all operator-valued symbols gy, η ∈ S µ;δ cl η n−1 × n−1 ; L 2 + ⊕ N − , ✁ + ⊕ N + such that g ∗ y, η ∈ S µ;δ cl η n−1 × n−1 ; L 2 + ⊕ N + , ✁ + ⊕ N − . Moreover, the space µ, d;δ G n−1 × n−1 ; N − , N + of Green symbols of order µ ∈ , type d ∈ ✁ and weight δ ∈ is defined to be the space of all operator families of the form gy, η = g y, η + d X j =1 g j y, η ∂ j t 29 for arbitrary g j ∈ µ− j,0;δ G n−1 × n−1 ; N − , N + , j = 0, . . . , d. P ROPOSITION 3. Every gy, η ∈ µ, d;δ G n−1 × n−1 ; N − , N + belongs to S µ;δ cl η n−1 × n−1 ; H s + ⊕ N − , ✁ + ⊕ N + for every real s d − 1 2 . Boundary value problems 319 The specific aspect in our symbol calculus near exits to infinity consists of classical el- ements, here with respect to y ∈ n−1 . Let µ, d;δ G,cl n−1 × n−1 ; N − , N + denote the subspace of all gy, η ∈ µ, d;δ G n−1 × n−1 ; N − , N + of the form 29 for g j y, η ∈ µ− j,0;δ G,cl n−1 × n−1 ; N − , N + , where µ, 0;δ G,cl n−1 × n−1 ; N − , N + is defined to be the space of all gy, η ∈ S µ;δ cl η; y n−1 × n−1 ; L 2 + ⊕ N − , ✁ + ⊕ N + with g ∗ y, η ∈ S µ;δ cl η; y n−1 × n−1 ; L 2 + ⊕ N + , ✁ + ⊕ N − . Similarly to Proposition 3 we have µ, d;δ G,cl n−1 × n−1 ; N − , N + ⊂ S µ;δ cl η; y n−1 × n−1 ; H s + ⊕ N − , ✁ + ⊕ N + 30 for all s d − 1 2 . Applying 18 we then get the triple of principal symbols σ g = σ ∂ g, σ e ′ g, σ ∂, e ′ g . 31 R EMARK 12. There is a direct analogue of Remark 5 in the framework of Green symbols that we do not repeat in this version in detail. Let us only observe that we can recover gy, η from 31 mod µ− 1,d;δ−1 G,cl n−1 × n−1 ; N − , N + by χ ησ ∂ gy, η+χ y{σ e ′ gy, η− χ ησ ∂, e ′ gy, η} ∈ µ, d;δ G,cl n−1 × n−1 ; N − , N + . L EMMA 1. Let φ y, t ∈ C ∞ q × + and assume that for some δ ∈ the following estimates hold: sup t ∈ + |D α y φ y, t | ≤ chyi δ−|α| for all y ∈ q and all α ∈ ✁ q , with constants c = ca 0. Then the operator M φ y,t of multiplication by φ y, t fulfils the relation M φ y,t ∈ S 0;δ q × q ; L 2 + , L 2 + . 32 Proof. We have to check the symbol estimates κ −1 η n D α y D β η M φ y,t o κη ✂ L 2 + ≤ chyi δ−|α| for all α, β ∈ ✁ q and all y, η ∈ q × q with suitable c = cα, β 0. Because M φ y,t is in- dependent of η it suffices to consider β = 0. Using κ −1 η D α y M φ y,t κη = D α y M φ y,t hyi − 1 we get for u ∈ L 2 + κ −1 η n D α y D β η M φ y,t o κη ut L 2 + = D α y φ y, t hyi −1 ut L 2 + ≤ sup t ∈ + D α y φ y, t hyi −1 kuk L 2 + ≤ chyi δ−|α| kuk L 2 + . 320 D. Kapanadze – B.-W. Schulze L EMMA 2. Let φ y, t ∈ C ∞ q × + be a function such that there are constants m, δ ∈ such that sup t ∈ + D α y D M t φ y, t ≤ chyi δ−|α| ht i m−M for all y ∈ q , t ∈ + and all α ∈ ✁ q , M ∈ ✁ , with constants c = cα, M 0. Then we have M φ y,t ∈ S 0,δ q × q ; ✁ + , ✁ + . 33 Proof. Let us express the Schwartz space as a projective limit ✁ + = proj lim ht i −k H k + : k ∈ ✁ . An operator b is continuous in ✁ + if for every k ∈ ✁ there is an l = lk ∈ ✁ such that kbk ✂ ht i − l H l + ,h t i − k H k + ≤ c for certain c = ck, l 0. The symbol estimates for 33 require for every k ∈ ✁ an l = lk ∈ ✁ such that κ −1 η n D α y M φ y,t o κη u ht i − k H k + ≤ chyi δ−|α| kuk ht i − l H l + , 34 for constants c 0 depending on k, l, α, for all α and k. Similarly to the proof of Lemma 1 the η -derivatives may be ignored. Estimate 34 is equivalent to hti k D α y φ y, t hηi −1 ht i −l v H k + ≤ chyi δ−|α| kvk H l + 35 for all v ∈ H l + . Setting l = k + m + for m + = maxm, 0 we get 35 from the system of simpler estimates D j t n ht i −m + D α y φ y, t hηi −1 v t o L 2 + ≤ chyi δ−|α| kvk H l + for all 0 ≤ j ≤ k. The function D j t ht i −m + D α y φ y, t hηi −1 v t is a sum of expressions of the form v j 1 j 2 j 3 t = cht i −m + − j 1 hηi − j 2 D j 2 t D α y φ y, t hηi −1 D j 3 t v t for j 1 + j 2 + j 3 = j and constants c = c j 1 , j 2 , j 3 . We now employ the assumption on φ , namely sup t ∈ + D α y D j 2 t φ y, t hηi −1 ≤ hyi δ−|α| ht hηi −1 i m− j 2 . Using ht hηi −1 i m− j 2 ≤ ht i m− j 2 for m − j 2 ≥ 0 and ht hηi −1 i m− j 2 ≤ 1 for m − j 2 0 we immediately get kv j 1 j 2 j 3 t k L 2 + ≤ chηi δ−|α| D j 3 t v L 2 + for all y ∈ q , with different constants c 0. This gives us finally the estimates 35.

3.4. The algebra of boundary value problems