The algebra of boundary value problems σ σ σ σ e where σ ab = σ aσ b with componentwise multiplication.

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

D EFINITION 5. µ, d;δ cl n−1 × n−1 ; N − , N + for µ, d ∈ × ✁ , δ ∈ , is defined to be the set of all operator families ay, η = op + ay, η + gy, η for arbitrary ax, ξ ∈ S µ;δ cl ξ ; x n + × n tr,≍ and gy, η ∈ µ, d;δ G,cl n−1 × n−1 ; N − , N + . Boundary value problems 321 Observe that the components of σ a := σ ψ

a, σ

e

a, σ

ψ, e a; σ ∂

a, σ

e ′

a, σ

∂, e ′ a 36 for ay, η ∈ µ, d;δ cl n−1 × n−1 ; N − , N + , given by σ ψ a := σ ψ a , σ e a := σ e a , σ ψ, e a := σ ψ, e a , and σ ∂ a = σ ∂ op + a| t =0 + σ ∂ g , σ e ′ a = σ e ′ op + a| t =0 + σ e ′ g , σ ∂, e ′ a = σ ∂, e ′ op + a| t =0 + σ ∂, e ′ g are uniquely determined by ay, η, and that σ a = 0 implies ay, η ∈ µ− 1,d;δ−1 cl n−1 × n−1 ; N − , N + . Moreover, ay, η ∈ µ, d;δ cl n−1 × n−1 ; N , N + and by, η ∈ ν, e;̺ cl n−1 × n−1 ; N − , N implies aby, η ∈ µ+ν, h;δ+̺ cl n−1 × n−1 ; N − , N + for h = maxν +

d, e where σ ab = σ aσ b with componentwise multiplication.

Next we define spaces of smoothing operators in the half-space. The space −∞,0;−∞ n + ; N − , N + is defined to be the set of all block matrix operators ✁ = A K T C : ✁ + ⊕ ✁ n−1 , N − −→ ✁ + ⊕ ✁ n−1 , N + , where i Auy, t = RR n + ay, t, y ′ , t ′ uy ′ , t ′ dy ′ dt ′ for certain ay, t, y ′ , t ′ ∈ ✁ n + × n + = ✁ n × n | n + × n + , u ∈ ✁ n + , ii K vy, t = P N − l=1 K l v l y, t for K l v l y, t = R n−1 k l y, t, y ′ v l y ′ dy ′ for certain k l y, t, y ′ ∈ ✁ n + × n−1 = ✁ n × n−1 n + × n−1 , for v = v l l=1,...,N − ∈ ✁ n−1 , N − , iii T uy = T m uy m=1,...,N + for T m uy = RR n + b m y, y ′ , t ′ uy ′ , t ′ dy ′ dt ′ for cer- tain b m y, y ′ , t ′ ∈ ✁ n−1 × n + = ✁ n−1 × n n−1 × n + , m = 1, . . . , N + for u ∈ ✁ n + , iv Cvy = P N − l=1 R c lm y, y ′ v l y ′ dy ′ m=1,...,N + for certain c lm y, y ′ ∈ ✁ n−1 × n−1 , l = 1, . . . , N − , m = 1, . . . , N + . −∞,d;−∞ n + ; N − , N + for d ∈ ✁ is the space of all operators ✂ = ✂ + d X j =1 ✂ j ∂ j t 322 D. Kapanadze – B.-W. Schulze for arbitrary ✂ j ∈ −∞,0;−∞ n + ; N − , N + , j = 0, . . . , d. Let L µ;δ cl n ⌣ denote the subspace of all P ∈ L µ;δ cl n such that there is an R 0 with φ Pψ = 0 for all φ, ψ ∈ C ∞ n with supp φ, supp ψ ⊆ n \ T R , cf. 28. Moreover, we set L µ;δ cl n + ⌣ = P = e P| n + : e P ∈ L µ;δ cl n ⌣ . For P = e P| n + , e P ∈ L µ;δ cl n ⌣ we define σ P = σ ψ e P n + × n \0 , σ e e P n + \0 × n , σ ψ, e e P n + \0 × n \0 37 D EFINITION 6. The space µ, d;δ cl n + ; N − , N + for µ, d ∈ × ✁ , δ ∈ , is defined to be the set of all operators ✁ = Op a + + ✂ 38 for arbitrary ay, η ∈ µ, d;δ cl n−1 × n−1 ; N − , N + , ∈ P 0 with P ∈ L µ;δ cl n ⌣ and ✂ ∈ −∞,d;−∞ n + ; N − , N + . Moreover, we set B µ, d;δ cl n + = u.l.c µ, d;δ cl n + ; N − , N + . 39 Similarly, we get the subspaces of so-called Green operators of order µ, type d, and weight δ µ, d;δ G,cl n + ; N − , N + and B µ, d;δ G,cl n + when we require amplitude functions to belong to µ, d;δ G,cl n−1 × n−1 ; N − , N + and R µ, d;δ G,cl n−1 × n−1 , respectively. For ✁ ∈ µ, d;δ cl n + ; N − , N + we write ord ✁ = µ; δ. Note that particularly simple elements in B µ, 0;δ cl n + are differential operators A = X |α|≤µ a α xD α x 40 with coefficients a α = ˜a α | n + where ˜ a α x ∈ S δ cl n x . T HEOREM 4. For every µ ∈ the space B µ, 0;0 cl n + cf. the notation 39 contains an element R µ that induces isomorphisms R µ : H s;̺ n + −→ H s−µ;̺ n + 41 for all s, ̺ ∈ as well as isomorphisms R µ : ✁ n + −→ ✁ n + , 42 where R −µ := R µ −1 ∈ B −µ,0;0 cl n + . This is a well-known result for ̺ = 0, proved in this form for all s ∈ in Grubb [7]; note that for s ≤ − 1 2 we have to compose the pseudo-differential operators from the right by an extension operator l : H s n + → H s n , while for s − 1 2 we can take e + . Let us mention Boundary value problems 323 for completeness that order reductions for s µ + − 1 2 have been constructed before by Boutet de Monvel [3]. The symbols from [7] have the form r µ − ξ = χ τ ahηi hηi − i τ µ 43 ξ = η, τ ∈ n for a sufficiently large constant a 0 and a function χ ∈ ✁ with F −1 χ supported in − and χ 0 = 1. It was proved in [20], Section 5.3, that 43 is a classical symbol in ξ . In other words, we have r µ − ξ ∈ S µ; cl ξ ; x n + × n tr , and we can set R µ u = r + Op r µ − l where l = e + for s − 1 2 . It is now trivial that R µ induces isomorphisms for all s, ̺, because the operators with symbols 43 in n belong to L µ; cl n , cf. 9. Note that the operator R µ is elliptic of order µ; 0 in the sense of Definition 7 below. Writing ✁ ∈ µ, d;δ cl n + ; N − , N + in the form 38 we set σ