Parametrices and Fredholm property

Boundary value problems 329 T HEOREM 7. For every N ∈ ✁ there exist elliptic elements ✁ + N ∈ 0,0;0 cl n + ; 0, N and ✁ − N ∈ 0,0;0 cl n + ; N , 0 that induce isomorphisms ✁ + N : H s n + −→ H s n + ⊕ H s n−1 , N , ✁ − N : H s n + ⊕ H s n−1 , N −→ H s n + for all s − 1 2 , where ✁ − N = ✁ + N −1 . Proof. Let us start from the above operator ✁ 2 and form ✁ = s −2 ✁ 2 e −s ∈ 0,0;0 cl n + ; 0, 1 56 for any fixed s 2, where e := R 1 ∈ B 1,0;0 cl n + is the order reducing element from Theorem 4 and := diag R 1 , R ′ for R ′ = Op y hηi ⊗ id ✁ N . Then, setting ✁ + 1 = ✁ , we can form ✁ + N inductively by ✁ + N = A + N T + N :=    A + N −1 T + N −1 1    A + 1 T + 1 =    A + N −1 A + 1 T + N −1 A + 1 T + 1    . Here, ✁ + 1 = A + 1 T + 1 . Moreover, from the above construction of ✁ −1 2 and Theorem 4 it follows that we may set ✁ − N := ✁ + N −1 .

3.6. Parametrices and Fredholm property

T HEOREM 8. Let ✁ ∈ µ, d;δ cl n + ; N − , N + be elliptic. Then ✁ : H s;̺ n + ⊕ H s;̺ n−1 , N − −→ H s−µ;̺−δ n + ⊕ H s−µ;̺−δ n−1 , N + 57 is a Fredholm operator for every s maxµ, d − 1 2 and every ̺ ∈ , and ✁ has a parametrix ∈ −µ,d−µ + ;−δ cl n + ; N + , N − where d l = maxµ, d and d r = d −µ + cf. the notation in Definition 8. The proof of this theorem will be given below after some preparations. R EMARK 18. Applying Remark 15 we can reduce the proof of Theorem 8 to the case δ = 0. In other words, it suffices to consider the operator ✁ ̺−δ ✁ ✁ −̺ ∈ µ, d;0 cl n + ; N − , N + . Furthermore, we can reduce orders and pass to ✁ := s −µ + ✁ ̺−δ ✁ ✁ −̺ −s − ∈ 0,0;0 cl n + ; N − , N + 330 D. Kapanadze – B.-W. Schulze for any choice of s maxµ, d, where ± = diag R 1 , R ′ N ± for R ′ N ± := Op hηi ⊗ id ✁ N± , cf. similarly 56. Clearly, the ellipticity of ✁ is equivalent to that of ✁ , and the construction of a parametrix for ✁ gives us immediately a parametrix of ✁ , namely = ✁ −̺ −s − s −µ + ✁ ̺−δ . 58 So we mainly concentrate on the case ✁ ∈ 0,0;0 cl n + ; N − , N + . Let px, ξ ∈ S 0;0 cl n + × n tr,≍ be a symbol with σ ψ p 6= 0 for all x, ξ ∈ n + × n \ 0 , 59 σ e p 6= 0 for all x, ξ ∈ n + \ 0 × n , 60 σ ψ, e p 6= 0 for all x, ξ ∈ n + \ 0 × n \ 0 . 61 Set b ′ 11 y, η := op + p| t =0 y, η , 62 and consider the operator families σ ∂ b ′ 11 y, η , σ e ′ b ′ 11 y, η , σ ∂, e ′ b ′ 11 y, η : L 2 + −→ L 2 + , 63 σ ∂ b ′ 11 for y, η ∈ n−1 × n−1 \ 0 , σ e ′ b ′ 11 for y, η ∈ n−1 \ 0 × n−1 , σ ∂, e ′ b ′ 11 for y, η ∈ n−1 \ 0 × n−1 \ 0 . These are families of Fredholm operators parametrized by the corresponding sets of y, η- variables. P ROPOSITION 4. For every ε 0 there exists an R = R ε 0 such that σ ∂ b ′ 11 y, η − σ ∂, e ′ b ′ 11 y, η ✂ L 2 + ε 64 for all |y| R and η ∈ n−1 \ 0, σ ∂ b ′ 11 y, η − σ e ′ b ′ 11 y, η ✂ L 2 + ε 65 for all |y| R and |η| R, σ e ′ b ′ 11 y, η − σ ∂, e ′ b ′ 11 y, η ✂ L 2 + ε 66 for all |y| ∈ n−1 \ 0 and |η| R. Proof. Let us first verify 64. Both op σ ψ p| t =0 y, η and op σ ψ, e p| t =0 y, η can be regarded as parameter-dependent families of pseudo-differential operators L 2 + → L 2 + with parameter y ∈ n−1 , smoothly dependent on η with |η| = 1. But op σ ψ p| t =0 − σ ψ, e p| t =0 y, η 67 is of order −1 in the parameter. A well-known result on operator norms of parameter-dependent pseudo-differential operators, cf., e.g., [30], Section 1.2.2, tells us that the ✁ L 2 + -norm of Boundary value problems 331 67 tends to zero for |y| → ∞, in this case uniformly for |y| = 1. Thus, composing 67 from the right with e + and from the left with r + we get relation 64 for all |y| ≥ R, R = R ε , and |η| = 1. In a similar way we can argue for 66, now with η ∈ n−1 \ 0 as parameter and smooth dependence on y with |y| = 1. This gives us relation 66. Estimate 65 is then an obvious consequence of 64 and 66. C OROLLARY 1. Under the conditions of Proposition 4 there exists an R = R ε 0 such that the Fredholm families σ ∂ b ′ 11 y, η : L 2 + −→ L 2 + for 0 ≤ |y| ≤ R , |η| = R and σ e ′ b ′ 11 y, η : L 2 + −→ L 2 + for |y| = R , 0 ≤ |η| ≤ R , satisfy σ ∂ b ′ 11 y, η − σ e ′ b ′ 11 y, η ✂ L 2 + ε for all |y| = |η| = R. Let ε 0, and set T ε = n y, η ∈ 2n−1 : |y| = |η| = R ε o , D ε = T ε × [0, 1] , and form Z j ε = n y, η ∈ 2n−1 : |y| ≤ R ε + j , |η| = R ε o , H j ε = n y, η ∈ 2n−1 : |y| = R ε , |η| ≤ R ε + j o for j = 0, 1, ∞. Define the spaces j ε = Z j ε ∪ d H j ε ∪ b D ε ∼ , where ∪ d is the disjoint union, while ∪ b is the disjoint union combined with the projection to the quotient space, given by natural identifications T ε ∩ Z j ε ∼ = T ε × {0}, T ε ∩ H j ε ∼ = T ε × {1}. Write for abbreviation Z ε = Z ε , H ε = H ε , ε = ε . Moreover, let D ε,τ := T ε × [0, τ ] and ε,τ := Z ε ∪ d H ε ∪ b D ε,τ , 0 ≤ τ ≤ 1, where ∪ b is defined by means the identifications T ε ∩ Z ε ∼ = T ε × {0}, T ε ∩ H ε ∼ = T ε × {τ }. Thus ε = ε, 1 , and we set ✁ ε = ε, . Define an operator function Fm, m ∈ ε , by the following relations: Fy, η = σ ∂ b ′ 11 y, η for m = y, η ∈ Z ε , Fy, η = σ e ′ b ′ 11 y, η for m = y, η ∈ H ε , Fy, η, δ = δσ ∂ b ′ 11 y, η + 1 − δσ e ′ b ′ 11 y, η for m = y, η, δ ∈ D ε . From Corollary 1 we get Fy, η, δ − F y, η, δ ′ ✂ L 2 + ≤ |δ − δ ′ |ε 68 for all y, η, δ, y, η, δ ′ ∈ D ε , 0 ≤ δ, δ ′ ≤ 1. We have F ∈ C ε , ✁ L 2 + , 69 and F| Z ε , F| H ε are continuous families of Fredholm operators. Relation 68 shows that 69 is a family of Fredholm operators for all m ∈ ε , provided ε 0 is sufficiently small. We then get 332 D. Kapanadze – B.-W. Schulze an index element ind ε F ∈ K ε . Because of K ε,τ ∼ = K ε for all 0 ≤ τ ≤ 1, ind ε F represents, in fact, an element in K ✁ ε that we denote by ind ✁ ε σ ∂ b ′ 11 y, η, σ e ′ b ′ 11 y, η ∈ K ✁ ε . 70 Our next objective is to check, whether the operator family b ′ 11 y, η for an elliptic symbol px, ξ ∈ S 0;0 cl ξ ; x n + × n tr can be completed to a block matrix valued symbol b ′ y, η = b ′ 11 b 12 b 21 b 22 y, η ∈ S 0;0 cl η; y n−1 × n−1 ; E, e E , 71 E = L 2 + ⊕ N − , e E = L 2 + ⊕ N + 72 with suitable N − , N + , such that the homogeneous symbols σ ∂ b ′ y, η ∈ S η , y, η ∈ n−1 × n−1 \ 0 , 73 σ e ′ b ′ y, η ∈ S y , y, η ∈ n−1 \ 0 × n−1 , 74 σ ∂, e ′ b ′ y, η ∈ S 0;0 η; y , y, η ∈ n−1 \ 0 × n−1 \ 0 , 75 are isomorphisms. T HEOREM 9. Let px, ξ ∈ S 0;0 cl ξ ; x n + × n tr be σ ψ , σ e , σ ψ, e -elliptic, i.e., relations 59, 60 and 61 are fulfilled. Then the following conditions are equivalent: i The families of Fredholm operators L 2 + → L 2 + σ ∂ b ′ 11 y, η for y, η ∈ n−1 × n−1 \ 0 , 76 σ e ′ b ′ 11 y, η for y, η ∈ n−1 \ 0 × n−1 , 77 σ ∂, e ′ b ′ 11 y, η for y, η ∈ n−1 \ 0 × n−1 \ 0 78 can be completed to D 0,0 + ; N − , N + -valued families of isomorphisms 73, 74 and 75, respectively. ii ind ✁ ε σ ∂ b ′ 11 , σ e ′ b ′ 11 ∈ π ∗ + K {+} , 79 where π + : ✁ ε → {+} is the projection of ✁ ε to a single point {+}, K {+} = . Proof. i ⇒ ii : In the construction of the proof we choose ε 0 sufficiently small. Assume that we have isomorphism-valued symbols 73, 74 and 75, associated with the given upper left corners 76, 77 and 78. Then the above Fredholm family Fm on ε , associated with σ ∂ b ′ 11 , σ e ′ b ′ 11 has the property ind ε F = N + − N − , i.e., ind ε F ∈ which implies ind ✁ ε σ ∂ b ′ 11 , σ e ′ b ′ 11 ∈ ∼ = π ∗ + K {+}. ii ⇒ i : Condition ind ✁ ε σ ∂ b ′ 11 , σ e ′ b ′ 11 ∈ π ∗ + K {+} implies the existence of numbers N ± ∈ ✁ with ind ✁ ε σ ∂ b ′ 11 , σ e ′ b ′ 11 = N + − N − . Replacing N ± by N ± + Boundary value problems 333 M for sufficiently large M and denoting the enlarged numbers again by N ± we find operator families km : N − −→ L 2 + , t m : L 2 + −→ N + , qm : N − −→ N + , such that f m := Fm km t m qm : L 2 + ⊕ N − −→ L 2 + ⊕ N + 80 is a family of isomorphisms, continuously parametrized by ε . It is evident that they can be cho- sen as D 0;0 + ; N − , N + -valued functions, similarly to the construction of bijective boundary symbols in the local algebra of boundary value problems with the transmission property. In addition it is clear that the functions km, t m and qm can be chosen to be smooth in y, η. Let us now define a Fredholm family F 1 m for m ∈ 1 ε by F 1 m = Fm for m ∈ ε , F 1 m = 1 − λσ ∂ b ′ 11 y, η + λσ ∂, e ′ b ′ 11 y, η for R ε ≤ |y| ≤ R ε + 1, |η| = R ε , where λ = R ε − |y|, F 1 m = 1 − λσ e ′ b ′ 11 y, η + λσ ∂, e ′ b ′ 11 y, η for |y| = R ε , R ε ≤ |η| ≤ R ε + 1, where λ = R ε − |η|. Estimates 64 and 66 show that F 1 is a family of Fredholm operators on 1 ε , provided ε 0 is sufficiently small. We can construct a family of isomorphisms f 1 m := F 1 m k 1 m t 1 m q 1 m : L 2 + ⊕ N − −→ L 2 + ⊕ N + , 81 m ∈ 1 ε , similarly as f m if necessary, we take N − , N + larger than before, where f 1 | ε = f . Since Fm is a-priori given on ∞ ε , we can also form ˜ f 1 m := Fm k 1 m t 1 m q 1 m : L 2 + ⊕ N − −→ L 2 + ⊕ N + , m ∈ 1 ε . Due to 64 and 66 this is a family of Fredholm operators. Clearly, we may choose ˜ f 1 m in such a way that ˜ f 1 | Z 1 ε and ˜ f 1 | H 1 ε are smooth in y, η. Let us finally look at ∞ ε . The operator function f 1 , first given on 1 ε , canonically extends to ∞ ε by homogeneity of order zero to ∞ ε \ 1 ε in y and η. Let f ∞ denote this extension, f ∞ m := F ∞ m k ∞ m t ∞ m q ∞ m 82 i.e., f ∞ | 1 ε = f 1 . Since f ∞ is obtained by homogeneous extension of a family of isomor- phisms, it is again isomorphism-valued. Moreover, we can also form ˜ f ∞ m := Fm k ∞ m t ∞ m q ∞ m 334 D. Kapanadze – B.-W. Schulze which is a family of isomorphisms because of the corresponding property of 82 and relations 64 and 66. Then, to get 73, 74 and 75, it suffices to define σ ∂ b ′ y, η as the extension by homo- geneity 0 in η of ˜ f ∞ | Z ∞ ε to n−1 × n−1 \ 0 , σ e ′ b ′ y, η as the extension by homogeneity 0 in y of ˜ f ∞ | H ∞ ε to n−1 \ 0 × n−1 and σ ∂, e ′ b ′ y, η as the extension by homogeneity 0 in y and η of ˜ f ∞ | {|y|=R ε +1,|η|=R ε +1} to n−1 \ 0 × n−1 \ 0 . To justify the notation in 73, 74 and 75 i.e., to generate the latter homogeneous functions in terms of a symbol 71 we can first form b ′′ y, η = χ ησ ∂ b ′ y, η + χ y σ e ′ b ′ y, η − χ ησ ∂, e ′ b ′ y, η ∈ S 0;0 cl η; y n−1 × n−1 ; E, e E , cf. the second part of Remark 5, and then define b ′ y, η by replac- ing the upper left entry of b ′′ y, η by b ′ 11 y, η. R EMARK 19. Notice that Theorem 9 is an analogue of the Atiyah-Bott condition for the existence of elliptic boundary conditions to an elliptic operator A, cf. also Section 4.4 below. The canonical projection T ∗ n−1 → n−1 restricted to the subset ✁ ε ⊂ T ∗ n−1 gives us a projection π ε : ✁ ε → B ε := y ∈ n−1 : |y| ≤ R ε . Condition 79 can equivalently be written ind ✁ ε σ ∂ b ′ 11 , σ e ′ b ′ 11 ∈ π ∗ ε K B ε , since B ε is contractible to a point {+}. C OROLLARY 2. Given a symbol px, ξ ∈ S 0;0 cl ξ ; x n + × n tr that is σ ψ , σ e , σ ψ, e - elliptic, under the condition 79 for b ′ 11 y, η = op + p| t =0 y, η we find a by, η ∈ 0,0;0 cl n−1 × n−1 ; N − , N + for suitable N − , N + ∈ ✁ , such that 76, 77 and 78 are isomorphims L 2 + ⊕ N − → L 2 + ⊕ N + , cf. Definition 7. To construct by, η it suffices to define by, η by replacing the upper left entry of b ′′ y, η by op + p ≍ y, η for p ≍ x, ξ = χ ≍ x px, ξ with some global admissible cut-off function χ ≍ , cf. Definition 3. P ROPOSITION 5. Let G ∈ B 0,0;0 G,cl n + be an operator such that A := 1 + G is elliptic in the sense of Definition 7. Then there is a e G ∈ B 0,0;0 G,cl n + such that e A := 1 + e G is a parametrix of A, i.e., A e A − 1, e A A − 1 ∈ B −∞,0;−∞ n + . Proof. Let us first observe that for every g ∈ Ŵ + i.e., g ∈ ✁ L 2 + with g, g ∗ : L 2 + → ✁ + being continuous, cf. Section 3.1, we have ag, ga ∈ Ŵ + for every a ∈ ✁ L 2 + . Then, if 1 + g : L 2 + → L 2 + for a g ∈ ✁ L 2 + is invertible, we have a = 1 + g −1 ∈ ✁ L 2 + and a1 + g = 1 = a + ag, i.e., a = 1 + ˜ g for ˜g = −ag ∈ Ŵ + . Analogous conclusions are valid for the symbols σ ∂ 1 + G, σ e ′ 1 + G and σ ∂, e ′ 1 + G. Then, setting ˜g ∂ y, η := σ ∂ 1 + G −1 y, η − 1 , ˜g e ′ y, η := σ e ′ 1 + G −1 y, η − 1 , ˜g ∂, e ′ y, η := σ ∂, e ′ 1 + G −1 y, η − 1 , we can form ˜ gy, η := χ η ˜ g ∂ y, η + χ y ˜ g e ′ y, η − χ η ˜ g ∂, e ′ y, η ∈ R 0,0;0 G,cl n−1 × n−1 cf. Remark 5. For e G 1 = Op y ˜ g we then have 1 + G 1 + e G 1 = 1 + e G 2 where Boundary value problems 335 e G 2 ∈ B −1,0;−1 G,cl n + . Then e G j 2 ∈ B − j,0;− j G,cl n + for all j , and we can carry out the asymptotic sum P ∞ j =0 − 1 j e G j 2 in the class of operators 1 + B −1,0;−1 G,cl n + which is just a version of the formal Neumann series argument in our operator class. In other words, we can find a e G 3 ∈ B −1,0;−1 G,cl n + such that 1 + e G 2 1 + e G 3 = 1 + C for C ∈ B −∞,0;−∞ n + . Because 1 + e G 1 1 + e G 3 = 1 + e G for some e G ∈ B 0,0;0 G,cl n + we get 1 + G 1 + e G = 1 + C. Similar arguments from the left yield a e e G ∈ B 0,0;0 G,cl n + with 1 + e e G 1 + G − 1 ∈ B −∞,0;−∞ n + . Then a standard algebraic argument gives us e G = e e G mod B −∞,0;−∞ n + . In other words e A = 1 + e G is as desired. Proof of Theorem 8. As noted in Remark 18 we may content ourselves with the case µ = d = δ = 0. The ellipticity of ✁ with respect to σ ψ ✁ , σ e ✁ , σ ψ, e ✁ allows us to form a symbol px, ξ = χ ξ σ ψ ✁ −1 x, ξ + χ x σ e ✁ −1 x, ξ − χ ξ σ ψ, e ✁ −1 x, ξ ∈ S 0;0 cl ξ ; x n + × n tr , where σ ψ p, σ e p, σ ψ, e p = σ ψ ✁ −1 , σ e ✁ −1 , σ ψ, e ✁ −1 . We now observe that px, ξ meets the assumption of Theorem 9. In fact, the original symbol ax, ξ belonging to ✁ satisfies these conditions because the assumed bijectivities just correspond to the ellipticity of ✁ with respect to σ ∂ ✁ , σ e ′ ✁ , σ ∂, e ′ ✁ . Hence relation 80 with respect to ax, ξ is fulfilled. This implies the corresponding relation with respect to px, ξ because the index element in π ∗ + K {+} is just the inverse of that for ax, ξ . By construction we have px, ξ ax, ξ = 1 + r x, ξ for an r x, ξ ∈ S −1;−1 cl ξ ; x n + × n tr . This yields px, ξ ax, ξ = 1 + ˜rx, ξ 83 for an ˜ r x, ξ ∈ S −1;−1 cl ξ ; x n + × n tr . A formal Neumann series argument, applied to 1+ ˜ r x, ξ in terms of the Leibniz multiplication gives us a symbol ˜ qx, ξ ∈ S −1;−1 cl ξ ; x n + × n tr such that 1 + ˜ qx, ξ 1 + ˜ r x, ξ = 1 mod S −∞;−∞ n + × n . Setting ˜ px, ξ = 1 + ˜ qx, ξ px, ξ from relation 83 we get ˜ px, ξ ax, ξ = 1 mod S −∞;−∞ n + × n . Applying Corollary 2 to ˜ px, ξ we can generate a by, η of the asserted kind, more precisely by, η ∈ 0,0;0 cl n−1 × n−1 . Then the operator e := Op b + R for R := 1 − χ ≍ xOp x ˜ p, cf. Definition 3, has the property ✁ e = + for some ∈ 0,0;0 cl n + ; N + , N + . Since ✁ and e are both elliptic also + is elliptic. Applying Theorem 7 to N = N + we can pass to the elliptic operator ✁ N + + ✁ −1 N + that has the form 1 + G for a G ∈ B 0,0;0 G,cl n + . Proposition 5 gives us a e G ∈ B 0,0;0 G,cl n + such that 1 + G 1 + e G = 1 + C for a C ∈ B −∞,0;−∞ n + . It follows that ✁ N + ✁ e ✁ −1 N + 1 + e G = 1 + C for an element C ∈ B −∞,0;−∞ n + . This yields ✁ e ✁ −1 N + 1 + e G ✁ N + = ✁ −1 N + 1 + C ✁ N + = 1 + ✂ for a remainder ✂ ∈ −∞,0;−∞ n + ; N + , N + . Hence, := e ✁ −1 N + 1 + e G ✁ N + ∈ 0,0;0 cl n + ; N + , N − 336 D. Kapanadze – B.-W. Schulze is a right parametrix of ✁ . In an analogous manner we can construct a parametrix from the left; then a standard argument shows that is also a left parametrix. In other words, when we go back to the original orders of Theorem 8, we get a parametrix by formula 58, where its type is d − µ + and the types d l and d r of remainders are an immediate consequence of Theorem 5. The Fredholm property of 57 follows from the fact that the remainders are compact operators in the respective spaces, since they improve smoothness and weight. This completes the proof of Theorem 8. 4. The global theory 4.1. Boundary value problems on smooth manifolds