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The symbol C∞ S ,c indicates bounded support. Given p ∈ [1, +∞[, we define the Xp norm of a map u ∈ L1 loc (Σ,�n ) by � |u(z)| p |z − w| p/2−1 ds dt, �u� p Xp (Σ) := �u�p Lp � (Σ\Br(S)) + w∈S Σ∩Br(w) where r < 1 is less than half of the minimal distance between pairs of distinct points in S . This is just a weighted L p norm, where the weight |z − w| p/2−1 comes from the identities (88), (89), and (91) of the last section. Note that when p > 2 the X p norm is weaker than the L p norm, when p < 2 the X p norm is stronger than the L p norm, and when p = 2 the two norms are equivalent. The space X p S (Σ,�n ) is the space of locally integrable�n -valued maps on Σ whose X p norm is finite. The X p norm makes it a Banach space. We view it as a real Banach space. The space X 1,p S (Σ,�n ) is defined as the completion of the space C ∞ S ,c (Σ,�n ) with respect to the norm �u� p X1,p (Σ) := �u�p Xp (Σ) + �Du�p Xp (Σ) . It is a Banach space with the above norm. Equivalently, it is the space of maps in X p (Σ,�n ) whose distributional derivative is also in X p . The space X 1,p S ,V ,V ′(Σ,�n ) is defined as the closure in X 1,p S (Σ,�n ) of the space of all u ∈ C ∞ S ,c (Σ,�n ) such that u(s) ∈ N ∗Vj ∀s ∈ [sj, sj+1], j = 0, . . .,k, u(s + i) ∈ N ∗V ′ j ∀s ∈ [s ′ j , s′ j+1 ], j = 0, . . .,k′ . Equivalently, it can be defined in terms of the trace of u on the boundary of Σ. Let A :Ê×[0, 1] → L(Ê2n ,Ê2n ) be continuous and bounded. For every ∈ [1, +∞[, the linear operator ∂A : X 1,p S (Σ,�n ) → X p S (Σ,�n ), ∂Au := ∂u + Au, is bounded. Indeed, ∂ is a bounded operator because of the inequality |∂u| ≤ |Du|, while the multiplication operator by A is bounded because �Au� X p (Σ) ≤ �A�∞�u� X p (Σ). We wish to prove that if p > 1 and A(z) satisfies suitable asymptotics for Re z → ±∞ the operator ∂A restricted to the space X 1,p S ,V ,V ′(Σ,�n ) of maps satisfying the boundary conditions (92) is Fredholm. Assume that A ∈ C 0 (Ê×[0, 1], L(Ê2n ,Ê2n )) is such that A(±∞, t) ∈ Sym(2n,Ê) for every t ∈ [0, 1]. Define Φ + , Φ − : [0, 1] → Sp(2n) to be the solutions of the linear Hamiltonian systems Then we have the following: (92) d dt Φ± (t) = iA(±∞, t)Φ ± (t), Φ ± (0) = I. (93) 5.9. Theorem. Assume that Φ− (1)N ∗V0 ∩ N ∗V ′ 0 = (0) and Φ+ (1)N ∗Vk ∩ N ∗V ′ k ′ = (0). Then the boundedÊ-linear operator is Fredholm of index − 1 2 ∂A : X 1,p S ,V ,V ′(Σ,�n ) → X p S (Σ,�n ), ∂Au = ∂u + Au, ind∂A = µ(Φ − N ∗ V0, N ∗ V ′ 0) − µ(Φ + N ∗ Vk, N ∗ V ′ k ′) k� (dim Vj−1 + dimVj − 2 dimVj−1 ∩ Vj) − 1 k 2 ′ � (dim V ′ j−1 + dimV ′ j − 2 dimV ′ j−1 ∩ V ′ j ). j=1 60 j=1 (94)
The proof of the Fredholm property for Cauchy-Riemann type operators is based on local estimates. By a partition of unity argument, the proof that ∂A is Fredholm reduces to the Calderon-Zygmund estimates of Lemmas 5.6, 5.8, and to the invertibility of ∂A when A does not depend on Re z and there are no jumps in the boundary conditions. Details are contained in the next section. The index computation instead is based on homotopy arguments together with a Liouville type result stating that in a particular case with one jump the operator ∂A is an isomorphism. 5.4 The Fredholm property The elliptic estimates of section 5.2 have the following consequence: 5.10. Lemma. For every p ∈]1, +∞[, there exist constants c0 = c0(p, n, S ) and c1 = c1(p, n, k + k ′ ) such that for every u ∈ C ∞ S ,c (Σ,�n ) such that for every j. Proof. Let {ψ1, ψ2} ∪ {ϕj} k+k′ j=1 By Lemma 5.8, �Du�Xp ≤ c0�u�X p + c1�∂u�X p, u(s) ∈ N ∗ Vj ∀s ∈ [sj−1, sj], u(s + i) ∈ N ∗ V ′ j ∀s ∈ [s ′ j−1 , s′ j ] be a smooth partition of unity on�satisfying: suppψ1 ⊂ {z ∈�|Im z < 2/3} \ B r/2(S ), suppψ2 ⊂ {z ∈�|Im z > 1/3} \ B r/2(S ), suppϕj ⊂ Br(sj) ∀j = 1, . . .,k, suppϕk+j ⊂ Br(s ′ j + i) ∀j = 1, . . .,k′ . �D(ϕju)� X p (Σ) ≤ c(p, n)�∂(ϕju)� X p (Σ) ≤ c(p, n)(�∂ϕj�∞�u� X p (Σ) + �∂u� X p (Σ)), 1 ≤ j ≤ k + k ′ . Since the X p norm is equivalent to the L p norm on the subspace of maps whose support does not meet B r/2(S ), the standard Calderon-Zygmund estimates on the half-plane (see Theorem 5.3) imply �D(ψju)� X p (Σ) ≤ c(p, n)�∂(ψju)� X p (Σ) ≤ c(p, n)(�∂ψj�∞�u� X p (Σ) + �∂u� X p (Σ)), ∀j = 1, 2. We conclude that �Du�Xp ≤ �D(ψ1u)� k+k Xp (Σ) + �D(ψ2u)�X p (Σ) + ′ � �D(ϕju)�X p (Σ) ≤ c0�u�X p (Σ) + c1�∂u�X p (Σ), with as claimed. j=0 j=1 � k+k c0 := c(p, n) �∂ψ1�∞ + �∂ψ2�∞ + ′ �+1 � �∂ϕj�∞ , c1 := (k + k ′ + 2)c(p, n), The next result we need is the following theorem, proved in [RS95, Theorem 7.1]. Consider two continuously differentiable Lagrangian paths λ, ν :Ê→L (n), assumed to be constant on [−∞, −s0] and on [s0, +∞], for some s0 > 0. Denote by W 1,p λ,ν (Σ,�n ) the space of maps u ∈ W 1,p (Σ,�n ) such that u(s, 0) ∈ λ(s) and u(s, 1) ∈ ν(s), for every s ∈Ê(in the sense of traces). Let A ∈ C0 (Ê×[0, 1], L(Ê2n ,Ê2n )) be such that A(±∞, t) ∈ Sym(2n,Ê) for any t ∈ [0, 1], and define Φ− , Φ + : [0, 1] → Sp(2n) by (93). 61 (95)
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5.8 Non-local boundary conditions .
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M into Λ(M) as the space of consta
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and converging to Hamiltonian orbit
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1.2 The Chas-Sullivan loop product
Page 12 and 13: 1.1. Remark. The loop product was d
Page 14 and 15: 2.2 Functoriality Let (M1, g1) and
Page 16 and 17: The orbits of −grad(f1 ⊕f2) are
Page 18 and 19: this intersection is a finite set o
Page 20 and 21: (L0) Ω every solution γ ∈ P
Page 22 and 23: transverse to the stable manifold o
Page 24 and 25: 2.7. Proposition. The homomorphism
Page 26 and 27: A submanifold L ⊂ T ∗ M is call
Page 28 and 29: The L 2 -negative gradient equation
Page 30 and 31: The Riemann surface Σ Ω Υ s = 0
Page 32 and 33: In the periodic case, we consider H
Page 34 and 35: 3.6. Definition. The Maslov index
Page 36 and 37: The Riemann surface ΣG is obtained
Page 38 and 39: 3.12. Proposition. For a generic ch
Page 40 and 41: The following compactness statement
Page 42 and 43: 4.2. Proposition. For a generic cho
Page 44 and 45: A standard gluing argument shows th
Page 46 and 47: In the following two sections we wi
Page 48 and 49: We shall apply the above lemma to t
Page 50 and 51: Since the inclusion c : M ֒→ Λ
Page 52 and 53: 4.5 The right-hand square is homoto
Page 54 and 55: commute. Our first aim in this sect
Page 56 and 57: Moreover, �Dαn(βnvn)�0,p;Ê
Page 58 and 59: We identify the product T ∗Ên ×
Page 60 and 61: Up to multiplying u by U, we can re
Page 64 and 65: 5.11. Theorem. (Cauchy-Riemann oper
Page 66 and 67: 5.15. Lemma. Let p > 1 and α ∈Ê
Page 68 and 69: Proof. By Proposition 5.14 the oper
Page 70 and 71: Then ind∂A = ind∂A1 + ind∂A1.
Page 72 and 73: (V0, . . . , Vk), by V ′ the (k
Page 74 and 75: is bounded and Fredholm of index in
Page 76 and 77: so the transversality hypotheses of
Page 78 and 79: 5.9 Coherent orientations As notice
Page 80 and 81: compact subsets of Σ Ω Υ , for
Page 82 and 83: where Ã(z) = CA(z)C ⊕ A(z), C de
Page 84 and 85: By the additivity property of the M
Page 86 and 87: The space M Υ GE . Let x1 ∈ P(H1
Page 88 and 89: The space M Θ Φ . Let γ ∈ PΘ
Page 90 and 91: The embedding Q ֒→ÊN induces an
Page 92 and 93: Given periodic orbits x1 ∈ P(H1),
Page 94 and 95: Since u and ∂u are integrable ove
Page 96 and 97: with non-local boundary conditions
Page 98 and 99: to ũ0 uniformly on Σ, we may loca
Page 100 and 101: we see that associating ũ to u pro
Page 102 and 103: The energy of a solution u ∈ M K
Page 104 and 105: Since we have applied a conformal r
Page 106 and 107: 6.9. Lemma. There exists a positive
Page 108 and 109: [ChS04] M. Chas and D. Sullivan, Cl
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