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By John M. Lee
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Extra info for Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds (draft)
Example text
We will show that p ∈ P. 4 that u ∈ Hδk,p (M ; E), which will prove case (a). Claim 1: If p1 ∈ P, then (1, p1 ] ⊂ P. To prove this, assume p1 ∈ P and 1 < p < p1 . The fact that p1 ∈ P means that there is some δ1 with |δ1 +n/p1 −n/2| < R 1 such that u ∈ Hδ0,p (M ; E). 6, u ∈ Hδ0,p (M ; E) for any δ such that 1 δ1 + n/p1 > δ + n/p . Choosing δ so that δ + n/p is sufficiently close to δ1 + n/p1 , we can ensure that |δ + n/p − n/2| < R. This implies that p ∈ P as desired. 12) where ε=δ+ n n − + R > 0, p 2 1 then p2 ∈ P.
3. For any real numbers p, q, r such that p + 1 > 0 and r > q + 1 > 0, there exists a constant C depending only on p, q, r such that the following estimate holds for all u ∈ [0, 1): 1 p 0 t (1 − t)q dt ≤ C(1 − u)q+1−r . (1 − ut)r Proof. We use the following standard integral representation for hypergeometric functions [27, p. 59]: F (α, β, γ; z) = Γ(γ) Γ(β)Γ(γ − β) 1 β−1 (1 − t)γ−β−1 dt, (1 − tz)α t 0 which is valid if Re γ > Re β > 0 and |z| < 1. The hypergeometric function F (α, β, γ; z) is analytic for |z| < 1 and satisfies a second-order ODE that has a regular singular point at z = 1 with characteristic exponents 0 and γ − α − β [17, p.
From this, it follows easily that (ρ∂i )∗ = −ρ∂i + (n − 2r)δin+1 − ρbi , 28 4. ELLIPTIC OPERATORS where bi = ∂i b(g) is C l−1,β up to ∂M . jk (θ, 0)∗ u(θ, 0) = = 0≤k≤m ji =···=jk =n+1 In−2r−s (P )∗ u, which was to be proved. 5. If P is a formally self-adjoint geometric operator of order m ≤ l, then the set of characteristic exponents of P is symmetric about the line Re s = n/2 − r. Proof. The preceding proposition shows that Is (P ) = Is (P ∗ ) = In−2r−s (P )∗ . Thus if s is a characteristic exponent of P , then In−2r−s (P )∗ , and hence also In−2r−s (P ), is singular, which means that s = n − 2r − s is also a characteristic exponent.