Download Fredholm Operators and Einstein Metrics on Conformally by John M. Lee PDF

Download Fredholm Operators and Einstein Metrics on Conformally by John M. Lee PDF

By John M. Lee

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Because the different choices of (θ, ρ) coordinates are all uniformly bounded in l+1,β C(0) (Ω) with respect to each other, and |dρ|g , |dθα |g are both in C1l,β (M ) with norms independent of p, it follows that the functions ψpb,r are uniformly bounded in C l,β (Ω), independently of p and r. 2, there is a number N such that for any r > 0 we can choose (necessarily finitely many) points {p1 , . . , pm } ⊂ ∂M such that the sets {Zr/2 (pi )} cover Ar/2 = {p ∈ M : ρ(p) < r/2} and no more than N of the sets {Zr (pi )} intersect nontrivially at any point.

If u ∈ Hδ0,p (M ; E) and P u ∈ Hδk−m,p (M ; E), then u ∈ Hδk,p (M ; E). (b) Suppose that 0 < α < 1, m < k +α ≤ l +β, |δ −n/2| < R, and |δ −n/2| < R. If u ∈ Cδ0,0 (M ; E) and P u ∈ Cδk−m,α (M ; E), then u ∈ Cδk,α (M ; E). Proof. 6(b), so assume δ > δ. Consider part (a). 8, it suffices to show that u ∈ Hδ0,p (M ; E). For any small r > 0, by means of a bump function we can write u = u0 + u∞ , where supp u0 is compact and supp u∞ ⊂ Ar . Local elliptic regularity gives u0 ∈ Hδk,p (M ; E). 3, if r is small enough, u∞ = QP u∞ − T u∞ ∈ Hδm,p (M ; E) + Hδ1,p (M ; E) 1 ⊂ Hδ0,p (M ; E), 1 6.

8), we conclude that for any ε > 0 there is a constant C such that |fj (t)| ≤ Ctn/2−r+R−ε for t away from 1. This in turn implies |K0 (ξ)u0 |g˘ ≤ Cρ(ξ)r |K0 (ξ)u0 |g ≤ C ρ(ξ)r ρ(ξ)n/2−r+R−ε |u0 |g = C ρ(ξ)n/2+R−ε |u0 |g˘ whenever d(ξ, 0) ≥ 1, which was to be proved. The next two lemmas give some estimates that will be needed to use our decay estimate for proving mapping properties of P −1 . 5. 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 .

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