Proposition 2.1 . [02AZ] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes 1 original proof heading/text diagnostics lack independently established complete proof boundaries; diagnostic occurrences may overlap and are not a count of distinct proofs. Complete original source context Β· Original author HTML
Proposition 2.1 .
(1)
There are constants K 0 , K 1 K_{0},K_{1} , depending only on n , c , V n,c,V such that if X X is in π¦ β‘ ( n , c , v ) {\mathcal{K}}(n,c,v) and s s is a holomorphic section of L k L^{k} (for any k > 0 k>0 ) we have
β s β L β , β― β€ K 0 β β s β L 2 , β― , β β s β L β , β― β€ K 1 β β s β L 2 , β― . . \|s\|_{L^{\infty,\sharp}}\leq K_{0}\|s\|_{L^{2,\sharp}}\ \ ,\|\nabla s\|_{L^{\infty,\sharp}}\leq K_{1}\|s\|_{L^{2,\sharp}}..
(2)
If X X is in π¦ β‘ ( n , c , V ) {\mathcal{K}}(n,c,V) then for any k > 0 k>0 the Laplacian Ξ β Β― \Delta_{\overline{\partial}} on Ξ© 0 , 1 β ( L k ) \Omega^{0,1}(L^{k}) is invertible and Ξ β Β― β 1 β€ 2 \Delta_{\overline{\partial}}^{-1}\leq 2 .