aa r X i v : . [ m a t h . C V ] M a y EXTENSION OF L HOLOMORPHIC FUNCTIONS
LUCA BARACCO
Abstract.
The purpose of this note is to show that the ¯ ∂ -estimate which is needed inthe Ohsawa-Takegoshi Extension Theorem [6] is a direct consequence of the H¨ormander-Kohn-Morrey weigthed inequality. In this inequality, the Donnelly-Fefferman argumentis not required and a single 1-parameter family of non-singular weights is used. Thispaper is the furtherst step of a great deal of work devoted to the simplification of theoriginal proof of Ohsawa-Takegoshi Theorem; among other papers on the subject, wemention [1] and [8] which are based on “twisted” basic estimates and, in recent time, [3]and [9].MSC: 32F10, 32F20, 32N15, 32T25 Through an approximation argument, the extension theorem relies on Theorem 1 below.
Theorem 1.
Let D ⊂⊂ C n be a bounded smooth pseudoconvex domain with diameter ≤ , ψ a plurisubharmonic function on D , α a ¯ ∂ -closed form in L ψ ( D ) of degree ≥ suchthat α J = 0 for / ∈ J and supp α ⊂ { z : | z | < δ } . Then there is a solution u = u δ ∈ L ψ to the problem (1) ( ¯ ∂u = α, k z u k ψ ≤ cδ k α k ψ , for c independent of δ , ψ and D. Proof.
We first assume ψ = 0. We set(2) ϕ (= ϕ δ ) = − log( − log( | z | + δ )) , a (= a δ ) = − log( | z | + δ );they are related by a = e − ϕ . Their key properties are(3) ϕ δ = − ¯ z log( | z | + δ )( | z | + δ ) , ϕ δ = | z | − δ log( | z | + δ )log ( | z | + δ )( | z | + δ )) . This readily implies(4) − a ≥ , − a > ∼ δ for | z | < δ. The contraction of the gradient with a multivector v is defined by ∂a | v = ( a i ) | v = P i a i v iK and the action of the Levi form is ∂ ¯ ∂a ( v, v ) = ( a i ¯ j )( v, v ) = P ′| K | = k − P ij =1 ,...,N a i ¯ j v iK ¯ v jK .From the identity ¯ ∂ ∗ ϕ = ¯ ∂ ∗ + ∂ϕ | , we get the estimate(5) k ¯ ∂ ∗ ϕ v k ϕ ≤ k ¯ ∂ ∗ v k ϕ + k ∂ϕ | v k ϕ + 2 (cid:12)(cid:12)(cid:12) Z D e − ϕ ( ∂ϕ | v ) · ( ¯ ∂ ∗ v ) dV (cid:12)(cid:12)(cid:12) . We also have2 (cid:12)(cid:12)(cid:12) Z D e − ϕ ∂ϕ | v · ¯ ∂ ∗ v dV (cid:12)(cid:12)(cid:12) = (2) (cid:12)(cid:12)(cid:12) Z D ∂a | v · ¯ ∂ ∗ vdV (cid:12)(cid:12)(cid:12) ≤ Cauchy-Schwarz k v k + k| ∂a | ¯ ∂ ∗ v k < ∼ Pseudoconvexity c ( k ¯ ∂v k + k ¯ ∂ ∗ v k + k| ∂a | ¯ ∂ ∗ v k ) , (6)where c only depends on the diameter of D . Recall that a = e − ϕ and k·k ϕ = k a ·k . Wemay then conclude Z D − ( a i ¯ j )( v, v ) dV = Z e − ϕ ϕ i ¯ j ( v, v ) dV − k ∂ϕ | u k ϕ ≤ basic k ¯ ∂v k ϕ + k ¯ ∂ ∗ ϕ v k ϕ − k ∂ϕ | u k ϕ ≤ (5) k ¯ ∂v k ϕ + k ¯ ∂ ∗ v k ϕ + 2 (cid:12)(cid:12)(cid:12) Z D e − ϕ ∂ϕ | v · ¯ ∂ ∗ v dV (cid:12)(cid:12)(cid:12) ≤ c ( k (1 + a ) ¯ ∂v k + k (1 + a + | ∂a | ) ¯ ∂ ∗ v k ) . (7)With (7) in our hands, we define a functional in L by putting(8) (1 + a + | ∂a | ) ¯ ∂ ∗ v ( v, α ) for v ∈ D ¯ ∂ ∗ . If v ∈ (ker ¯ ∂ ) ⊥ , we have ( v, α ) = 0. If, instead, v ∈ ker ¯ ∂ , then | ( v, α ) | = (cid:12)(cid:12)(cid:12) Z D ( − a i ¯ j )( v, ( − a ) − α ) dV (cid:12)(cid:12)(cid:12) ≤ Cauchy-Schwarz (cid:18)Z D ( − a i ¯ j )( v, v ) dV (cid:19) k ( − a ) − α k ≤ (4) (cid:16) Z D ( − a i ¯ j )( v, v ) dV (cid:17) ( δ k α k ) < ∼ (7) k (1 + a + | ∂a | ) ¯ ∂ ∗ v k δ k α k , (9)where, in order to apply Cauchy-Schwarz, we have used that ( − a i ¯ j ) ≥
0. Thus, by RieszRepresentation Theorem, there exists w ∈ L of norm k w k < ∼ δ k α k which represents(8). Setting u := (1 + a + | ∂a | ) w , we get ¯ ∂u = α ; also, from the trivial inequality(1 + a + | ∂a | ) ≤ | z | − , we conclude k z u k < ∼ δ k α k . (Note that we replace (1 + a + | ∂a | )by | z | − in the conclusion but not in the proof, especially in (9), because | z | − is not in L .) XTENSION OF L HOLOMORPHIC FUNCTIONS 3
When ψ is smooth in ¯ D , the proof above can be repeated verbatim, with the weight ϕ replaced by ϕ + ψ in the basic estimate (7) and with the Levi form ( ψ ij ) dropped down be-cause of its positivity. For a general plurisubharmonic ψ , we take a smooth approximationof ψ from above and an exhaustion of D from inside. A weak limit of solutions on thesesubdomains yields the solution on D . For this approximation we need that L ψ ⊂ L . Butthis follows from the local boundedness of ψ from above which is in turn a consequenceof its upper semicontinuity. (cid:3) References [1]
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Dipartimento di Matematica, Universit`a di Padova, via Trieste 63, 35121 Padova, Italy
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