L2 Extension for jets of holomorphic sections of a Hermitian line Bundle

Let (X, ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ X a closed submanifold defined by some holomorphic section of a vector bundle over X, and L a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer k > 0, any section of the jet sheaf which satisfies a certain L2 condition, can be extended into a global holomorphic section of L over X whose L2 growth on an arbitrary compact subset of X is under control. In particular, if Y is merely a point, this gives the existence of a global holomorphic function with an L2 norm under control and with prescribed values for all its derivatives up to order k at that point. This result generalizes the L2 extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

Linear Isometries of Spaces of Absolutely Continuous Functions

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequalityimplies that([7], p. 221, 223).