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Abstract
We provide necessary and sufficient conditions for a 1jet (f,G):E→R×X to admit an extension (F,∇F) for some F∈C^{1,ω}(X). Here E stands for an arbitrary subset of a Hilbert space X and ω is a modulus of continuity. As a corollary, in the particular case X=R^{n}, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension n. Furthermore, we construct extensions (F,∇F) in such a way that: (1) the (nonlinear) operator (f,G)↦(F,∇F) is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) F is given by an explicit formula; (3) (F,∇F) depend continuously on the given data (f,G); (4) if f is bounded (resp. if G is bounded) then so is F (resp. F is Lipschitz). We also provide similar results on superreflexive Banach spaces.
Original language  English 

Article number  107928 
Journal  ADVANCES IN MATHEMATICS 
Volume  389 
DOIs  
Publication status  Published  8 Oct 2021 
MoE publication type  A1 Journal articlerefereed 
Keywords
 C functions
 Hilbert space
 Whitney extension theorem
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Geometric Analysis and Conformal Structures
Astala, K., Prause, I., Brustad, K., Mudarra DíazMalaguilla, C., Vavilov, A., Evdoridis, S. & Hitruhin, L.
01/09/2018 → 30/04/2021
Project: Academy of Finland: Other research funding