Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

Given X a Hilbert space, ω a modulus of continuity, E an arbitrary subset of X, and functions f:E→R, G:E→X, we provide necessary and sufficient conditions for the jet (f,G) to admit an extension (F,∇F) with F:X→R convex and of class C1,ω(X), by means of a simple explicit formula. As a consequence of this result, if ω is linear, we show that a variant of this formula provides explicit C1,1 extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes Cconv1,α(X).