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Extension of \(C^{m, \omega}\)-smooth functions by linear operators. (English) Zbl 1173.46014
Let \(C^{m,\omega}(\mathbb{R}^n)\) be the Banach space of functions on \(\mathbb{R}^n\) whose \(m\)th derivatives have modulus of continuity majorised by a nondecreasing concave function \(\omega:\mathbb{R}_+\to \mathbb{R}_+\) at zero. For \(E\subset\mathbb{R}^n\), let \(C^{m, \omega}(E)\) be the space of all restrictions to \(E\) of functions in \(C^{m,\omega}(\mathbb{R}^n)\). The main result is as follows.
Theorem. There exists a linear extension operator \(T:C^{m,\omega}(E)\to C^{m,\omega} (\mathbb{R}^n)\) whose norm is bounded by a constant depending only on \(n,m\).
The author also proves a generalization of this result related to the notion of Whitney convexity. This, in turn, allows to simplify the proof of his extremely difficult result from [Ann. Math. 166, No. 3, 779–835 (2007; Zbl 1161.46013)], on \(C^m\) extension by linear operators.

MSC:
46E15 Banach spaces of continuous, differentiable or analytic functions
49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
52A35 Helly-type theorems and geometric transversal theory
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