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Extension operators for real analytic functions on compact subvarieties of \(\mathbb R^d\). (English) Zbl 1133.46014

In this profound paper, the author characterizes those compact coherent real analytic subvarieties \(X\subset \mathbb{R}^d\) admitting a continuous linear extension operator from the real analytic functions on \(X\) to the the real analytic functions on \(\mathbb{R}^d\) by the fact that \(X\) is of type \((PL)\), that is, in every point of \(X\) the local complexification of \(X\) satisfies Hörmander’s local Phragmen–Lindelöf condition. Notice that this condition is purely local. The proof is reduced to the question if there is an extension operator from \(A(X)\) into \(A(D)\) if \(X\) is contained in the closed Euclidean ball \(D\subset \mathbb{R}^d\). In this case, the sufficiency of the above condition is proved via the tame splitting theorem of M.Poppenberg and D.Vogt [Math.Z.219, 141–161 (1995; Zbl 0823.46002)], while the necessity part of the proof uses an interpolation lemma of \((\underline{DN})\)-\((\overline{\Omega})\)-type [see R.Meise and D.Vogt, “Introduction to functional analysis” (Oxford Graduate Texts in Mathematics 2, Oxford:Clarendon Press) (1997; Zbl 0924.46002)].
Several interesting examples are treated in the spirit of the paper. If \(X\) is the variety of an irreducible homogeneous polynomial \(P\) satisfying the strong dimension condition, then \(X\) is of type \((PL)\) iff \(P(D)\) is surjective in \(A(\mathbb{R}^d)\) iff \(P(D)\) admits a continuous linear right inverse in \(C^{\infty}(\mathbb{R}^d)\).

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
32C05 Real-analytic manifolds, real-analytic spaces
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
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