On the homology of algebras of Whitney functions over subanalytic sets

Jean-Paul Brasselet; Markus Pflaum

doi:10.4007/annals.2008.167.1

On the homology of algebras of Whitney functions over subanalytic sets

Annals of Mathematics ◽

10.4007/annals.2008.167.1 ◽

2008 ◽

Vol 167 (1) ◽

pp. 1-52 ◽

Cited By ~ 5

Author(s):

Jean-Paul Brasselet ◽

Markus Pflaum

Keyword(s):

Subanalytic Sets ◽

Whitney Functions

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Arc-smooth functions on closed sets

Compositio Mathematica ◽

10.1112/s0010437x19007097 ◽

2019 ◽

Vol 155 (4) ◽

pp. 645-680 ◽

Cited By ~ 2

Author(s):

Armin Rainer

Keyword(s):

Holomorphic Function ◽

Smooth Function ◽

Smooth Curve ◽

Smooth Functions ◽

Closed Sets ◽

Open Set ◽

Connected Set ◽

Subanalytic Sets ◽

Real Analytic ◽

Analytic Curves

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb{R}^{d}$ is smooth (${\mathcal{C}}^{\infty }$) if and only if it is arc-smooth, that is, $f\,\circ \,c$ is smooth for every smooth curve $c:\mathbb{R}\rightarrow U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X\subseteq \mathbb{R}^{d}$ is any such set and $f:X\rightarrow \mathbb{R}$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb{R}^{d}$. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if $f:X\rightarrow \mathbb{R}$ is the restriction of a smooth function on $\mathbb{R}^{d}$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb{C}^{d}$. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

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Spaces of Whitney Functions on Cantor-Type Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-2002-007-3 ◽

2002 ◽

Vol 54 (2) ◽

pp. 225-238 ◽

Cited By ~ 5

Author(s):

Bora Arslan ◽

Alexander P. Goncharov ◽

Mefharet Kocatepe

Keyword(s):

Extension Property ◽

Compact Set ◽

Compact Sets ◽

Whitney Functions ◽

Type Sets

AbstractWe introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces Ɛ(K) can be described for Cantor-type compact sets.

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