In Example 1, we describe a subset X of the plane and a function on X which has a -extension to the whole for each finite, but has no -extension to . In Example 2, we construct a similar example of a subanalytic subset of ; much more sophisticated than the first one. The dimensions given here are smallest possible.
A continuous linear extension operator, different from Whitney’s, for -Whitney fields (p finite) on a closed o-minimal subset of is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.
A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any...
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
For any subanalytic -Whitney field (k finite), we construct its subanalytic -extension to . Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.
A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.
This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local...
A local-global version of the implicit function theorem in o-minimal structures and a generalization of the theorem of Wilkie on covering open sets by open cells are proven.
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