On real analytic functions on closed subanalytic domains

TL;DR

The paper establishes conditions under which a function on a specific geometric domain is real analytic.

Contribution

It provides a novel characterization of real analyticity using polynomial curve composites and boundary regularity.

Findings

A function is real analytic if composites with polynomial curves of degree 2 suffice when the boundary is Lipschitz.

Smoothness is not required for analyticity in the Lipschitz case when using quadratic polynomial maps.

Abstract

We show that a function \documentclass[12pt]{minimal}
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\begin{document}$$f: X \rightarrow {\mathbb {R}}$$\end{document}f:X→R defined on a closed uniformly polynomially cuspidal set X in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document}Rn is real analytic if and only if f is smooth and all its composites with germs of polynomial curves in X are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of X. For instance, if the boundary of X is locally Lipschitz, then polynomial curves of degree 2 suffice. In this Lipschitz case, we also prove that a function \documentclass[12pt]{minimal}
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\begin{document}$$f: X \rightarrow {\mathbb {R}}$$\end{document}f:X→R is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in X are real analytic; here it is not necessary to assume that f is smooth.