Abstract
In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^2, \mathbb{R}^{D})$. If $\mathcal{H}$ has a section, then it has a semialgebraic section.
Funding Statement
The first author is supported by the Air Force Office of Scientific Research (AFOSR), under award FA9550-18-1-0069, the National Science Foundation (NSF), under grant DMS-1700180, and the US-Israel Binational Science Foundation (BSF), under grant 2014055. The second author is supported by NSF Grant DMS-1554733 and the UC Davis Chancellor's Fellowship.
Citation
Charles FEFFERMAN. Garving K. LULI. "$C^{m}$ semialgebraic sections over the plane." J. Math. Soc. Japan 74 (3) 759 - 811, July, 2022. https://doi.org/10.2969/jmsj/86258625
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