Łojasiewicz inequality at singular points
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- by Anna Valette PDF
- Proc. Amer. Math. Soc. 147 (2019), 1109-1117 Request permission
Abstract:
We prove a generalized version of Łojasiewicz’s inequality. The famous Łojasiewicz inequality asserts that if $f$ is a ${\mathcal C}^1$ globally subanalytic function in a neighborhood of a point $a \in \mathbb {R}^n$, there is a neighborhood $U$ of $a$ and a rational number $\theta \in [0,1)$ as well as a constant $C$ such that $|f(x)-f(a)|^\theta \le C |\nabla _x f|$ for $x\in U$ (where $\nabla _x f$ stands for the gradient of the function $f$ at $x$). We give an inequality of the same type that applies to the case where $a$ is not an interior point of the domain of $f$.References
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Additional Information
- Anna Valette
- Affiliation: Instytut Matematyki Uniwersytetu Jagiellońskiego, ul. S. Łojasiewicza, Kraków, Poland
- MR Author ID: 693690
- Email: anna.valette@im.uj.edu.pl
- Received by editor(s): September 22, 2017
- Received by editor(s) in revised form: June 4, 2018
- Published electronically: November 16, 2018
- Additional Notes: This research was partially supported by the NCN grant 2014/13/B/ST1/00543.
- Communicated by: Michael Wolf
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1109-1117
- MSC (2010): Primary 32B20, 58K05, 14P10
- DOI: https://doi.org/10.1090/proc/14329
- MathSciNet review: 3896060