*Submitted Paper*

**Inserted:** 8 apr 2023

**Last Updated:** 15 jul 2023

**Year:** 2023

**Abstract:**

In this note, our aim is to show that families of smooth hypersurfaces of ${\mathbb R}^{n+1}$ which are all $C^1$-close enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and "geometric" Calderón-Zygmund inequalities. This technical result is quite useful, in particular, in the analysis of the geometric flows of hypersurfaces, when one studies the behavior of the hypersurfaces close (in some norm, for instance in $C^1$-norm) to critical ones (possibly "stable") or the asymptotic limits of flows existing for all times.

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