We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties \begin{itemize} \item[-] $\mathrm{tr}(\mathrm{Hess}f)=\Delta f$ on $U\subset\X$ for every $f$ sufficiently regular, \item[-] $\mm=c\haus^n$ on $U\subset\X$ for some $c>0$, \end{itemize} where $U\subset\X$ is open and $\X$ is a - possibly collapsed - RCD space of essential dimension $n$.

Weakly non-collapsed RCD spaces are strongly non-collapsed / Brena, Camillo; Gigli, Nicola; Honda, Shouhei; Zhu, Xingyu. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 1435-5345. - 2023:794(2022), pp. 215-252. [10.1515/crelle-2022-0071]

Weakly non-collapsed RCD spaces are strongly non-collapsed

Brena, Camillo;Gigli, Nicola;
2022-01-01

Abstract

We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties \begin{itemize} \item[-] $\mathrm{tr}(\mathrm{Hess}f)=\Delta f$ on $U\subset\X$ for every $f$ sufficiently regular, \item[-] $\mm=c\haus^n$ on $U\subset\X$ for some $c>0$, \end{itemize} where $U\subset\X$ is open and $\X$ is a - possibly collapsed - RCD space of essential dimension $n$.
2022
2023
794
215
252
https://www.degruyter.com/document/doi/10.1515/crelle-2022-0071/pdf
Brena, Camillo; Gigli, Nicola; Honda, Shouhei; Zhu, Xingyu
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135495
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