TY - JOUR
T1 - Weakly non-collapsed RCD spaces are strongly non-collapsed
AU - Brena, Camillo
AU - Gigli, Nicola
AU - Honda, Shouhei
AU - Zhu, Xingyu
N1 - Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston 2023.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - 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 tr (Hess f) = Δ f on U ⊆ X for every f sufficiently regular, m = c H n on U ⊆ X for some c > 0, where U ⊆ X is open and X is a - possibly collapsed - RCD space of essential dimension n.
AB - 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 tr (Hess f) = Δ f on U ⊆ X for every f sufficiently regular, m = c H n on U ⊆ X for some c > 0, where U ⊆ X is open and X is a - possibly collapsed - RCD space of essential dimension n.
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U2 - 10.1515/crelle-2022-0071
DO - 10.1515/crelle-2022-0071
M3 - Article
AN - SCOPUS:85143728195
SN - 0075-4102
VL - 2023
SP - 215
EP - 252
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
IS - 794
ER -