TY - JOUR
T1 - Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter
AU - Magnabosco, Mattia
AU - Rigoni, Chiara
AU - Sosa, Gerardo
N1 - Funding Information:
The authors thank Professor Karl-Theodor Sturm for suggesting them the problem and for many valuable discussions. Many thanks are due to Lorenzo Dello Schiavo, for many enlightening conversations on measure theory. Finally, we thank the anonymous reviewer for their careful reading of our manuscript and their many insightful comments and suggestions. The second author gratefully acknowledges support by the European Union through the ERC-AdG 694405 RicciBounds. This research was partially funded by the Austrian Science Fund (FWF) [F65, ESP 224]. A large part of this work was written while the third author was employed in the group of Professor Karl-Theodor Sturm and furthermore he very gratefully acknowledges support by the European Union through the ERC-AdG 694405 RicciBounds. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
Funding Information:
The second author gratefully acknowledges support by the European Union through the ERC-AdG 694405 RicciBounds. This research was partially funded by the Austrian Science Fund (FWF) [ F65, ESP 224 ]. A large part of this work was written while the third author was employed in the group of Professor Karl-Theodor Sturm and furthermore he very gratefully acknowledges support by the European Union through the ERC-AdG 694405 RicciBounds. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
Publisher Copyright:
© 2023 The Author(s)
PY - 2023/12
Y1 - 2023/12
N2 - We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the CD(K,N) condition for N<0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance diKRW, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K,N) condition with N<0 is converging with respect to the distance diKRW to some metric measure space, then this limit structure is still a CD(K,N) space.
AB - We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the CD(K,N) condition for N<0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance diKRW, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K,N) condition with N<0 is converging with respect to the distance diKRW to some metric measure space, then this limit structure is still a CD(K,N) space.
KW - Convergence
KW - Curvature-dimension condition
KW - Metric measure space
KW - Negative dimension
KW - Stability result
UR - http://www.scopus.com/inward/record.url?scp=85170270863&partnerID=8YFLogxK
U2 - 10.1016/j.na.2023.113366
DO - 10.1016/j.na.2023.113366
M3 - Article
AN - SCOPUS:85170270863
VL - 237
JO - Nonlinear Analysis: Theory, Methods & Applications
JF - Nonlinear Analysis: Theory, Methods & Applications
SN - 0362-546X
M1 - 113366
ER -