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 -