scholarly journals Characterizations of monotonicity of vector fields on metric measure spaces

2018 ◽  
Vol 57 (5) ◽  
Author(s):  
Bang-Xian Han
Keyword(s):  
Vector Fields ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abimbola Abolarinwa ◽  
Akram Ali ◽  
Ali Alkhadi
Keyword(s):  
Lower Bound ◽  
Vector Fields ◽  
First Eigenvalue ◽  
Metric Measure Spaces ◽  
Cheeger Constant ◽  
Weighted Mean Curvature ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Geometric Constant ◽  
Weighted Manifolds

AbstractWe establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

scholarly journals Lagrangian calculus for nonsymmetric diffusion operators

2020 ◽  
Vol 13 (4) ◽  
pp. 361-383
Author(s):  
Christian Ketterer
Keyword(s):  
Ricci Curvature ◽  
Vector Fields ◽  
Probabilistic Approach ◽  
Gradient Estimates ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Maximal Diameter ◽  
Measure Spaces ◽  
Evolution Variational Inequality ◽  
Dual Semigroup

AbstractWe characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the {L^{2}}-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

scholarly journals Regularity of Lagrangian flows over RCD*(K, N) spaces

2020 ◽  
Vol 2020 (765) ◽  
pp. 171-203 ◽  
Author(s):  
Elia Brué ◽  
Daniele Semola
Keyword(s):  
Vector Fields ◽  
Additional Assumption ◽  
Natural Extension ◽  
Regularity Result ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Regularity Results ◽  
Symmetric Derivative

AbstractThe aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

2020 ◽  
Vol 126 (2) ◽  
pp. 259-275
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Vector Fields ◽  
Integral Functional ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Regular Measure ◽  
Local Boundedness ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Bounded Open Subset ◽  
Hörmander’S Condition

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

Jump processes and nonlinear fractional heat equations on metric measure spaces

2006 ◽  
Vol 279 (1-2) ◽  
pp. 150-163 ◽  
Author(s):  
Jiaxin Hu ◽  
Martina Zähle
Keyword(s):  
Jump Processes ◽  
Heat Equations ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals A note on the extension of BV functions in metric measure spaces

2008 ◽  
Vol 340 (1) ◽  
pp. 197-208 ◽  
Author(s):  
Annalisa Baldi ◽  
Francescopaolo Montefalcone
Keyword(s):  
Metric Measure Spaces ◽  
Bv Functions ◽  
Measure Spaces

scholarly journals Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

2014 ◽  
Vol 7 (5) ◽  
pp. 1179-1234 ◽  
Author(s):  
Luigi Ambrosio ◽  
Dario Trevisan
Keyword(s):  
Well Posedness ◽  
Metric Measure Spaces ◽  
Continuity Equations ◽  
Measure Spaces
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