First-Order Calculus on Metric Measure Spaces

2020 ◽  
pp. 97-122
Author(s):  
Nicola Gigli ◽  
Enrico Pasqualetto
Keyword(s):  
Metric Measure Spaces ◽  
First Order ◽  
Measure Spaces

scholarly journals First order Poincaré inequalities in metric measure spaces

2013 ◽  
Vol 38 ◽  
pp. 287-308 ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Jesús A. Jaramillo ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Measure Spaces ◽  
Poincaré Inequalities ◽  
First Order ◽  
Poincare Inequalities ◽  
Measure Spaces

scholarly journals Density and Extension of Differentiable Functions on Metric Measure Spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 254-268
Author(s):  
Rafael Espínola García ◽  
Luis Sánchez González
Keyword(s):  
Banach Spaces ◽  
Metric Measure Spaces ◽  
Differentiable Structure ◽  
First Order ◽  
Differentiable Functions ◽  
Measure Spaces ◽  
The Given ◽  
Vector Valued

Abstract We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.

scholarly journals Newtonian Spaces Based on Quasi-Banach Function Lattices

2016 ◽  
Vol 119 (1) ◽  
pp. 133 ◽  
Author(s):  
Lukáš Malý
Keyword(s):  
Absolute Continuity ◽  
General Setting ◽  
Sobolev Type ◽  
Metric Measure Spaces ◽  
First Order ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Sobolev Capacity

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

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

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 Discretization of integrals on compact metric measure spaces

2021 ◽  
Vol 381 ◽  
pp. 107602
Author(s):  
Martin D. Buhmann ◽  
Feng Dai ◽  
Yeli Niu
Keyword(s):  
Metric Measure Spaces ◽  
Measure Spaces
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