## On the Differential Structure of Metric Measure Spaces and Applications (Memoirs of the American Mathematical Society)

### Nicola Gigli

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The main goals of this paper are:(i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative.(ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where $g$ is a function and $\mu$ is a measure.(iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

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Nicola Gigli, University of Bordeaux 1, France.

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## 1.On the Differential Structure of Metric Measure Spaces and Applications (Paperback)

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Book Description American Mathematical Society, United States, 2015. Paperback. Condition: New. Language: English. Brand new Book. The main goals of this paper are:(i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative.(ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $Delta g=mu$, where $g$ is a function and $mu$ is a measure.(iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail. Seller Inventory # AAN9781470414207

US$85.48 Convert currency Shipping: FREE From United Kingdom to U.S.A. Destination, rates & speeds ## 2.On the Differential Structure of Metric Measure Spaces and Applications (Paperback) Published by American Mathematical Society, United States (2015) ISBN 10: 1470414201 ISBN 13: 9781470414207 New Paperback Quantity Available: 1 Seller: Book Depository International (London, United Kingdom) Rating Book Description American Mathematical Society, United States, 2015. Paperback. Condition: New. Language: English. Brand new Book. The main goals of this paper are:(i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative.(ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like$Delta g=mu$, where$g$is a function and$mu$is a measure.(iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail. Seller Inventory # AAN9781470414207 Buy New US$ 90.29
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## 5.ON THE DIFFERENTIAL STRUCTURE OF METRIC MEASURE SPACES AND APPLICATIONS (MEMO/236/1113)

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