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Sobolev spaces.

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This thesis by Lukáš Malý presents a study of Newtonian spaces based on quasi-Banach function lattices. It includes two papers that explore the generalization of Sobolev spaces in abstract metric measure spaces. The work discusses weak derivatives, weak upper gradients, and the techniques available for analyzing these spaces. It also covers the absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces. The thesis aims to broaden the theory of Newtonian spaces, making it applicable to more general metric spaces.

Sobolev spaces. --- Metric spaces.

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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.

Differential calculus. --- Sobolev spaces.

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Sobolev spaces --- Sobolev, Espaces de

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Besov spaces. --- Hardy spaces. --- Sobolev spaces.