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Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. (Nonlinear Potential Theory)
(English)In: Revista matemática iberoamericana, ISSN 0213-2230Article in journal (Refereed) Accepted
Abstract [en]

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (Hölder) continuous. Particular focus is put on the borderline case when the degree of integrability equals the “dimension of the measure”. If Lipschitz functions are dense in a Newtonian space on a proper metric space, then locally Lipschitz functions are proven dense in the corresponding Newtonian space on open subsets, where no hypotheses (besides being open) are put on these sets.

Place, publisher, year, edition, pages
Zürich: European Mathematical Society Publishing House.
Keyword [en]
Newtonian space, Sobolev-type space, metric measure space, Banach function lattice, Sobolev capacity, quasi-continuity, outer capacity, locally Lipschitz function, continuity, doubling measure, Poincaré inequality
National Category
Mathematical Analysis
URN: urn:nbn:se:liu:diva-105615OAI: diva2:708830
Available from: 2014-03-30 Created: 2014-03-30 Last updated: 2014-09-19
In thesis
1. Sobolev-Type Spaces: Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces
Open this publication in new window or tab >>Sobolev-Type Spaces: Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s.

In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting.

Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied.

Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike -smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well.

Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 22 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1591
Newtonian space, Sobolev-type space, metric measure space, upper gradient, Sobolev capacity, Banach function lattice, quasi-normed space, rearrangement-invariant space, maximal operator, Lipschitz function, regularization, weak boundedness, density of Lipschitz functions, quasi-continuity, continuity, doubling measure, Poincaré inequality
National Category
Mathematical Analysis
urn:nbn:se:liu:diva-105616 (URN)10.3384/diss.diva-105616 (DOI)978-91-7519-354-0 (print) (ISBN)
Public defence
2014-05-27, Nobel (BL32), B-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Available from: 2014-04-16 Created: 2014-03-30 Last updated: 2016-05-04Bibliographically approved

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