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Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. (Nonlinear Potential Theory)
Abstract [en]

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a p-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on rearrangement-invariant spaces are established and applied.

Keyword [en]
Newtonian space, Sobolev-type space, metric measure space, upper gradient, Banach function lattice, rearrangement-invariant space, maximal operator, Lipschitz function, regularization, weak boundedness, density of Lipschitz functions
National Category
Mathematical Analysis
Identifiers
OAI: oai:DiVA.org:liu-105614DiVA: diva2:708829
Available from: 2014-03-30 Created: 2014-03-30 Last updated: 2014-04-16
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 $\mathcal{C}^1$-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.

Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1591
Keyword
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
Identifiers
urn:nbn:se:liu:diva-105616 (URN)10.3384/diss.diva-105616 (DOI)978-91-7519-354-0 (ISBN)
Supervisors
Available from: 2014-04-16 Created: 2014-03-30 Last updated: 2016-05-04Bibliographically approved

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Malý, Lukáš

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