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Newtonian Spaces Based on Quasi-Banach Function Lattices
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. (Nonlinear Potential Theory)
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted.

The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established.

The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2012. , 9 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1543
Keyword [en]
Newtonian space, upper gradient, weak upper gradient, Banach function lattice, quasi-normed space, metric measure space
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-79166ISBN: 978-91-7519-839-2 (print)OAI: oai:DiVA.org:liu-79166DiVA: diva2:538662
Presentation
2012-09-18, Alan Turing, E-huset, Campus Valla, Linköping University, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2012-08-13 Created: 2012-07-01 Last updated: 2016-05-04Bibliographically approved
List of papers
1. Newtonian spaces based on quasi-Banach function lattices
Open this publication in new window or tab >>Newtonian spaces based on quasi-Banach function lattices
2016 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, 133-160 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Mathematica Scandinavica, 2016
Keyword
Newtonian space, upper gradient, weak upper gradient, Sobolev-type space, Banach function lattice, quasi-normed space, metric measure space
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-79164 (URN)10.7146/math.scand.a-24188 (DOI)
Available from: 2012-07-01 Created: 2012-07-01 Last updated: 2017-12-07Bibliographically approved
2. Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices
Open this publication in new window or tab >>Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 2, 727-745 p.Article in journal (Refereed) Published
Abstract [en]

Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

Place, publisher, year, edition, pages
Suomalainen Tiedeakatemia, 2013
Keyword
Newtonian space, upper gradient, weak upper gradient, Banach function lattice, quasi-normed space, metric measure space
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-79165 (URN)10.5186/aasfm.2013.3831 (DOI)000322091900020 ()
Available from: 2012-07-01 Created: 2012-07-01 Last updated: 2017-12-07Bibliographically approved

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Malý, Lukáš

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