liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Minimal weak upper gradients in 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)
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. Vol. 38, no 2, 727-745 p.
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-79165DOI: 10.5186/aasfm.2013.3831ISI: 000322091900020OAI: oai:DiVA.org:liu-79165DiVA: diva2:538661
Available from: 2012-07-01 Created: 2012-07-01 Last updated: 2017-12-07Bibliographically approved
In thesis
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
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
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-79166 (URN)978-91-7519-839-2 (ISBN)
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
2. 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.
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)
Public defence
2014-05-27, Nobel (BL32), B-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2014-04-16 Created: 2014-03-30 Last updated: 2016-05-04Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text

Authority records BETA

Malý, Lukáš

Search in DiVA

By author/editor
Malý, Lukáš
By organisation
Mathematics and Applied MathematicsThe Institute of Technology
In the same journal
Annales Academiae Scientiarum Fennicae Mathematica
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 224 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf