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Newtonian spaces based on quasi-Banach function lattices
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. (Nonlinear Potential Theory)
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. Vol. 119, no 1, 133-160 p.
##### Keyword [en]
Newtonian space, upper gradient, weak upper gradient, Sobolev-type space, Banach function lattice, quasi-normed space, metric measure space
##### National Category
Mathematical Analysis
##### Identifiers
OAI: oai:DiVA.org:liu-79164DiVA: diva2:538660
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)
##### 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 $\mathcal{C}^1$-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.

##### 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)
##### 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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