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Sobolev-Type Spaces: Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. (Nonlinear Potential Theory)
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. , p. 22
##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1591
##### Keyword [en]
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
ISBN: 978-91-7519-354-0 (print)OAI: oai:DiVA.org:liu-105616DiVA, id: diva2:708831
##### 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
##### 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, p. 133-160Article 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
Institut for Matematik Aarhus Universitet, 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)000383815600008 ()
##### Note

Funding Agencies|NordForsk Research Network "Analysis and Applications" [080151]

Available from: 2012-07-01 Created: 2012-07-01 Last updated: 2018-02-23Bibliographically 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, p. 727-745Article 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
3. Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
Open this publication in new window or tab >>Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
(English)Manuscript (preprint) (Other academic)
##### 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
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
urn:nbn:se:liu:diva-105614 (URN)
Available from: 2014-03-30 Created: 2014-03-30 Last updated: 2014-04-16
4. Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces
Open this publication in new window or tab >>Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces
2016 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 32, no 1, p. 219-255Article in journal (Refereed) Published
##### 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, 2016
##### Keyword
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
##### Identifiers
urn:nbn:se:liu:diva-105615 (URN)10.4171/RMI/884 (DOI)000373379500006 ()2-s2.0-84960380046 (Scopus ID)
Available from: 2014-03-30 Created: 2014-03-30 Last updated: 2017-05-03Bibliographically approved

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Malý, Lukáš

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