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Journal articles on the topic 'Sobolev spaces'

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Published: 4 June 2021
Last updated: 30 July 2024

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1

Ichinose, Takashi, and Yoshimi Saito. "Dirac–Sobolev Spaces and Sobolev Spaces." Funkcialaj Ekvacioj 53, no. 2 (2010): 291–310. http://dx.doi.org/10.1619/fesi.53.291.

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2

Pustylnik, Evgeniy. "Sobolev type inequalities in ultrasymmetric spaces with applications to Orlicz-Sobolev embeddings." Journal of Function Spaces and Applications 3, no. 2 (2005): 183–208. http://dx.doi.org/10.1155/2005/254184.

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LetDkfmean the vector composed by all partial derivatives of orderkof a functionf(x),x∈Ω⊂ℝn. Given a Banach function spaceA, we look for a possibly small spaceBsuch that‖f‖B≤c‖|Dkf|‖Afor allf∈C0k(Ω). The estimates obtained are applied to ultrasymmetric spacesA=Lφ,E,B=Lψ,E, giving some optimal (or rather sharp) relations between parameter-functionsφ(t)andψ(t)and new results for embeddings of Orlicz-Sobolev spaces.
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3

Evans, W. D. "SOBOLEV SPACES." Bulletin of the London Mathematical Society 19, no. 1 (January 1987): 95–96. http://dx.doi.org/10.1112/blms/19.1.95.

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4

Johnson, Raymond. "Sobolev spaces." Acta Applicandae Mathematicae 8, no. 2 (February 1987): 199–205. http://dx.doi.org/10.1007/bf00046713.

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5

OHNO, TAKAO, and TETSU SHIMOMURA. "SOBOLEV INEQUALITIES FOR RIESZ POTENTIALS OF FUNCTIONS IN OVER NONDOUBLING MEASURE SPACES." Bulletin of the Australian Mathematical Society 93, no. 1 (November 12, 2015): 128–36. http://dx.doi.org/10.1017/s0004972715001331.

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6

Chen, Shutao, Changying Hu, and Charles Xuejin Zhao. "Extreme points and rotundity of Orlicz-Sobolev spaces." International Journal of Mathematics and Mathematical Sciences 32, no. 5 (2002): 293–99. http://dx.doi.org/10.1155/s0161171202202252.

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It is well known that Sobolev spaces have played essential roles in solving nonlinear partial differential equations. Orlicz-Sobolev spaces are generalized from Sobolev spaces. In this paper, we present sufficient and necessary conditions of extreme points of Orlicz-Sobolev spaces. A sufficient and necessary condition of rotundity of Orlicz-Sobolev spaces is obtained.
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7

Yang, Dachun, and Yong Lin. "SPACES OF LIPSCHITZ TYPE ON METRIC SPACES AND THEIR APPLICATIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 3 (October 2004): 709–52. http://dx.doi.org/10.1017/s0013091503000907.

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AbstractNew spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99
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8

Gröger, Konrad, and Lutz Recke. "Preduals of Sobolev-Campanato spaces." Mathematica Bohemica 126, no. 2 (2001): 403–10. http://dx.doi.org/10.21136/mb.2001.134016.

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9

Gaczkowski, Michał, and Przemysław Górka. "Variable Hajłasz-Sobolev spaces on compact metric spaces." Mathematica Slovaca 67, no. 1 (February 1, 2017): 199–208. http://dx.doi.org/10.1515/ms-2016-0258.

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Abstract We study variable exponent Sobolev spaces on compact metric spaces. Without the assumption of log-Hölder continuity of the exponent, the compact Sobolev-type embeddings theorems for these spaces are shown.
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10

Evans, W. D. "WEIGHTED SOBOLEV SPACES." Bulletin of the London Mathematical Society 18, no. 2 (March 1986): 220–21. http://dx.doi.org/10.1112/blms/18.2.220.

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11

Zhikov, V. V. "Weighted Sobolev spaces." Sbornik: Mathematics 189, no. 8 (August 31, 1998): 1139–70. http://dx.doi.org/10.1070/sm1998v189n08abeh000344.

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12

Simas, Alexandre B., and Fábio J. Valentim. "W-Sobolev spaces." Journal of Mathematical Analysis and Applications 382, no. 1 (October 2011): 214–30. http://dx.doi.org/10.1016/j.jmaa.2011.04.043.

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13

Hassib, Moulay Cherif, and Youssef Akdim. "Weighted Variable Exponent Sobolev spaces on metric measure spaces." Moroccan Journal of Pure and Applied Analysis 4, no. 2 (December 1, 2018): 62–76. http://dx.doi.org/10.1515/mjpaa-2018-0007.

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AbstractIn this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.
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14

Menovschikov, Alexander, and Alexander Ukhlov. "Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings." Ukrainian Mathematical Bulletin 18, no. 2 (July 9, 2021): 209–25. http://dx.doi.org/10.37069/1810-3200-2021-18-2-5.

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In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with $\BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $\BMO$-spaces, we prove that $\BMO$-quasiconformal mappings generate bounded composition operators from Hardy--Sobolev spaces to Sobolev spaces.
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15

CHO, HONG RAE, and SOOHYUN PARK. "FRACTIONAL FOCK–SOBOLEV SPACES." Nagoya Mathematical Journal 237 (March 6, 2018): 79–97. http://dx.doi.org/10.1017/nmj.2018.11.

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Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$. The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.
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16

Fabec, R. C. "Multipliers Between Sobolev Spaces." Canadian Mathematical Bulletin 34, no. 4 (December 1, 1991): 465–73. http://dx.doi.org/10.4153/cmb-1991-075-7.

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AbstractA sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.
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17

Evans, Mogoi N., Samuel B. Apima, and Amos Otieno Wanjara. "Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications." Advances in Research 25, no. 3 (April 10, 2024): 148–53. http://dx.doi.org/10.9734/air/2024/v25i31060.

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This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.
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18

Nguyen, Hoai-Minh, and Marco Squassina. "On anisotropic Sobolev spaces." Communications in Contemporary Mathematics 21, no. 01 (January 28, 2019): 1850017. http://dx.doi.org/10.1142/s0219199718500177.

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We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain–Brezis–Mironescu formula, including the magnetic case both for Sobolev and BV functions.
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19

Mikhailets, Vladimir A., and Aleksandr A. Murach. "Interpolation Hilbert Spaces Between Sobolev Spaces." Results in Mathematics 67, no. 1-2 (July 11, 2014): 135–52. http://dx.doi.org/10.1007/s00025-014-0399-x.

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20

Ambrosio, Luigi, Andrea Pinamonti, and Gareth Speight. "Weighted Sobolev spaces on metric measure spaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 746 (January 1, 2019): 39–65. http://dx.doi.org/10.1515/crelle-2016-0009.

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Abstract We investigate weighted Sobolev spaces on metric measure spaces {(X,\mathrm{d},\mathfrak{m})} . Denoting by ρ the weight function, we compare the space {W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} (which always coincides with the closure {H^{1,p}(X,\mathrm{d},\rho\mathfrak{m})} of Lipschitz functions) with the weighted Sobolev spaces {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} and {H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})} defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that {W^{1,p}(X,\mathrm{d},\rho\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{% m})} . We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality {W^{1,p}_{\rho}(X,\mathrm{d},\mathfrak{m})=H^{1,p}_{\rho}(X,\mathrm{d},% \mathfrak{m})} .
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21

GOL’DSHTEIN, VLADIMIR, EVGENY SEVOST’YANOV, and ALEXANDER UKHLOV. "COMPOSITION OPERATORS ON SOBOLEV SPACES AND." Mathematical Reports 26(76), no. 2 (June 15, 2024): 101–13. http://dx.doi.org/10.59277/mrar.2024.26.76.2.101.

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In this paper, we study connections between composition operators on Sobolev spaces and mappings defined by p-moduli inequalities (p-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on corresponding Sobolev spaces and conversely, that composition operators on Sobolev spaces imply weighted moduli inequalities.
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22

Strichartz, Robert S. "$H^p$ Sobolev spaces." Colloquium Mathematicum 60, no. 1 (1990): 129–39. http://dx.doi.org/10.4064/cm-60-61-1-129-139.

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23

Kerman, Ron, and Luboš Pick. "Optimal Sobolev imbedding spaces." Studia Mathematica 192, no. 3 (2009): 195–217. http://dx.doi.org/10.4064/sm192-3-1.

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24

Ostrovskii, M. I. "Sobolev spaces on graphs." Quaestiones Mathematicae 28, no. 4 (December 2005): 501–23. http://dx.doi.org/10.2989/16073600509486144.

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25

Marcos, Miguel Andres. "On Newton--Sobolev spaces." Publicationes Mathematicae Debrecen 90, no. 1-2 (January 1, 2017): 107–24. http://dx.doi.org/10.5486/pmd.2017.7486.

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26

Hedberg, Lars Inge. "Book Review: Sobolev spaces." Bulletin of the American Mathematical Society 18, no. 1 (January 1, 1988): 87–89. http://dx.doi.org/10.1090/s0273-0979-1988-15613-3.

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27

Ludwig, Monika. "Valuations on Sobolev spaces." American Journal of Mathematics 134, no. 3 (2012): 824–42. http://dx.doi.org/10.1353/ajm.2012.0019.

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28

Brezis, Haïm, YanYan Li, Petru Mironescu, and Louis Nirenberg. "Degree and Sobolev spaces." Topological Methods in Nonlinear Analysis 13, no. 2 (June 1, 1999): 181. http://dx.doi.org/10.12775/tmna.1999.009.

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29

Brezis, Haı̈m, and YanYan Li. "Topology and Sobolev spaces." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331, no. 5 (September 2000): 365–70. http://dx.doi.org/10.1016/s0764-4442(00)01656-6.

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30

Bourgain, Jean, Haim Brezis, and Petru Mironescu. "Lifting in Sobolev spaces." Journal d'Analyse Mathématique 80, no. 1 (December 2000): 37–86. http://dx.doi.org/10.1007/bf02791533.

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31

Salomonsen, Gorm. "Equivalence of Sobolev Spaces." Results in Mathematics 39, no. 1-2 (March 2001): 115–30. http://dx.doi.org/10.1007/bf03322679.

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32

Chua, Seng-Kee. "On Weighted Sobolev Spaces." Canadian Journal of Mathematics 48, no. 3 (June 1, 1996): 527–41. http://dx.doi.org/10.4153/cjm-1996-027-5.

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AbstractWe study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains𝓓when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the MuckenhouptApclass locally in 𝓓. Moreover, when the weightswi(x) are of the form dist(x,Mi)αi,αi∈ ℝ,Mi⊂ 𝓓that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.
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33

Brezis, Haim, and Yanyan Li. "Topology and Sobolev Spaces." Journal of Functional Analysis 183, no. 2 (July 2001): 321–69. http://dx.doi.org/10.1006/jfan.2000.3736.

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34

Biegert, Markus, and Robin Nittka. "Isometries between Sobolev spaces." Mathematische Nachrichten 285, no. 17-18 (June 22, 2011): 2059–77. http://dx.doi.org/10.1002/mana.201190020.

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35

HÄSTÖ, PETER, YOSHIHIRO MIZUTA, TAKAO OHNO, and TETSU SHIMOMURA. "SOBOLEV INEQUALITIES FOR ORLICZ SPACES OF TWO VARIABLE EXPONENTS." Glasgow Mathematical Journal 52, no. 2 (November 25, 2009): 227–40. http://dx.doi.org/10.1017/s0017089509990292.

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AbstractOur aim in this paper is to deal with Sobolev's embeddings for Sobolev–Orlicz functions with ∇u ∈ Lp(·) logLq(·)(Ω) for Ω ⊂ n. Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.
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36

Behzadan, Ali, and Michael Holst. "Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited." Mathematics 10, no. 3 (February 7, 2022): 522. http://dx.doi.org/10.3390/math10030522.

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In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.
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37

Eryilmaz, İlker. "Sobolev type spaces based on Lorentz-Karamata spaces." Filomat 30, no. 11 (2016): 3023–32. http://dx.doi.org/10.2298/fil1611023e.

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In this paper, firstly Lorentz-Karamata-Sobolev spaces Wk,(p,q,b) (Rn) of integer order are introduced and some of their important properties are emphasized. Also, Banach spaces Ak,L(p,q,b)(Rn) = L1(Rn)? Wk,L(p,q,b)(Rn) (Lorentz-Karamata-Sobolev algebras) are studied. Using a result of H.C.Wang, it is showed that Banach convolution algebras AkL(p,q,b)(Rn) don?t have weak factorization and the multiplier algebra of Ak,L(p,q,b)(Rn) coincides with the measure algebra M(Rn) for 1 < p < 1 and 1 ? q < 1.
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38

Guo, Benqi, and Ivo Babuška. "Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 1 (1997): 77–125. http://dx.doi.org/10.1017/s0308210500023520.

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This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.
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39

Bojarski, Bogdan, Juha Kinnunen, and Thomas Zürcher. "Higher Order Sobolev-Type Spaces on the Real Line." Journal of Function Spaces 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/261565.

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This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.
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40

Pavlov, S. V. "Grand Sobolev Spaces on Metric Measure Spaces." Siberian Mathematical Journal 63, no. 5 (September 2022): 956–66. http://dx.doi.org/10.1134/s0037446622050147.

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41

Ahern, Patrick, and William Cohn. "Besov spaces, Sobolev spaces, and Cauchy integrals." Michigan Mathematical Journal 39, no. 2 (1992): 239–61. http://dx.doi.org/10.1307/mmj/1029004520.

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42

Lu, Yufeng, Dachun Yang, and Wen Yuan. "Morrey-Sobolev Spaces on Metric Measure Spaces." Potential Analysis 41, no. 1 (September 11, 2013): 215–43. http://dx.doi.org/10.1007/s11118-013-9370-9.

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43

Abdulla, Ugur G. "Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces." AIMS Mathematics 8, no. 9 (2023): 20700–20717. http://dx.doi.org/10.3934/math.20231055.

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<abstract><p>It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.</p></abstract>
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44

Evans, Mogoi N., and Samuel B. Apima. "Interplay of Sobolev Spaces on Compact Manifolds: Embedding Theorems, Inequalities, and Compactness." Advances in Research 25, no. 1 (January 8, 2024): 21–25. http://dx.doi.org/10.9734/air/2024/v25i11014.

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This research paper explores various properties of Sobolev spaces on compact manifolds, focusing on embedding theorems, compactness, and inequalities. We establish the compact embedding of Sobolev spaces into continuous and Lebesgue spaces, as well as the continuity and compactness of embeddings between different Sobolev spaces. We also derive inequalities involving the Laplacian and gradients of functions, providing insights into their behavior on manifolds. These results contribute to our understanding of the interplay between function smoothness, continuity, and distribution on compact manifolds.
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45

Hardy, Grahame. "Demicontinuity of Nemitsky operators on Orlicz-Sobolev spaces." Bulletin of the Australian Mathematical Society 37, no. 1 (February 1988): 29–42. http://dx.doi.org/10.1017/s0004972700004123.

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46

Bonk, Mario, and Eero Saksman. "Sobolev spaces and hyperbolic fillings." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 737 (April 1, 2018): 161–87. http://dx.doi.org/10.1515/crelle-2015-0036.

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AbstractLetZbe an AhlforsQ-regular compact metric measure space, where{Q>0}. For{p>1}we introduce a new (fractional) Sobolev space{A^{p}(Z)}consisting of functions whose extensions to the hyperbolic filling ofZsatisfy a weak-type gradient condition. IfZsupports aQ-Poincaré inequality with{Q>1}, then{A^{Q}(Z)}coincides with the familiar (homogeneous) Hajłasz–Sobolev space.
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47

Krejčí, Pavel, and Vladimír Lovicar. "Continuity of hysteresis operators in Sobolev spaces." Applications of Mathematics 35, no. 1 (1990): 60–66. http://dx.doi.org/10.21136/am.1990.104387.

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48

CHIRON, DAVID. "ON THE DEFINITIONS OF SOBOLEV AND BV SPACES INTO SINGULAR SPACES AND THE TRACE PROBLEM." Communications in Contemporary Mathematics 09, no. 04 (August 2007): 473–513. http://dx.doi.org/10.1142/s0219199707002502.

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The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space Ws,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the [Formula: see text] regularity of traces of maps in Ws,p (0 < s ≤ 1 < sp).
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49

Yang, Dachun. "Some Function Spaces Relative to Morrey-Campanato Spaces on Metric Spaces." Nagoya Mathematical Journal 177 (2005): 1–29. http://dx.doi.org/10.1017/s002776300000903x.

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In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.
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50

Behzadan, A., and M. Holst. "On the Space of Locally Sobolev-Slobodeckij Functions." Journal of Function Spaces 2022 (July 18, 2022): 1–30. http://dx.doi.org/10.1155/2022/9094502.

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The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature.
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