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Journal articles on the topic 'Hölder and Sobolev spaces'

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Published: 4 June 2021

Last updated: 2 February 2022

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1

Gaczkowski, Michał, and Przemysław Górka. "Variable Hajłasz-Sobolev spaces on compact metric spaces." Mathematica Slovaca 67, no. 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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2

Björn, Anders, Jana Björn, and Nageswari Shanmugalingam. "Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces." Canadian Journal of Mathematics 59, no. 6 (2007): 1135–53. http://dx.doi.org/10.4153/cjm-2007-049-7.

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AbstractWe study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

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3

Lappalainen, Vesa, and Ari Lehtonen. "Embedding of Orlicz-Sobolev spaces in Hölder spaces." Annales Academiae Scientiarum Fennicae Series A I Mathematica 14 (1989): 41–46. http://dx.doi.org/10.5186/aasfm.1989.1417.

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4

BUCKLEY, STEPHEN M., and ALEXANDER STANOYEVITCH. "WEAK SLICE CONDITIONS AND HÖLDER IMBEDDINGS." Journal of the London Mathematical Society 64, no. 3 (2001): 690–706. http://dx.doi.org/10.1112/s0024610701002654.

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5

Harjulehto, Petteri, Juha Kinnunen, and Katja Tuhkanen. "Hölder Quasicontinuity in Variable Exponent Sobolev Spaces." Journal of Inequalities and Applications 2007 (2007): 1–19. http://dx.doi.org/10.1155/2007/32324.

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6

Heikkinen, Toni, Juha Lehrbäck, Juho Nuutinen, and Heli Tuominen. "Fractional Maximal Functions in Metric Measure Spaces." Analysis and Geometry in Metric Spaces 1 (May 28, 2013): 147–62. http://dx.doi.org/10.2478/agms-2013-0002.

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Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

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7

Baladi, Viviane, and Masato Tsujii. "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms." Annales de l’institut Fourier 57, no. 1 (2007): 127–54. http://dx.doi.org/10.5802/aif.2253.

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8

de Faria, Edson, Peter Hazard, and Charles Tresser. "Infinite entropy is generic in Hölder and Sobolev spaces." Comptes Rendus Mathematique 355, no. 11 (2017): 1185–89. http://dx.doi.org/10.1016/j.crma.2017.10.016.

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9

Bojarski, Bogdan, Piotr Hajłasz, and Paweł Strzelecki. "Sard's theorem for mappings in Hölder and Sobolev spaces." manuscripta mathematica 118, no. 3 (2005): 383–97. http://dx.doi.org/10.1007/s00229-005-0590-1.

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10

Almeida, Alexandre, and Stefan Samko. "Embeddings of variable Hajłasz–Sobolev spaces into Hölder spaces of variable order." Journal of Mathematical Analysis and Applications 353, no. 2 (2009): 489–96. http://dx.doi.org/10.1016/j.jmaa.2008.12.034.

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11

Ibrahim, H. "A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class." Journal of Function Spaces and Applications 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/148706.

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In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

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12

Dao, Nguyen Anh, Jesus Ildefonso Díaz, and Quoc-Hung Nguyen. "Generalized Gagliardo–Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces." Nonlinear Analysis 173 (August 2018): 146–53. http://dx.doi.org/10.1016/j.na.2018.04.001.

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13

El Baraka, A., and M. Masrour. "Regularity results for solutions of linear elliptic degenerate boundary-value problems." Arabian Journal of Mathematics 9, no. 3 (2020): 545–66. http://dx.doi.org/10.1007/s40065-020-00278-x.

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Abstract We give an a-priori estimate near the boundary for solutions of a class of higher order degenerate elliptic problems in the general Besov-type spaces $$B^{s,\tau }_{p,q}$$ B p , q s , τ . This paper extends the results found in Hölder spaces $$C^s$$ C s , Sobolev spaces $$H^s$$ H s and Besov spaces $$B^s_{p,q}$$ B p , q s , to the more general framework of Besov-type spaces.

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14

Cianchi, Andrea, and Luboš Pick. "Sobolev embeddings into spaces of Campanato, Morrey, and Hölder type." Journal of Mathematical Analysis and Applications 282, no. 1 (2003): 128–50. http://dx.doi.org/10.1016/s0022-247x(03)00110-0.

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15

Lytchak, Alexander, Stefan Wenger, and Robert Young. "Dehn functions and Hölder extensions in asymptotic cones." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 763 (2020): 79–109. http://dx.doi.org/10.1515/crelle-2018-0041.

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AbstractThe Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones.

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16

Seuret, S., and F. Vigneron. "MULTIFRACTAL ANALYSIS OF FUNCTIONS ON HEISENBERG AND CARNOT GROUPS." Journal of the Institute of Mathematics of Jussieu 16, no. 1 (2015): 1–38. http://dx.doi.org/10.1017/s1474748015000092.

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In this article, we investigate the pointwise behaviors of functions on the Heisenberg group. We find wavelet characterizations for the global and local Hölder exponents. Then we prove some a priori upper bounds for the multifractal spectrum of all functions in a given Hölder, Sobolev, or Besov space. These upper bounds turn out to be optimal, since in all cases they are reached by typical functions in the corresponding functional spaces. We also explain how to adapt our proof to extend our results to Carnot groups.

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17

Ben Slimane, Mourad Ben, Moez Ben Ben Abid, Ines Ben Omrane, and Borhen Halouani. "Directional Thermodynamic Formalism." Symmetry 11, no. 6 (2019): 825. http://dx.doi.org/10.3390/sym11060825.

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The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly

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18

DURÁN, RICARDO G., та FERNANDO LÓPEZ GARCÍA. "SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS". Mathematical Models and Methods in Applied Sciences 20, № 01 (2010): 95–120. http://dx.doi.org/10.1142/s0218202510004167.

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If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev s

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19

Duduchava, R., and F. O. Speck. "Singular Integral Equations in Special Weighted Spaces." Georgian Mathematical Journal 7, no. 4 (2000): 633–42. http://dx.doi.org/10.1515/gmj.2000.633.

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Abstract We prove the boundedness of the Cauchy singular integral operator in special weighted Sobolev and Hölder-Zygmund spaces for large values of the smoothness parameter, which is an integer m ≥ 0, when the underlying contour is piecewise-smooth with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise-continuous coefficients and complex conjugation in the spaces and provided that the underlying contour has only angular points but no cusps. The Fredholm property and the index turn out to be independent of the sm

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20

Wang, Zhiyong, Chuanhong Sun, and Pengtao Li. "Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups." Journal of Function Spaces 2020 (October 10, 2020): 1–21. http://dx.doi.org/10.1155/2020/8851287.

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In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related w

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21

Gogatishvili, Amiran, Susana D. Moura, Júlio S. Neves, and Bohumír Opic. "Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $$k$$ k -modulus of smoothness." Annali di Matematica Pura ed Applicata (1923 -) 194, no. 2 (2013): 425–50. http://dx.doi.org/10.1007/s10231-013-0383-1.

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22

Mitina, O. A., and V. M. Tyurin. "On invertibility of linear partial differential operators in Hölder and Sobolev spaces." Sbornik: Mathematics 194, no. 5 (2003): 733–44. http://dx.doi.org/10.1070/sm2003v194n05abeh000736.

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23

Ren, Jiagang, and Michael Röckner. "Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications." Probability Theory and Related Fields 117, no. 2 (2000): 201–20. http://dx.doi.org/10.1007/s004400050004.

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24

Chkadua, George. "Mixed type boundary value problems for polymetaharmonic equations." Georgian Mathematical Journal 23, no. 4 (2016): 489–510. http://dx.doi.org/10.1515/gmj-2016-0042.

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AbstractIn the paper we consider three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation ${(\Delta+k^{2}_{1})(\Delta+k^{2}_{2})u=0}$. We investigate these problems by means of the potential method and the theory of pseudodifferential equations. We prove the existence and uniqueness theorems in Sobolev–Slobodetskii spaces, analyse the asymptotic properties of solutions and establish the best Hölder smoothness results for solutions.

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25

BLATT, SIMON. "THE ENERGY SPACES OF THE TANGENT POINT ENERGIES." Journal of Topology and Analysis 05, no. 03 (2013): 261–70. http://dx.doi.org/10.1142/s1793525313500131.

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In this note, we will give a necessary and sufficient condition under which the tangent point energies introduced by von der Mosel and Strzelecki in [J. Geom. Anal., pp. 1–55 (2011), J. Knot Theory Ramifications21 (2012) 1250044] are bounded. We show that an admissible submanifold has bounded 𝔈q-energy if and only if it is injective and locally agrees with the graph of functions that belong to Sobolev–Slobodeckij space [Formula: see text]. The known Morrey embedding theorems of von der Mosel and Strzelecki can then be interpreted as standard Morrey embedding theorems for these spaces. Especial

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26

Fortunato, Westher Manricky Bernardes, and Dassael Fabricio dos Reis Santos. "An Application of the Browder-Minty Theorem in a Problem of Partial Differential Equations." REMAT: Revista Eletrônica da Matemática 6, no. 1 (2019): 1–12. http://dx.doi.org/10.35819/remat2020v6i1id3524.

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In this work, we will show existence of weak solution for a semilinear elliptic problem using as main tool the Browder-Minty Theorem. First, we will make a brief introduction about basic theory of the Sobolev Spaces to support our study and provide sufficient tools for the development of our work. Then we will take a quick approach on the Browder-Minty Theorem and use this result to show the existence of at least one weak solution to an elliptic Partial Differential Equations (PDE) problem whose nonlinearity, denoted by f, is a known function. For this, in addition to the already mentioned res

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27

Martínez, Ángel D., and Daniel Spector. "An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces." Advances in Nonlinear Analysis 10, no. 1 (2020): 877–94. http://dx.doi.org/10.1515/anona-2020-0157.

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Abstract It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality $$\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$ for all $\|f\|_{L^{N/\alpha,q}(\Omega)

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28

Gogatishvili, Amiran, Júlio S. Neves, and Bohumír Opic. "Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by L-modulus of smoothness." Journal of Functional Analysis 276, no. 2 (2019): 636–57. http://dx.doi.org/10.1016/j.jfa.2018.10.023.

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29

Porretta, Alessio. "A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁". Advanced Nonlinear Studies 20, № 2 (2020): 361–71. http://dx.doi.org/10.1515/ans-2020-2086.

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AbstractIt is known that the Sobolev space {W^{1,p}(\mathbb{R}^{N})} is embedded into {L^{Np/(N-p)}(\mathbb{R}^{N})} if {p<N} and into {L^{\infty}(\mathbb{R}^{N})} if {p>N}. There is usually a discontinuity in the proof of those two different embeddings since, for {p>N}, the estimate {\lVert u\rVert_{\infty}\leq C\lVert Du\rVert_{p}^{N/p}\lVert u\rVert_{p}^{1-N% /p}} is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the {L^{\infty}}-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate {\lVert u\rVert

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30

Viswanathan, P., and M. A. Navascués. "A Fractal Operator on Some Standard Spaces of Functions." Proceedings of the Edinburgh Mathematical Society 60, no. 3 (2017): 771–86. http://dx.doi.org/10.1017/s0013091516000316.

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AbstractThrough appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as anα-fractal operator on, the space of all real-valued continuous functions defined on a compact intervalI. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general

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31

Holík, Miloslav. "Reduction theorems for Sobolev embeddings into the spaces of Hölder, Morrey and Campanato type." Mathematische Nachrichten 289, no. 13 (2016): 1626–35. http://dx.doi.org/10.1002/mana.201500043.

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32

Mikulevičius, R., and H. Pragarauskas. "On the cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces." Lithuanian Mathematical Journal 32, no. 2 (1992): 238–64. http://dx.doi.org/10.1007/bf02450422.

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33

Cox, Sonja G., and Kristin Kirchner. "Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields." Numerische Mathematik 146, no. 4 (2020): 819–73. http://dx.doi.org/10.1007/s00211-020-01151-x.

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AbstractWe analyze several types of Galerkin approximations of a Gaussian random field $$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$ Z : D × Ω → R indexed by a Euclidean domain $$\mathscr {D}\subset \mathbb {R}^d$$ D ⊂ R d whose covariance structure is determined by a negative fractional power $$L^{-2\beta }$$ L - 2 β of a second-order elliptic differential operator $$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$ L : = - ∇ · ( A ∇ ) + κ 2 . Under minimal assumptions on the domain $$\mathscr {D}$$ D , the coefficients $$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$ A

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34

Prilepko, A. I., A. B. Kostin, and V. V. Solov’ev. "Inverse Source and Coefficient Problems for Elliptic and Parabolic Equations in Hölder and Sobolev Spaces." Journal of Mathematical Sciences 237, no. 4 (2019): 576–94. http://dx.doi.org/10.1007/s10958-019-04184-2.

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35

KOHR, MIRELA, G. P. RAJA SEKHAR, and WOLFGANG L. WENDLAND. "BOUNDARY INTEGRAL EQUATIONS FOR A THREE-DIMENSIONAL STOKES–BRINKMAN CELL MODEL." Mathematical Models and Methods in Applied Sciences 18, no. 12 (2008): 2055–85. http://dx.doi.org/10.1142/s0218202508003297.

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The purpose of this paper is to prove the existence and uniqueness of the solution in Sobolev or Hölder spaces for a cell model problem which describes the Stokes flow of a viscous incompressible fluid in a bounded region past a porous particle. The flow within the porous particle is described by the Brinkman equation. In order to obtain the desired existence and uniqueness result, we use an indirect boundary integral formulation and potential theory for both Brinkman and Stokes equations. Some special cases, which refer to the cell model for a porous particle with large permeability, or to th

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36

Bousquet, Pierre, and Gyula Csató. "The equation divu + 〈a,u〉 = f." Communications in Contemporary Mathematics 22, no. 04 (2019): 1950035. http://dx.doi.org/10.1142/s0219199719500354.

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We study the solutions [Formula: see text] to the equation [Formula: see text] where [Formula: see text] and [Formula: see text] are given. We significantly improve the existence results of [G. Csató and B. Dacorogna, A Dirichlet problem involving the divergence operator, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 829–848], where this equation has been considered for the first time. In particular, we prove the existence of a solution under essentially sharp regularity assumptions on the coefficients. The condition that we require on the vector field [Formula: see text] is necessary an

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37

Ligocka, Ewa. "Estimates in Sobolev norms $∥ ·∥^{s}_{p}$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions." Studia Mathematica 86, no. 3 (1987): 255–71. http://dx.doi.org/10.4064/sm-86-3-255-271.

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38

Pavlovic, Miroslav. "Definition and Properties of the Libera Operator on Mixed Norm Spaces." Scientific World Journal 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/590656.

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We consider the action of the operatorℒg(z)=(1-z)-1∫z1‍f(ζ)dζon a class of “mixed norm” spaces of analytic functions on the unit disk,X=Hα,νp,q, defined by the requirementg∈X⇔r↦(1-r)αMp(r,g(ν))∈Lq([0,1],dr/(1-r)), where1≤p≤∞,0<q≤∞,α>0, andνis a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expressionℒgneed not be defined forganalytic in the unit disk, even forg∈X. A sufficient, but not necessary, condition is that∑n=0∞‍|g^(n)|/(n+1)<∞. We identify the indicesp,q,α, andνfor which1∘ℒis well

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39

CAZENAVE, THIERRY, FLÁVIO DICKSTEIN, and FRED B. WEISSLER. "NON-REGULARITY IN HÖLDER AND SOBOLEV SPACES OF SOLUTIONS TO THE SEMILINEAR HEAT AND SCHRÖDINGER EQUATIONS." Nagoya Mathematical Journal 226 (September 9, 2016): 44–70. http://dx.doi.org/10.1017/nmj.2016.35.

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In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear

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40

Amann, Herbert. "Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds." Journal of Evolution Equations 17, no. 1 (2016): 51–100. http://dx.doi.org/10.1007/s00028-016-0347-1.

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41

Sadovnichaya, I. V. "Equiconvergence theorems in Sobolev and Hölder spaces of eigenfunction expansions for Sturm-Liouville operators with singular potentials." Doklady Mathematics 83, no. 2 (2011): 169–70. http://dx.doi.org/10.1134/s1064562411020128.

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42

Brezis, Haïm, and Hoai-Minh Nguyen. "On the distributional Jacobian of maps from $\mS^N$ into $\mS^N$ in fractional Sobolev and Hölder spaces." Annals of Mathematics 173, no. 2 (2011): 1141–83. http://dx.doi.org/10.4007/annals.2011.173.2.15.

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43

Neelima and David Šiška. "$$L^p$$-estimates and regularity for SPDEs with monotone semilinearity." Stochastics and Partial Differential Equations: Analysis and Computations 8, no. 2 (2019): 422–59. http://dx.doi.org/10.1007/s40072-019-00150-w.

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Abstract Semilinear stochastic partial differential equations on bounded domains $${\mathscr {D}}$$D are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are $$L^p$$Lp-estimates for such equations. The $$L^p$$Lp-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such

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44

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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45

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

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46

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

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47

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

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48

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

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49

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

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50

Ponce, Rodrigo. "Hölder continuous solutions for Sobolev type differential equations." Mathematische Nachrichten 287, no. 1 (2013): 70–78. http://dx.doi.org/10.1002/mana.201200168.

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