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Title: On embedding theorems (English)
Author: Kolyada, Viktor I.
Language: English
Journal: Nonlinear Analysis, Function Spaces and Applications
Volume: Vol. 8
Issue: 2006
Year:
Pages: 35-94
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Category: math
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Summary: This paper is devoted to embedding theorems for classes of functions of several variables. One of our main objectives is to give an analysis of some basic embeddings as well as to study relations between them. We also discuss some methods in this theory that were developed in the last decades. These methods are based on non-increasing rearrangements of functions, iterated rearrangements, estimates of sections of functions, related mixed norms, and molecular decompositions. (English)
Keyword: Rearrangements
Keyword: embeddings
Keyword: modulus of continuity
Keyword: Sobolev spaces
Keyword: Besov spaces
Keyword: mixed norms
MSC: 46E30
MSC: 46E35
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Date available: 2009-10-08T09:51:46Z
Last updated: 2013-10-18
Stable URL: http://hdl.handle.net/10338.dmlcz/702492
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Reference: [1] Aubin T.: Nonlinear Analysis on Manifolds. Monge-Ampère Equations.Grundlehren der Mathematischen Wissenschaften, Bd. 252. Springer Verlag, New York, 1982. Zbl 0512.53044, 85j:58002. Zbl 0512.53044, MR 0681859
Reference: [2] Bastero J., Milman M., Blasco F. J. Ruíz: A note on $L(\infty ,q)$ spaces and Sobolev embeddings.Indiana Univ. Math. J. 52 (2003), no. 5, 1215–1230. Zbl 1098.46023, MR 2004h:46025. MR 2010324, 10.1512/iumj.2003.52.2364
Reference: [3] Bennett C., DeVore R., Sharpley R.: Weak-$L^\infty $ and BMO.Ann. Math. 113 (1981), 601–611. Zbl 0465.42015, MR 82h:46047. MR 0621018
Reference: [4] Bennett C., Sharpley R.: Interpolation of Operators.Pure and Applied Mathematics, 129. Academic Press, Inc., Boston, MA, 1988. Zbl 0647.46057,MR 89e:46001. Zbl 0647.46057, MR 0928802
Reference: [5] Besov O. V., in V. P. Il,’ skii S. M. Nikol,’ : Integral Representations of Functions and Imbedding Theorems. Vol. 1–2.Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C., John Wiley & Sons, New York, 1978, 1979. Zbl 0392.46022, 0392.46023, MR 80f:46030a, 80f:46030b. MR 0521808
Reference: [6] Blozinski A. P.: Multivariate rearrangements and Banach function spaces with mixed norms.Trans. Amer. Math. Soc. 263 (1981), no. 1, 149–167. Zbl 0462.46020, 81k:46023. Zbl 0462.46020, MR 0590417, 10.1090/S0002-9947-1981-0590417-X
Reference: [7] Bochkarev S. V.: A Fourier series in an arbitrary bounded orthonormal system that diverges on a set of positive measure.Mat. Sb. 98, no. 3 (1975), 436–449; English transl. in Math. USSR-Sb. 27 (1975), 393–405. Zbl 0371.42010, MR 52#11459 Zbl 0335.42011, MR 0390634, 10.1070/SM1975v027n03ABEH002521
Reference: [8] Bourdaud G., Meyer Y.: Fonctions qui opèrent sur les espaces de Sobolev.J. Funct. Anal. 97 (1991), no. 2, 351–360. Zbl 0737.46011, MR 92e:46062. Zbl 0737.46011, MR 1111186, 10.1016/0022-1236(91)90006-Q
Reference: [9] Bourgain J., Brezis H., Mironescu P.: Another look at Sobolev spaces.Optimal Control and Partial Differential Equations. In honour of Professor Alain Bensoussan’s 60th Birthday. Proceedings of the conference, Paris, France, December 4, 2000 (J. L. Menaldi, E. Rofman, A. Sulem, eds.). IOS Press, Amsterdam, 2001, 439–455. Zbl 1103.46310. Zbl 1103.46310
Reference: [10] Bourgain J., Brezis H., Mironescu P.: Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications.J. Anal. Math. 87 (2002), 77–101. Zbl 1029.46030, MR 2003k:46035. MR 1945278, 10.1007/BF02868470
Reference: [11] Brezis H.: How to recognize constant functions. Connections with Sobolev spaces.(Russian) Uspekhi Mat. Nauk 57 (2002), no. 4(346), 59–74; English transl. in Russian Math. Surveys 57 (2002), no. 4, 693–708. Zbl 1072.46020, MR 2003m:46047. Zbl 1072.46020, MR 1942116
Reference: [12] Brezis H., Mironescu P.: Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces.J. Evol. Equ. 1 (2001), no. 4, 387–404. Zbl 1023.46031, MR 2002k:46073. Zbl 1023.46031, MR 1877265, 10.1007/PL00001378
Reference: [13] Brudnyi, Yu. A.: Moduli of continuity and rearrangements.Mat. Zametki 18 (1975), 63–66; English transl. in Math. Notes 18 (1975), 619–621. Zbl 0322.26002, MR 52 #3442. MR 0382559
Reference: [14] Budagov A. A.: Peano curves and moduli of continuity.Mat. Zametki 50 (1991), no. 2, 20–27; English transl. in Math. Notes 50 (1991), no. 1–2, 783–789. Zbl 0743.26008, MR 92k:26028. Zbl 0743.26008, MR 1139695
Reference: [15] Cianchi A.: Second order derivatives and rearrangements.Duke Math. J. 105 (2000), 355–385. Zbl 1017.46023, MR 2002e:46035. Zbl 1017.46023, MR 1801766, 10.1215/S0012-7094-00-10531-5
Reference: [16] Cianchi A.: Rearrangements of functions in Besov spaces.Math. Nachr. 230 (2001), 19–35. Zbl 1022.46021, MR 2002h:46052. Zbl 1022.46021, MR 1854875, 10.1002/1522-2616(200110)230:1<19::AID-MANA19>3.0.CO;2-D
Reference: [17] Cianchi A.: Symmetrization and second-order Sobolev inequalities.Ann. Mat. Pura Appl., IV. Ser. 183 (2004), no. 1, 45–77. Zbl pre05058531, MR 2005b:46067. Zbl 1223.46033, MR 2044332, 10.1007/s10231-003-0080-6
Reference: [18] Cianchi A., Pick L.: Sobolev embeddings into BMO, VMO, and $L^\infty $.Ark. Mat. 36 (1998), no. 2, 317–340. Zbl 1035.46502, MR 99k:46052. MR 1650446, 10.1007/BF02384772
Reference: [19] Chong K. M., Rice N. M.: Equimeasurable rearrangements of functions.Queen’s Papers in Pure and Applied Mathematics 28, Queen’s University, Kingston, Ont., 1971. Zbl 0275.46024, MR 51 #8357. Zbl 0275.46024, MR 0372140
Reference: [20] Cohen A., Dahmen W., Daubechies I., DeVore R.: Harmonic analysis of the space $BV$.Rev. Mat. Iberoamericana 19 (2003), no. 1, 235–263. Zbl 1044.42028, MR 2004f:42051. Zbl 1044.42028, MR 1993422, 10.4171/RMI/345
Reference: [21] Fournier J.: Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality.Ann. Mat. Pura Appl., IV. Ser. 148 (1987), 51–76. Zbl 0639.46034, MR 89e:46037. Zbl 0639.46034, MR 0932758, 10.1007/BF01774283
Reference: [22] Gagliardo E.: Proprietà di alcune classi di funzioni in più variabili.Ricerche Mat. 7 (1958), 102–137. Zbl 0089.09401, MR 21 #1526 Zbl 0089.09401, MR 0102740
Reference: [23] Garsia A. M.: Combinatorial inequalities and smoothness of functions.Bull. Amer. Math. Soc. 82 (1976), 157–170. Zbl 0351.26005, MR 58 #28362. Zbl 0351.26005, MR 0582776, 10.1090/S0002-9904-1976-13975-4
Reference: [24] Garsia A. M., Rodemich E.: Monotonicity of certain functionals under rearrangement.Ann. Inst. Fourier (Grenoble) 24, no. 2 (1974), 67–116. Zbl 0274.26006, MR 54 #2894. Zbl 0274.26006, MR 0414802, 10.5802/aif.507
Reference: [25] dman M. L. Gol,’ : Embedding of generalized Nikol’skii-Besov spaces into Lorentz spaces.Trudy Mat. Inst. Steklov 172 (1985), 128–139; English transl. in Proc. Stekolov Inst. Math. 172 (1985), 143–154. MR 87e:46047. MR 0810423
Reference: [26] Hardy G. H., Littlewood J. E.: Some properties of fractional integrals. I.Math. Z. 27 (1928), no. 1, 565–606. JFM 54.0275.05, MR 1544927. MR 1544927, 10.1007/BF01171116
Reference: [27] Hardy G. H., Littlewood J. E.: A convergence criterion for Fourier series.Math. Z. 28 (1928), no. 1, 612–634. JFM 54.0301.03, MR 1544980. MR 1544980, 10.1007/BF01181186
Reference: [28] Kashin B. S.: Remarks on the estimation of Lebesgue functions of orthonormal systems.Mat. Sb. 106 (1978), no. 3, 380–385; English transl. in Math. USSR Sb. 35 (1979), no. 1, 57–62. Zbl 0417.42014, MR 58 #29787. MR 0619470
Reference: [29] Klimov V. S.: Embedding theorems for Orlicz spaces and their applications to boundary value problems.Sib. Mat. Zh. 13 (1972), 334–348; English transl. in Siberian Math. J. 13 (1972), 231–240. Zbl 0246.46022, MR 48 #12033. MR 0333708, 10.1007/BF00971611
Reference: [30] Kolyada V. I.: The embedding of certain classes of functions of several variables.Sib. Mat. Zh. 14 (1973), 766–790; English transl. in Siberian Math. J. 14 (1973). Zbl 0281.46027, MR 48 #12034. MR 0333709
Reference: [31] Kolyada V. I.: On imbedding in classes $\varphi (L)$.Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 418–437; English transl. in Math. USSR Izv. 9 (1975), 395–413. Zbl 0334.46034, MR51 #11084. MR 0374888
Reference: [32] Kolyada V. I.: On embedding of classes $H_p^{\omega _1,\dots ,\omega _\nu }$.Mat. Sb. 127 (1985), no. 3, 352–383; English transl. in Math. USSR-Sb. 55 (1986), no. 2, 351–381. Zbl 0581.41030, MR 87d:46040. MR 0798382
Reference: [33] Kolyada V. I.: Estimates of rearrangements and embedding theorems.Mat. Sb. 136 (1988), 3–23; English transl. in Math. USSR-Sb. 64 (1989), no. 1, 1–21. Zbl 0693.46030. MR 0945897
Reference: [34] Kolyada V. I.: On relations between moduli of continuity in different metrics., Trudy Mat. Inst. Steklov 181 (1988), 117–136; English transl. in Proc. Steklov Inst. Math. 181 (1989), 127–148. Zbl 0716.41018, MR 90k:41021. MR 0945427
Reference: [35] Kolyada V. I.: Rearrangements of functions and embedding theorems.Uspekhi Matem. Nauk 44 (1989), no. 5, 61–95; English transl. in Russian Math. Surveys 44 (1989), no. 5, 73–118. MR 91i:46029. MR 1040269
Reference: [36] Kolyada V. I.: On the differential properties of the rearrangements of functions.In: Progress in Approximation Theory (A. A. Gonchar and E. B. Saff, eds.). Springer Ser. Comput. Math. 19, Springer-Verlag, Berlin, 1992, 333–352. Zbl 0848.26013, MR 95j:26025. MR 1240790
Reference: [37] Kolyada V. I.: On the embedding of Sobolev spaces.Mat. Zametki 54 (1993), no. 3, 48–71; English transl. in Math. Notes 54 (1993), no. 3, 908–922. Zbl 0821.46043, MR 94j:46042. MR 1248284
Reference: [38] Kolyada V. I.: Rearrangement of functions and embedding of anisotropic spaces of Sobolev type.East J. Approx. 4 (1998), no. 2, 111–199. Zbl 0917.46019, MR 99g:46043b. Zbl 0917.46019, MR 1638343
Reference: [39] Kolyada V. I.: Embeddings of fractional Sobolev spaces and estimates of Fourier transforms.Mat. Sb. 192 (2001), no. 7, 51–72; English transl. in Sb. Math. 192 (2001), no. 7, 979–1000. Zbl 1031.46040, MR 2002k:46080. MR 1861373
Reference: [40] Kolyada V. I.: Inequalities of Gagliardo–Nirenberg type and estimates for the moduli of continuity.Uspekhi Mat. Nauk 60 (2005), no. 6, 139–156; English transl. in Russian Math. Surveys 60 (2005), no. 6, 1147–1164. MR 2007b:26026. Zbl 1145.26010, MR 2215758
Reference: [41] Kolyada V. I.: Mixed norms and Sobolev type inequalities.Approximation and probability. Papers of the conference held on the occasion of the 70th anniversary of Prof. Zbigniew Ciesielski, Bedlewo, Poland, September 20–24, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publ. 72 (2006), 141–160. Zbl pre05082653. Zbl 1114.46024, MR 2325743
Reference: [42] Kolyada V. I., Lerner A. K.: On limiting embeddings of Besov spaces.Studia Math. 171 (2005), no. 1, 1–13. Zbl 1090.46026, MR 2006m:46042. Zbl 1090.46026, MR 2182269, 10.4064/sm171-1-1
Reference: [43] Krein S. G., Petunin, Yu. I., Semenov, and E. M.: Interpolation of linear operators.Nauka, Moscow 1978. Zbl 0499.46044, MR 81f:46086. English transl. in Translations of Mathematical Monographs 54, Amer. Math. Soc., Providence, 1982. Zbl 0493.46058, MR 84j:46103. MR 0649411
Reference: [44] Lieb E. H., Loss M.: Analysis.2nd ed. Graduate Studies in Mathematics, 14, Amer. Math. Soc., Providence, RI, 2001. Zbl 0966.26002, MR 2001i:00001. Zbl 0966.26002, MR 1817225
Reference: [45] Loomis L. H., Whitney H.: An inequality related to the isoperimetric inequality.Bull. Amer. Math. Soc. 55 (1949), 961–962. Zbl 0035.38302, MR 11,166d. Zbl 0035.38302, MR 0031538, 10.1090/S0002-9904-1949-09320-5
Reference: [46] Malý J., Pick L.: An elementary proof of sharp Sobolev embeddings.Proc. Amer. Math. Soc. 130 (2002), no. 2, 555–563. Zbl 0990.46022, MR 2002j:46042. MR 1862137, 10.1090/S0002-9939-01-06060-9
Reference: [47] ya V. Maz,’ Shaposhnikova T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces., J. Funct. Anal. 195 (2002), no. 2, 230–238. Zbl 1028.46050, MR 2003j:46051. MR 1940355, 10.1006/jfan.2002.3955
Reference: [48] ya V. Maz,’ Shaposhnikova T.: On the Brezis and Mironescu conjecture concerning a Gagliardo–Nirenberg inequality for fractional Sobolev norms.J. Math. Pures Appl. 81, no. 9 (2002), 877–884. Zbl 1036.46026, MR 2003j:46052. MR 1940371
Reference: [49] Milne S. C.: Peano curves and smoothness of functions.Adv. Math. 35 (1980), 129–157. Zbl 0449.26015, MR 82e:26017. Zbl 0449.26015, MR 0560132, 10.1016/0001-8708(80)90045-6
Reference: [50] Neil R. O,’ : Convolution operators and $L(p,q)$ spaces.Duke Math. J. 30 (1963), 129–142. Zbl 0178.47701, MR 26 #4193. MR 0146673, 10.1215/S0012-7094-63-03015-1
Reference: [51] Netrusov, Yu. V.: Embedding theorems for the Lizorkin-Triebel classes.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 159 (1987), 103–112 (in Russian); English transl. in J. Soviet Math. 47 (1989), 2896–2903. Zbl 0686.46027, MR 88h:46073. MR 0885079
Reference: [52] Netrusov, Yu. V.: Embedding theorems for spaces with a given majorant of the modulus of continuity.Ph.D. Thesis, LOMI AN SSSR, Leningrad, 1988 (Russian).
Reference: [53] skii S. M. Nikol,’ : Approximation of Functions of Several Variables and Imbedding Theorems.Die Grundlehren der mathematischen Wissenschaften 205. Springer–Verlag, Berlin, 1975. Zbl 0307.46024, MR 51 #11073. MR 0374877
Reference: [54] Oswald P.: On the moduli of continuity of equimeasurable functions in the classes $\varphi (L)$.Mat. Zametki 17 (1975), no. 3, 231–244; English transl. in Math. Notes 17 (1975), 134–141. MR 53 #8342. MR 0404542
Reference: [55] Oswald P.: Moduli of continuity of equimeasurable functions and approximation of functions by algebraic polynomials in $L^p$.Ph.D. Thesis, Odessa State University, Odessa, 1978 (Russian).
Reference: [56] Oswald P.: On the boundedness of the mapping $f\rightarrow |f|$ in Besov spaces.Comment. Math. Univ. Carolin. 33 (1992), no. 1, 57–66. Zbl 0766.46018, MR 93c:46052. Zbl 0766.46018, MR 1173747
Reference: [57] Peetre J.: Espaces d’interpolation et espaces de Soboleff.Ann. Inst. Fourier (Grenoble) 16 (1966), no. 1, 279–317. Zbl 0151.17903, MR 36 #4334. MR 0221282, 10.5802/aif.232
Reference: [58] Pelczyński A., Wojciechowski M.: Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm.Studia Math. 107 (1993), no. 1, 61–100. Zbl 0811.46028, MR 94h:46050. Zbl 0811.46028, MR 1239425
Reference: [59] Pérez F. J.: Embedding theorems for anisotropic Lipschitz spaces.Studia Math. 168 (2005), no. 1, 51–72. Zbl 1079.46025, MR 2006a:46037. Zbl 1079.46025, MR 2133387, 10.4064/sm168-1-4
Reference: [60] Poornima S.: An embedding theorem for the Sobolev space $W^{1,1}$.Bull. Sci. Math., II. Ser. 107 (1983), no. 3, 253–259. Zbl 0529.46025, MR 85b:46042. MR 0719267
Reference: [61] Runst T.: Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type.Anal. Math. 12 (1986), no. 4, 313–346. Zbl 0644.46022,MR 88f:46079. Zbl 0644.46022, MR 0877164, 10.1007/BF01909369
Reference: [62] Stein E. M.: Singular Integrals and Differentiability Properties of Functions.Princeton Univ. Press, 1970. Zbl 0207.13501, MR 44 #7280. Zbl 0207.13501, MR 0290095
Reference: [63] Stein E. M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces.Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, N.J., 1971. Zbl 0207.13501, MR 44 #7280. Zbl 0232.42007, MR 0304972
Reference: [64] yanov P. L. Ul,’ : The embedding of certain function classes $H_p^\omega $.Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 649–686; English transl. in Math. USSR Izv. 2 (1968), 601–637. Zbl 0181.13404, MR 37 #6749. MR 0231194
Reference: [65] yanov P. L. Ul,’ : Imbedding theorems and relations between best approximations (moduli of continuity) in various metrics.Mat. Sb. 81 (1970), no. 1, 104–131; English transl. in Math. USSR Sb. 10 (1970), no. 1, 103–126. Zbl 0215.17702, MR 54 #3393. MR 0415303
Reference: [66] Wik I.: The non-increasing rearrangement as extremal function.Report no. 7. Univ. Umeå, Dept. of Math., Umeå, 1974. Zbl 0337.26007, MR 1352013
Reference: [67] Wik I.: Symmetric rearrangement of functions and sets in $\R^n$.Preprint Univ. Umeå, Dept. of Math., no. 1, Umeå, 1977.
Reference: [68] Yatsenko A. A.: Iterative rearrangements of functions and the Lorentz spaces.Izv. Vyssh. Uchebn. Zaved. Mat. (1998), no. 5, 73–77; English transl. in Russian Mathematics (Iz. VUZ) 42 (1998), no. 5, 71–75. MR 99i:46019. MR 1639194
Reference: [69] Ziemer W. P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation.Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. Zbl 0692.46022, MR91e:46046. Zbl 0692.46022, MR 1014685
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