On embedding theorems

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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. (1982) Zbl0512.53044MR0681859
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. MR2010324
3. Bennett C., DeVore R., Sharpley R., Weak-$L^{\infty }$ and BMO, Ann. Math. 113 (1981), 601–611. Zbl 0465.42015, MR 82h:46047. (1981) MR0621018
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. (1988) Zbl0647.46057MR0928802
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. (1978) MR0521808
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. (1981) Zbl0462.46020MR0590417
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 (1975) Zbl0335.42011MR0390634
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. (1991) Zbl0737.46011MR1111186
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. Zbl1103.46310
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. MR1945278
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. Zbl1072.46020MR1942116
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. Zbl1023.46031MR1877265
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. (1975) MR0382559
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. (1991) Zbl0743.26008MR1139695
15. Cianchi A., Second order derivatives and rearrangements, Duke Math. J. 105 (2000), 355–385. Zbl 1017.46023, MR 2002e:46035. Zbl1017.46023MR1801766
16. Cianchi A., Rearrangements of functions in Besov spaces, Math. Nachr. 230 (2001), 19–35. Zbl 1022.46021, MR 2002h:46052. Zbl1022.46021MR1854875
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. Zbl1223.46033MR2044332
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. (1998) MR1650446
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. (1971) Zbl0275.46024MR0372140
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. Zbl1044.42028MR1993422
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. (1987) Zbl0639.46034MR0932758
22. Gagliardo E., Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137. Zbl 0089.09401, MR 21 #1526 (1958) Zbl0089.09401MR0102740
23. Garsia A. M., Combinatorial inequalities and smoothness of functions, Bull. Amer. Math. Soc. 82 (1976), 157–170. Zbl 0351.26005, MR 58 #28362. (1976) Zbl0351.26005MR0582776
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. (1974) Zbl0274.26006MR0414802
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. (1985) MR0810423
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. (1928) MR1544927
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. (1928) MR1544980
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. (1978) MR0619470
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. (1972) MR0333708
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. (1973) MR0333709
31. Kolyada V. I., On imbedding in classes $\varphi \left(L\right)$, 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. (1975) MR0374888
32. Kolyada V. I., On embedding of classes $H_p^{{\omega }_1,\cdots ,{\omega }_{\nu }}$, 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. (1985) MR0798382
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. (1988) MR0945897
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. (1988) MR0945427
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. (1989) MR1040269
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. (19,) MR1240790
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. (1993) MR1248284
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. (1998) Zbl0917.46019MR1638343
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. (192) MR1861373
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. Zbl1145.26010MR2215758
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. Zbl1114.46024MR2325743
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. Zbl1090.46026MR2182269
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. (1978) MR0649411
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. Zbl0966.26002MR1817225
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. (1949) Zbl0035.38302MR0031538
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. MR1862137
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. (195) MR1940355
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. MR1940371
49. Milne S. C., Peano curves and smoothness of functions, Adv. Math. 35 (1980), 129–157. Zbl 0449.26015, MR 82e:26017. (1980) Zbl0449.26015MR0560132
50. Neil R. O,’, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963), 129–142. Zbl 0178.47701, MR 26 #4193. (1963) MR0146673
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. (1987) MR0885079
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). (1988) 
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. (1975) MR0374877
54. Oswald P., On the moduli of continuity of equimeasurable functions in the classes $\varphi \left(L\right)$, Mat. Zametki 17 (1975), no. 3, 231–244; English transl. in Math. Notes 17 (1975), 134–141. MR 53 #8342. (1975) MR0404542
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). (1978) 
56. Oswald P., On the boundedness of the mapping $f\to \left|f\right|$ in Besov spaces, Comment. Math. Univ. Carolin. 33 (1992), no. 1, 57–66. Zbl 0766.46018, MR 93c:46052. (1992) Zbl0766.46018MR1173747
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. (1966) MR0221282
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. (1993) Zbl0811.46028MR1239425
59. Pérez F. J., Embedding theorems for anisotropic Lipschitz spaces, Studia Math. 168 (2005), no. 1, 51–72. Zbl 1079.46025, MR 2006a:46037. Zbl1079.46025MR2133387
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. (1983) MR0719267
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. (1986) Zbl0644.46022MR0877164
62. Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl 0207.13501, MR 44 #7280. (1970) Zbl0207.13501MR0290095
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. (1971) Zbl0232.42007MR0304972
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. (1968) MR0231194
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. (1970) MR0415303
66. Wik I., The non-increasing rearrangement as extremal function, Report no. 7. Univ. Umeå, Dept. of Math., Umeå, 1974. (1974) Zbl0337.26007MR1352013
67. Wik I., Symmetric rearrangement of functions and sets in ${ℝ}^n$, Preprint Univ. Umeå, Dept. of Math., no. 1, Umeå, 1977. (1977) 
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. (1998) MR1639194
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. (1989) Zbl0692.46022MR1014685