NAFSA
NAFSA 8: Proceedings of the 8th Spring School, Praha, 2006
Kolyada, Viktor I.
On embedding theorems. (English). In: Rákosník, Jiří (ed.): Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Prague, May 30-June 6, 2006. Vol. 8. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, 2007. pp. 35-94
MSC: 46E30, 46E35
Rearrangements; embeddings; modulus of continuity; Sobolev spaces; Besov spaces; mixed norms
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.
[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. MR 0681859 | Zbl 0512.53044
[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. DOI 10.1512/iumj.2003.52.2364 | MR 2010324
[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
[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. MR 0928802 | Zbl 0647.46057
[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
[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. DOI 10.1090/S0002-9947-1981-0590417-X | MR 0590417 | Zbl 0462.46020
[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 DOI 10.1070/SM1975v027n03ABEH002521 | MR 0390634 | Zbl 0335.42011
[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. DOI 10.1016/0022-1236(91)90006-Q | MR 1111186 | Zbl 0737.46011
[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
[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. DOI 10.1007/BF02868470 | MR 1945278
[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. MR 1942116 | Zbl 1072.46020
[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. DOI 10.1007/PL00001378 | MR 1877265 | Zbl 1023.46031
[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
[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. MR 1139695 | Zbl 0743.26008
[15] Cianchi A.: Second order derivatives and rearrangements. Duke Math. J. 105 (2000), 355–385. Zbl 1017.46023, MR 2002e:46035. DOI 10.1215/S0012-7094-00-10531-5 | MR 1801766 | Zbl 1017.46023
[16] Cianchi A.: Rearrangements of functions in Besov spaces. Math. Nachr. 230 (2001), 19–35. Zbl 1022.46021, MR 2002h:46052. DOI 10.1002/1522-2616(200110)230:1<19::AID-MANA19>3.0.CO;2-D | MR 1854875 | Zbl 1022.46021
[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. DOI 10.1007/s10231-003-0080-6 | MR 2044332 | Zbl 1223.46033
[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. DOI 10.1007/BF02384772 | MR 1650446
[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. MR 0372140 | Zbl 0275.46024
[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. DOI 10.4171/RMI/345 | MR 1993422 | Zbl 1044.42028
[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. DOI 10.1007/BF01774283 | MR 0932758 | Zbl 0639.46034
[22] Gagliardo E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7 (1958), 102–137. Zbl 0089.09401, MR 21 #1526 MR 0102740 | Zbl 0089.09401
[23] Garsia A. M.: Combinatorial inequalities and smoothness of functions. Bull. Amer. Math. Soc. 82 (1976), 157–170. Zbl 0351.26005, MR 58 #28362. DOI 10.1090/S0002-9904-1976-13975-4 | MR 0582776 | Zbl 0351.26005
[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. DOI 10.5802/aif.507 | MR 0414802 | Zbl 0274.26006
[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
[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. DOI 10.1007/BF01171116 | MR 1544927
[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. DOI 10.1007/BF01181186 | MR 1544980
[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
[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. DOI 10.1007/BF00971611 | MR 0333708
[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
[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
[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
[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
[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
[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
[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
[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
[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. MR 1638343 | Zbl 0917.46019
[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
[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. MR 2215758 | Zbl 1145.26010
[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. MR 2325743 | Zbl 1114.46024
[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. DOI 10.4064/sm171-1-1 | MR 2182269 | Zbl 1090.46026
[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
[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. MR 1817225 | Zbl 0966.26002
[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. DOI 10.1090/S0002-9904-1949-09320-5 | MR 0031538 | Zbl 0035.38302
[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. DOI 10.1090/S0002-9939-01-06060-9 | MR 1862137
[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. DOI 10.1006/jfan.2002.3955 | MR 1940355
[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
[49] Milne S. C.: Peano curves and smoothness of functions. Adv. Math. 35 (1980), 129–157. Zbl 0449.26015, MR 82e:26017. DOI 10.1016/0001-8708(80)90045-6 | MR 0560132 | Zbl 0449.26015
[50] Neil R. O,' : Convolution operators and $L(p,q)$ spaces. Duke Math. J. 30 (1963), 129–142. Zbl 0178.47701, MR 26 #4193. DOI 10.1215/S0012-7094-63-03015-1 | MR 0146673
[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
[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).
[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
[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
[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).
[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. MR 1173747 | Zbl 0766.46018
[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. DOI 10.5802/aif.232 | MR 0221282
[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. MR 1239425 | Zbl 0811.46028
[59] Pérez F. J.: Embedding theorems for anisotropic Lipschitz spaces. Studia Math. 168 (2005), no. 1, 51–72. Zbl 1079.46025, MR 2006a:46037. DOI 10.4064/sm168-1-4 | MR 2133387 | Zbl 1079.46025
[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
[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. DOI 10.1007/BF01909369 | MR 0877164 | Zbl 0644.46022
[62] Stein E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, 1970. Zbl 0207.13501, MR 44 #7280. MR 0290095 | Zbl 0207.13501
[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. MR 0304972 | Zbl 0232.42007
[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
[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
[66] Wik I.: The non-increasing rearrangement as extremal function. Report no. 7. Univ. Umeå, Dept. of Math., Umeå, 1974. MR 1352013 | Zbl 0337.26007
[67] Wik I.: Symmetric rearrangement of functions and sets in $\R^n$. Preprint Univ. Umeå, Dept. of Math., no. 1, Umeå, 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. MR 1639194
[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. MR 1014685 | Zbl 0692.46022