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@article{EMJ_2012_3_3_a6, author = {W. Sickel}, title = {Smoothness spaces related to {Morrey} spaces~-- {a~survey.~I}}, journal = {Eurasian mathematical journal}, pages = {110--149}, publisher = {mathdoc}, volume = {3}, number = {3}, year = {2012}, language = {en}, url = {http://geodesic.mathdoc.fr/item/EMJ_2012_3_3_a6/} }
W. Sickel. Smoothness spaces related to Morrey spaces~-- a~survey.~I. Eurasian mathematical journal, Tome 3 (2012) no. 3, pp. 110-149. http://geodesic.mathdoc.fr/item/EMJ_2012_3_3_a6/
[1] J.-M. Angeletti, S. Mazet, P. Tchamitchian, “Analysis of second order elliptic operators without boundary conditions and with VMO or Hölderian coefficients”, Multiscale wavelet methods for PDE's, Academic Press, Boston, 1997, 495–539 | MR
[2] R. Aulaskari, J. Xiao, R. H. Zhao, “On subspaces and subsets of BMOA and UBC”, Analysis, 15 (1995), 101–121 | DOI | MR | Zbl
[3] J. Bergh, J. Löfström, Interpolation spaces. An Introduction, Springer, Berlin, 1976 | MR | Zbl
[4] O. V. Besov, V. P. Il'in, S. M. Nikol'skij, Integral representations of functions and imbedding theorems, v. I, V. H. Winston Sons, Washington, D.C., 1978 ; v. II, Scripta Series in Mathematics, ed. M. H. Taibleson, 1979, Translated from Russian | Zbl | MR
[5] O. V. Besov, V. P. Il'in, S. M. Nikol'skij, Integralnye predstavleniya funktsii i teoremy vlozheniya, Second ed., Fizmatlit “Nauka”, Moscow, 1996 (Russian) | MR | Zbl
[6] G. Bourdaud, “$L^p$ estimates for certain nonregular pseudodifferential operators”, Comm. Partial Differential Equations, 7 (1982), 1023–1033 | DOI | MR | Zbl
[7] G. Bourdaud, “Ondelettes et espaces de Besov”, Rev. Mat. Iberoamericana, 11 (1995), 477–512 | DOI | MR | Zbl
[8] G. Bourdaud, W. Sickel, “Changes of variable in Besov spaces”, Math. Nachr., 198 (1999), 19–39 | DOI | MR | Zbl
[9] Bui Huy Qui, “Some aspects of weighted and non-weighted Hardy spaces”, Kokyuroku Res. Inst. Math. Sci., 383 (1980), 38–56
[10] Proceedings of the Steklov Institute of Mathematics, 150, 1981, 33–53 | MR | Zbl | Zbl
[11] G. Dafni, J. Xiao, “Some new tent spaces and duality theorem for fractional Carleson measures and $Q_\alpha(\mathbb R^n)$”, J. Funct. Anal., 208 (2004), 377–422 | DOI | MR | Zbl
[12] G. Dafni, J. Xia, “The dyadic structure and atomic decomposition of $Q$ spaces in several real variables”, Tohoku Math. J., 57 (2005), 119–145 | DOI | MR | Zbl
[13] D. Drihem, “Some embeddings and equivalent norms of the $\mathcal L^\lambda_{p,q}$ spaces”, Funct. Appro. Comment. Math., 41 (2009), 15–40 | DOI | MR | Zbl
[14] D. Drihem, “Some characterizations of function spaces connecting $\mathcal L^{2,\alpha}$ spaces”, Rev. Mat. Complut., 24 (2011), 323–333 | DOI | MR | Zbl
[15] D. Drihem, “Characterizations of Besov-type and Triebel–Lizorkin-type spaces by differences”, J. Funct. Spaces Appl., 2012 (2012), Article ID 328908, 24 | DOI | MR | Zbl
[16] A. El Baraka, “An embedding theorem for Campanato spaces”, Electron. J. Differential Equations, 66 (2002), 1–17 | MR
[17] A. El Baraka, “Function spaces of BMO and Campanato type”, Proc. of the 2002 Fez Conference on Partial Differential Equations (Southwest Texas State Univ., San Marcos, TX, 2002), Electron. J. Differ. Equ. Conf., 9, 2002, 109–115, (electronic) | MR
[18] A. El Baraka, “Littlewood–Paley characterization for Campanato spaces”, J. Funct. Spaces Appl., 4 (2006), 193–220 | DOI | MR | Zbl
[19] M. Essén, S. Janson, L. Peng, J. Xiao, “$Q$ spaces of several real variables”, Indiana Univ. Math. J., 49 (2000), 575–615 | DOI | MR | Zbl
[20] M. Essén, J. Xiao, “Some results on $Q_p$ spaces, $0
1$”, J. Reine Angew. Math., 485 (1997), 173–195 | MR | Zbl[21] W. Farkas, J. Johnsen, W. Sickel, “Traces of anisotropic Besov–Lizorkin–Triebel spaces – a complete treatment of the borderline cases”, Math. Bohemica, 125 (2000), 1–37 | MR | Zbl
[22] C. Fefferman, “Characterizations of bounded mean oscillation”, Bull. Amer. Math. Soc., 77 (1971), 587–588 | DOI | MR | Zbl
[23] J. Franke, “On the spaces $F^s_{p,q}$ of Triebel–Lizorkin type: pointwise multipliers and spaces on domains”, Math. Nachr., 125 (1986), 29–68 | MR | Zbl
[24] M. Frazier, Y.-S. Han, B. Jawerth, G. Weiss, “The T1 theorem for Triebel–Lizorkin spaces”, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., Springer, Berlin, 1384, 168–181 | MR
[25] M. Frazier, B. Jawerth, “Decomposition of Besov spaces”, Indiana Univ. Math. J., 34 (1985), 777–799 | DOI | MR | Zbl
[26] M. Frazier, B. Jawerth, “A discrete transform and decomposition of distribution spaces”, J. Funct. Anal., 93 (1990), 34–170 | DOI | MR | Zbl
[27] M. Frazier, R. Torres, G. Weiss, “The boundedness of Calderón–Zygmund operators on the spaces $\dot F^{\alpha,q}_p$”, Rev. Mat. Iberoamericana, 4 (1988), 41–72 | DOI | MR | Zbl
[28] D. Goldberg, “A local version of real Hardy spaces”, Duke Math. J., 46 (1979), 27–42 | DOI | MR | Zbl
[29] V. M. Gol'dstein, Yu. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Kluwer, Dordrecht, 1990
[30] L. Grafakos, R. H. Torres, “Pseudodifferential operators with homogeneous symbols”, Michigan Math. J., 46 (1999), 261–269 | DOI | MR | Zbl
[31] L. I. Hedberg, Y. V. Netrusov, An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Mem. Amer. Math. Soc., 882, 2007, 97 pp. | MR | Zbl
[32] E. Hernández, G. Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996 | DOI | MR | Zbl
[33] L. Hörmander, “Pseudodifferential operators of type 1,1”, Comm. Partial Differential Equations, 13 (1988), 1085–1111 | DOI | MR | Zbl
[34] L. Hörmander, “Continuity of pseudodifferential operators of type 1,1”, Comm. Partial Differential Equations, 14 (1989), 231–243 | DOI | MR | Zbl
[35] S. Janson, “On functions with conditions on the mean oscillation”, Ark. Mat., 4 (1976), 189–196 | DOI | MR
[36] S. Janson, “On the space $Q_p$ and its dyadic counterpart”, Complex Analysis and Differential Equations, Proceeding, Marcus Wallenberg Symposium in Honor of Matts Essén (Uppsala, 1997), Acta Univ. Upsaliensis, 64, eds. C. Kiselman, A. Vretblad, 1999, 194–205 | MR | Zbl
[37] B. Jawerth, “The trace of Sobolev and Besov spaces if $0
1$”, Studia Math., 62 (1978), 65–71 | MR | Zbl[38] F. John, L. Nirenberg, “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl
[39] J. Johnsen, W. Sickel, “On the trace problem for Lizorkin–Triebel spaces with mixed norms”, Math. Nachr., 281 (2008), 1–28 | DOI | MR
[40] J.-P. Kahane, P.-G. Lemarie-Rieuseut, Fourier series and wavelets, Gordon and Breach Publ., 1995
[41] G. A. Kalyabin, “Multiplier conditions of function spaces of Besov and Lizorkin–Triebel type”, Dokl. Acad. Nauk SSSR, 251:1 (1980), 25–26 | MR | Zbl
[42] G. A. Kalyabin, “Criteria of the multiplication property and the embeddings in $C$ of spaces of Besov–Lizorkin–Triebel type”, Mat. Zametki, 30:4 (1981), 517–526 | MR | Zbl
[43] H. Koch, W. Sickel, “Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions”, Rev. Mat. Iberoamericana, 18 (2002), 587–626 | DOI | MR | Zbl
[44] H. Kozono, M. Yamazaki, “Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data”, Comm. PDE, 19 (1994), 959–1014 | DOI | MR | Zbl
[45] A. Kufner, O. John, S. Fučik, Function Spaces, Academia, Prague, 1977 | MR | Zbl
[46] Y. Liang, Y. Sawano, T. Ullrich, D. Yang, W. Yuan, “New characterizations of Besov–Triebel–Lizorkin–Hausdorff spaces including coorbits and wavelets”, J. Fourier Anal. Appl., 18:5 (2012), 1067–1111 | DOI | MR | Zbl
[47] Y. Liang, Y. Sawano, T. Ullrich, D. Yang, W. Yuan, A new framework for generalized Besov–type and Triebel–Lizorkin–type spaces, Preprint, Beijing–Kyoto–Bonn, 2011, 122 pp. | MR
[48] C. C. Lin, K. Wang, “Equivalency between generalized Carleson measure spaces and Triebel–Lizorkin–type spaces”, Taiwanese J. Math., 15 (2011), 919–926 | MR | Zbl
[49] J. Marschall, “Some remarks on Triebel spaces”, Studia Math., 87 (1987), 79–92 | MR | Zbl
[50] J. Marschall, “Nonregular pseudo-differential operators”, Z. Anal. Anwendungen, 1 (1996), 109–148 | DOI | MR | Zbl
[51] V. G. Maz'ya, Sobolev spaces, Springer, Berlin, 1985 | MR | Zbl
[52] V. G. Maz'ya, T. O. Shaposnikova, Theory of multipliers in spaces of differentiable functions, Pitman, Boston, 1985 | Zbl
[53] V. G. Maz'ya, T. O. Shaposnikova, Theory of Sobolev multipliers, Springer, Berlin, 2009 | MR
[54] A. Mazzucato, “Decomposition of Besov–Morrey spaces”, Harmonic Analysis (Mount Holyoke, 2001), Contemporary Math., 320, 2003, 279–294 | DOI | MR | Zbl
[55] A. Mazzucato, “Function space theory and applications to non-linear PDE”, Trans. Amer. Math. Soc., 355 (2003), 1297–1369 | DOI | MR
[56] Y. Meyer, “Remarques sur un théorème de J. M. Bony”, Rend. Circ. Mat. Palermo, Serie II, 1981, Suppl. 1, 1–20 | MR | Zbl
[57] Y. Meyer, Wavelets and operators, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl
[58] Y. V. Netrusov, “Theorems on traces and multipliers for functions in Lizorkin–Triebel spaces”, Zapiski Nauchnykh Sem. Leningrad. Otdel. Mat. Inst. Steklov, 200, 1992, 132–138 | MR | Zbl
[59] S. M. Nikol'skij, “Inequalities for entire analytic functions of finite order and their application to the theory of differentiable functions of several variables”, Trudy Mat. Inst. Steklov, 38, 1951, 244–278
[60] S. M. Nikol'skij, Approximation of functions of several variables and imbedding theorems, Springer, Berlin, 1975 | MR | Zbl
[61] Y. Peetre, “On the theory of $\mathcal L_{p,\lambda}$ spaces”, J. Funct. Anal., 4 (1969), 71–87 | DOI | MR | Zbl
[62] J. Peetre, $H_p$ spaces, Lecture Notes, Lund, 1974
[63] Y. Peetre, The trace of Besov spaces – a limiting case, Technical Report, Lund, 1975
[64] J. Peetre, “On the spaces of Triebel–Lizorkin type”, Ark. Mat., 13 (1975), 123–130 | DOI | MR | Zbl
[65] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press, Durham, 1976 | MR | Zbl
[66] T. Runst, “Pseudo-differential operators of the “exotic” class $L^0_{1,1}$ in spaces of Besov and Triebel–Lizorkin type”, Annals Global Analysis Geometry, 3 (1985), 13–28 | DOI | MR | Zbl
[67] T. Runst, “Mapping properties of non-linear operators in spaces of Triebel–Lizorkin and Besov type”, Anal. Math., 12 (1986), 313–346 | DOI | MR | Zbl
[68] T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations, de Gruyter, Berlin, 1996 | MR | Zbl
[69] Y. Sawano, “Wavelet characterization of Besov–Morrey and Triebel–Lizorkin–Morrey spaces”, Funct. Approx. Comment. Math., 38 (2008), 93–107 | DOI | MR | Zbl
[70] Y. Sawano, “A note on Besov–Morrey and Triebel–Lizorkin–Morrey spaces”, Acta Math. Sin. (engl. series), 25:8 (2009), 1223–1242 | DOI | MR | Zbl
[71] Y. Sawano, H. Tanaka, “Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces”, Math. Z., 257 (2007), 871–905 | DOI | MR | Zbl
[72] Y. Sawano, D. Yang, W. Yuan, “New applications of Besov–type and Triebel–Lizorkin–type spaces”, J. Math. Anal. Appl., 363 (2010), 73–85 | DOI | MR | Zbl
[73] A. Seeger, “A note on Triebel–Lizorkin spaces”, Banach Center Publ., 22, PWN Polish Sci. Publ., Warsaw, 1989, 391–400 | MR
[74] W. Sickel, “On pointwise multipliers for $F^s_{p,q}(\mathbb R^d)$ in case $\sigma_{p,q}/p$”, Annali Mat. Pura Appl., 176 (1999), 209–250 | DOI | MR | Zbl
[75] W. Sickel, “Pointwise multipliers for Lizorkin–Triebel spaces”, Operator theory: Advances and Appl., 110 (1999), 295–321 | MR | Zbl
[76] W. Sickel, Smoothness spaces related to Morrey spaces – a survey. II, Preprint, Jena, 2012 | MR
[77] R. S. Strichartz, “Multipliers on fractional Sobolev spaces”, J. Math. Mechanics, 16 (1967), 1031–1060 | MR | Zbl
[78] L. Tang, J. Xu, “Some properties of Morrey type Besov–Triebel spaces”, Math. Nachr., 278 (2005), 904–917 | DOI | MR | Zbl
[79] R. H. Torres, “Continuity properties of pseudodifferential operators of type 1,1”, Comm. PDE, 15 (1990), 1313–1328 | DOI | MR | Zbl
[80] R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc., 90, no. 442, 1991, viii+172 pp. | MR
[81] H. Triebel, Spaces of Besov–Sobolev–Hardy type, Teubner Texte Math., 9, Teubner, Leipzig, 1978 | MR
[82] H. Triebel, Interpolation theory, function spaces, differential operators, North Holland, Amsterdam, 1978 | MR | Zbl
[83] H. Triebel, Theory of function spaces, Birkhäuser, Basel, 1983 | MR | Zbl
[84] H. Triebel, Theory of function spaces, v. II, Birkhäuser, Basel, 1992 | MR | Zbl
[85] H. Triebel, Fractals and spectra, Birkhäuser Verlag, Basel, 1997 | MR | Zbl
[86] H. Triebel, Theory of function spaces, v. III, Birkhäuser, Basel, 2006 | MR | Zbl
[87] H. Triebel, Function Spaces and wavelets on domains, EMS Publishing House, Zürich, 2008 | MR | Zbl
[88] S. K. Vodop'yanov, “Mappings of homogeneous groups and imbeddings of functional spaces”, Siberian Math. Zh., 30 (1989), 25–41 | DOI | MR | Zbl
[89] H. Wang, “Decomposition for Morrey type Besov–Triebel spaces”, Math. Nachr., 282 (2009), 774–787 | DOI | MR | Zbl
[90] P. Wojtasczyk, A mathematical introduction to wavelets, Cambridge Univ. Press, Cambridge, 1997 | MR
[91] Z. Wu, C. Xie, “Decomposition theorems for $Q_p$ spaces”, Ark. Mat., 40 (2002), 383–401 | DOI | MR | Zbl
[92] Z. Wu, C. Xie, “$Q$ spaces and Morrey spaces”, J. Funct. Anal., 201 (2003), 282–297 | DOI | MR | Zbl
[93] J. Xiao, Geometric $Q_p$ functions, Birkhäuser Verlag, Basel, 2006 | MR
[94] J. Xiao, “Homothetic variant of fractional Sobolev space with application to Navier–Stokes system”, Dyn. Partial Differ. Equ., 4 (2007), 227–245 | DOI | MR | Zbl
[95] J. Xiao, “The $Q_p$ Carleson measure problem”, Adv. Math., 217 (2008), 2075–2088 | DOI | MR | Zbl
[96] M. Yamazaki, “A quasi-homogeneous version of paradifferential operators. I: Boundedness on spaces of Besov type”, J. Fac. Sci. Univ. Tokyo, Section IA Math., 33 (1986), 131–174 | MR | Zbl
[97] D. Yang, W. Yuan, “A new class of function spaces connecting Triebel–Lizorkin spaces and $Q$ spaces”, J. Funct. Anal., 255 (2008), 2760–2809 | DOI | MR | Zbl
[98] D. Yang, W. Yuan, “New Besov–type spaces and Triebel–Lizorkin–type spaces including $Q$ spaces”, Math. Z., 265 (2010), 451–480 | DOI | MR | Zbl
[99] D. Yang, W. Yuan, “Characterizations of Besov–type spaces and Triebel–Lizorkin–type spaces via maximal functions and local means”, Nonlinear Anal., 73 (2010), 3805–3820 | DOI | MR | Zbl
[100] D. Yang, W. Yuan, “Relations among Besov–type spaces, Triebel–Lizorkin–type and generalized Carleson measure spaces”, Applicable Analysis, 92:3 (2013), 549–561 | DOI | MR | Zbl
[101] D. Yang, W. Yuan, C. Zhuo, “Fourier multipliers on Triebel–Lizorkin-type spaces”, J. Funct. Spaces Appl., 2012 (2012), Article ID 431016, 37 pp. | MR
[102] W. Yuan, W. Sickel, D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Note in Math., 2005, Springer, Berlin, 2010 | DOI | MR | Zbl
[103] H. Yue, G. Dafni, “A John–Nirenberg type inequality for $Q_\alpha(\mathbb R^n)$”, J. Math. Anal. Appl., 351 (2009), 428–439 | DOI | MR | Zbl