Besov-type spaces and differences

M. Hovemann ; W. Sickel

Eurasian mathematical journal, Tome 11 (2020) no. 1, pp. 25-56.

Résumé

We study characterizations by differences of the Besov-type spaces $B_{p,q}^{s,\tau}(\mathbb{R}^d)$. Our focus is on necessary and sufficient conditions on s for the validity of those characterizations.

Bibliographie

[1] O. V. Besov, “On a family of function spaces. Embedding theorems and extensions”, Dokl. Acad. Nauk SSSR, 126 (1959), 1163–1165 | MR | Zbl

[2] O. V. Besov, “On a family of function spaces in connection with embeddings and extensions”, Trudy Mat. Inst. Steklov, 60, 1961, 42–81 | MR | Zbl

[3] O. V. Besov, “On spaces of functions of smoothness zero”, Mat. Sbornik, 203:8 (2012), 1077–1090 | DOI | MR | Zbl

[4] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral representations of functions and embedding theorems, Transalated from Russian, v. I, Scripta Series in Mathematics, ed. by M. H. Taibleson, V. H. Winston Sons, Washington, D., 1978 ; v. II, 1979 | MR

[5] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integralnye predstavleniya funktsii i teoremy vlozheniya, Second ed., Fizmatlit “Nauka”, M., 1996 (Russian)

[6] V. I. Burenkov, E. D. Nursultanov, “Description of interpolation spaces for local Morrey-type spaces”, Trudy Math. Inst. Steklov, 269, 2010, 46–56 | DOI | MR | Zbl

[7] V. I. Burenkov, D. K. Chigambayeva, E. D. Nursultanov, “Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations”, Trudy Math. Inst. Steklov, 284, 2014, 97–128 | DOI | MR | Zbl

[8] F. Chiarenza, M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function”, Rend. Math., 7 (1987), 273–279 | MR | Zbl

[9] M. Christ, A. Seeger, “Necessary conditions for vector-valued operator inequalities in harmonic analysis”, Proc. London Math. Soc., 93:2 (2006), 447–473 | DOI | MR | Zbl

[10] M. Christ, A. Seeger, Necessary conditions for vector-valued operator inequalities in harmonic analysis, https://www.math.wisc.edu/s̃eeger/preprints.html | MR

[11] R. A. DeVore, G. Lorentz, Constructive approximation, Springer, Berlin, 1993 | MR | Zbl

[12] D. Drihem, “Characterizations of Besov-Type and Triebel-Lizorkin-type spaces by differences”, Journal Funct. Spaces Appl., 2012, 328908, 24 pp. | MR | Zbl

[13] A. El Baraka, “An embedding theorem for Campanato spaces”, Electron. J. Differential Equations, 66 (2002), 1–17 | MR

[14] A. El Baraka, “Function spaces of BMO and Campanato type” (Southwest Texas State Univ., San Marcos, TX), Electron. J. Differ. Equ. Conf., 9, Proc. of the 2002 Fez Conference on Partial Differential Equations (2002), 109–115 (electronic) | MR

[15] A. El Baraka, “Littlewood-Paley characterization for Campanato spaces”, J. Funct. Spaces Appl., 4 (2006), 193–220 | DOI | MR | Zbl

[16] H. F. Gonçalves, D. D. Haroske, L. Skrzypczak, Compact embeddings in Besov- and Triebel-Lizorkin-type spaces on bounded domains, arXiv: 2001.02046 | MR

[17] D. D. Haroske, S. D. Moura, L. Skrzypczak, “Smoothness Morrey Spaces of regular distributions, and some unboundedness property”, Nonlinear Analysis, 139 (2016), 218–244 | DOI | MR | Zbl

[18] 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

[19] F. Hirzebruch, W. Scharlau, Einführung in die Funktionalanalysis, Bibliographisches Institut, Mannheim, 1971 | MR | Zbl

[20] M. Hovemann, Triebel-Lizorkin-Morrey spaces and differences, Submitted, Jena, 2019, 32 pp.

[21] M. Hovemann, W. Sickel, Strichartz characterizations of Lizorkin-Triebel spaces, Preprint, Jena, 2019 | MR

[22] 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

[23] 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 (2012), 1067–1111 | DOI | MR | Zbl

[24] Y. Liang, Y. Sawano, T. Ullrich, D. Yang, W. Yuan, A new framework for generalized Besov-type and TriebelLizorkin-type spaces, Dissertationes Math. (Rozprawy Mat.), 489, 2013, 116 pp. | MR

[25] A. L. Mazzucato, “Besov-Morrey spaces: function space theory and applications to non-linear PDE”, Trans. Amer. Math. Soc., 355 (2003), 1297–1364 | DOI | MR | Zbl

[26] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces”, Proc. Harmonic Anal. Conf., ICM-90 Satell. Conf. Proc. (Sendai/Jap. 1990), 1991, 183–189 | DOI | MR | Zbl

[27] E. Nakai, “Hardy Littlewood maximal operator, singular integral operators, and the Riesz potential on generalized Morrey spaces”, Math. Nachr., 166 (1994), 95–103 | DOI | MR | Zbl

[28] S. Nakamura, T. Noi, Y. Sawano, “Generalized Morrey spaces and trace operator”, Science China Math., 59 (2016), 281–336 | DOI | MR | Zbl

[29] Y. V. Netrusov, “Some imbedding theorems for spaces of Besov-Morrey type”, Numerical methods and questions in the organisation of calculations, 7, Zapiski Nauchnykh Sem. Leningrad. Otdel. Mat. Inst. Steklov, 139, 1984, 139–147 (Russian) | MR | Zbl

[30] S. M. Nikol'skii, “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

[31] Springer Verlag, Berlin, 1975 | MR

[32] T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, Walter de Gruyter Co, Berlin, 1996 | MR

[33] Y. Sawano, Theory of Besov spaces, Springer, Singapore, 2018 | MR | Zbl

[34] Y. Sawano, H. Tanaka, “Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces”, Math. Z., 257 (2007), 871–905 | DOI | MR | Zbl

[35] C. Schneider, “On dilation operators in Besov spaces”, Rev. Mat. Complut., 22:1 (2009), 111–128 | DOI | MR | Zbl

[36] W. Sickel, “Smoothness spaces related to Morrey spaces a survey. I”, Eurasian Math. J., 3 (2012), 110–149 | MR | Zbl

[37] W. Sickel, H. Triebel, “Hölder inequalities and sharp embeddings in function spaces of $B^s_{p,q}$ and $F^s_{p,q}$ type”, Z. Anal. Anwendungen, 14 (1995), 105–140 | DOI | MR | Zbl

[38] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, 1970 | MR | Zbl

[39] L. Tang, J. Xu, “Some properties of Morrey type Besov-Triebel spaces”, Math. Nachr., 278 (2005), 904–917 | DOI | MR | Zbl

[40] H. Triebel, Theory of function spaces, Birkhäuser, Basel, 1983 | MR | Zbl

[41] H. Triebel, Theory of function spaces, v. II, Birkhäuser, Basel, 1992 | MR | Zbl

[42] H. Triebel, Theory of function spaces, v. III, Birkhäuser, Basel, 2006 | MR | Zbl

[43] H. Triebel, “Characterizations of some function spaces in terms of Haar wavelets”, Comment. Math., 53 (2013), 135–153 | MR | Zbl

[44] H. Triebel, Hybrid function spaces, heat and Navier-Stokes equations, EMS, Zürich, 2014 | MR

[45] H. Triebel, Theory of function spaces IV, Manuscript, Jena, 2019

[46] 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

[47] 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

[48] D. Yang, W. Yuan, “Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces”, Appl. Anal., 92 (2013), 549–561 | DOI | MR | Zbl

[49] W. Yuan, W. Sickel, D. Yang, Morrey, Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005, Springer, Berlin, 2010 | DOI | MR | Zbl

[50] W. Yuan, W. Sickel, D. Yang, “On the coincidence of certain approaches to smoothness spaces related to Morrey spaces”, Math. Nachr., 286 (2013), 1571–1584 | MR | Zbl

[51] W. Yuan, W. Sickel, D. Yang, Regularity properties of the Haar system with respect to Besov-type spaces, Studia Math. (online) | MR

[52] C. Zhou, W. Sickel, D. Yang, W. Yuan, “Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls”, Canad. Math. Bull., 60:3 (2017), 655–672 | DOI | MR