Smoothness spaces related to Morrey spaces~--- a survey.~II

W. Sickel

Eurasian mathematical journal, Tome 4 (2013) no. 1, pp. 82-124.

Résumé

We continue the discussion of different strategies of introducing smoothness spaces related to Morrey spaces.

Bibliographie

[1] D. R. Adams, J. L. Lewis, “On Morrey-Besov inequalities”, Studia Math., 74 (1982), 169–182 | MR | Zbl

[2] D. R. Adams, J. Xiao, “Nonlinear potential analysis on Morrey spaces and their capacities”, Indiana Univ. Math. J., 53 (2004), 1629–1662 | DOI | MR

[3] A. Akbulut, V. S. Guliyev, R. Mustafayev, On the boundedness of the maximal operator and singular integral operators in the generalized Morrey spaces, Preprint, Institute of Mathematics, AS CR, Prague, 2010, 26 pp. | MR

[4] J. Bergh, J. Löfström, Interpolation spaces. An introduction, Springer, Berlin, 1976 | MR | Zbl

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

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

[7] O. Blasco, A. Ruiz, L. Vega, “Non-interpolation in Morrey–Campanato and block spaces”, Ann. Scuola Norm. Sup. Pisa CI Sci. (4), 28 (1999), 31–40 | MR | Zbl

[8] H. Brezis, P. Mironescu, “Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces”, J. Evol. Eq., 1 (2001), 387–404 | DOI | MR | Zbl

[9] V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces, I”, Eurasian Math. J., 3:3 (2012), 11–32 | MR | Zbl

[10] V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces, II”, Eurasian Math. J., 4:1 (2013), 21–45 | Zbl

[11] V. I. Burenkov, D. K. Darbayeva, E. D. Nursultanov, “Description of interpolation spaces for general local Morrey-type spaces”, Eurasian Math. J., 4:1 (2013), 46–53 | Zbl

[12] V. I. Burenkov, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces”, Studia Math., 163 (2004), 157–176 | DOI | MR | Zbl

[13] V. I. Burenkov, H. V. Guliyev, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the fractional maximal operator in local Morrey-type spaces”, Dokl. Acad. Nauk, 409 (2006), 443–447 | MR

[14] V. I. Burenkov, H. V. Guliyev, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the fractional maximal operator in local Morrey-type spaces”, J. Comput. and Appl. Math., 208 (2007), 280–301 | DOI | MR | Zbl

[15] V. I. Burenkov, V. S. Guliyev, “Necessary and sufficient conditions for boundedness of the Riesz potential in local Morrey-type spaces”, Potential Anal., 30 (2009), 211–249 | DOI | MR | Zbl

[16] Proceedings Steklov Inst. Math., 269, 2010, 46–56 | DOI | MR | Zbl

[17] P. L. Butzer, K. Scherer, Approximationsprozesse und Interpolationsmethoden, Mannheim–Zurich, 1968 | MR

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

[19] A. Clop, D. Faraco, A. Ruiz, “Stability of Calderon's inverse conductivity problem in the plane for discontinuous conductivities”, Inverse Probl. Imaging, 4 (2010), 49–91 | DOI | MR | Zbl

[20] F. Cobos, D. L. Fernandez, “Hardy-Sobolev spaces and Besov spaces with a function parameter”, Proc. Lund Conf. (1986), Lect. Notes Math., 1302, Springer, Berlin, 1986, 158–170 | DOI | MR

[21] G. T. Dchumakaeva, “On the continuity of functions with a mixed derivative”, Proc. VII Int. Conf. in Math. and Mechanics (Karaganda, 1981), 19–20

[22] G. T. Dchumakaeva, “Criteria for the embedding of Sobolev–Morrey spaces in the space $C$”, Mat. Zametki, 37 (1985), 399–406 | MR

[23] R. A. DeVore, V. Popov, “Interpolation spaces and nonlinear approximation”, Function Spaces and Applications, Lecture Notes in Mathematics, 1302, eds. M. Cwikel et al., Springer, Berlin, 1988, 191–205 | DOI | MR

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

[25] D. E. Edmunds, H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge Univ. Press, Cambridge, 1996 | MR

[26] D. Faraco, K. M. Rotgers, The Sobolev norm of characteristic functions with applications to the Calderon inverse problem, Preprint, Madrid, 2011, 14 pp. | Zbl

[27] W. Farkas, H. G. Leopold, “Characterisations of function spaces of generalised smoothness”, Annali di Mat., 185 (2006), 162 | MR

[28] J. Franke, “On the spaces $F_{p,q}^s$ of Triebel–Lizorkin type: pointwise multipliers and spaces on domains”, Math. Nachr., 125 (1986), 29–68 | MR | Zbl

[29] M. Frazier, B. Jawerth, “A discrete transform and decomposition of distribution spaces”, J. Funct. Anal., 93 (1990), 34–170 | DOI | MR | Zbl

[30] M. L. Goldman, “A description of the traces of some function spaces”, Trudy Mat. Inst. Steklov, 150, 1979, 99–127 | MR | Zbl

[31] M. L. Goldman, “A method of coverings for describing general spaces of Besov type”, Trudy Mat. Inst. Steklov, 156, 1980, 47–81 | MR | Zbl

[32] M. L. Goldman, “Imbedding theorems for anisotropic Nikol'skii–Besov spaces with moduli of continuity of general type”, Trudy Mat. Inst. Steklov, 170, 1984, 86–104 | MR | Zbl

[33] M. L. Goldman, “On imbedding generalized Nikol'skii–Besov spaces in Lorentz spaces”, Trudy Mat. Inst. Steklov, 172, 1985, 128–139 | MR | Zbl

[34] V. S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces”, Journal of Inequalities and Applications, 2009, 503948, 20 pages pp. | DOI | MR | Zbl

[35] V. S. Guliyev, S. S. Aliyev, T. Karaman, “Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces”, Abstract and Applied Analysis, 2011, 356041, 18 pages pp. | MR | Zbl

[36] V. S. Guliyev, S. S. Aliyev, T. Karaman, P. S. Shukurov, “Boundedness of sublinear operators and commutators on generalized Morrey spaces”, Integr. Equ. Oper. Theory, 71 (2011), 327–355 | DOI | MR | Zbl

[37] H. Hajaiej, L. Molinet, T. Ozawa, B. Wang, “Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and application to Navier–Stokes and generalized boson equations”, RIMS Kôkyûroku Bessatsu B, 26 (2011), 159–175 | MR

[38] D. D. Haroske, L. Skrzypczak, “Continuous embeddings of Besov–Morrey function spaces”, Acta Math. Sin. (Engl. series), 28 (2012), 1307–1328 | DOI | MR | Zbl

[39] 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

[40] M. Izuki, Y. Sawano, H. Tanaka, “Weighted Besov–Morrey spaces and Triebel–Lizorkin spaces”, RIMS Kôyûroku Bessatsu B, 22 (2010), 21–60 | MR

[41] 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), eds. C. Kiselman, A. Vretblad ; Acta Univ. Upsaliensis, 64 (1999), 194–205 | MR | MR | Zbl

[42] B. Jawerth, “Some observations on Besov and Lizorkin–Triebel spaces”, Math. Scand., 40 (1977), 94–104 | MR | Zbl

[43] E. A. Kalita, “Dual Morrey spaces”, Doklady Math., 58 (1998), 85–87 | MR | Zbl

[44] G. A. Kalyabin, “Characterization of spaces of generalized Liouville differentiation”, Mat. Sbornik Nov. Ser., 104 (1977), 42–48 | MR | Zbl

[45] Proc. Steklov Inst. Math., no. 2 (156), 1983 | MR | Zbl

[46] G. A. Kalyabin, P. I. Lizorkin, “Spaces of functions of generalized smoothness”, Math. Nachr., 133 (1987), 7–32 | DOI | MR | Zbl

[47] Y. Komori, S. Shirai, “Weighted Morrey spaces and a singular integral operator”, Math. Nachr., 282 (2009), 219–231 | DOI | MR | Zbl

[48] 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

[49] H. Kozono, M. Yamazaki, “The stability of small stationary solutions in Morrey spaces of the Navier–Stokes equation”, Indiana Univ. Math. J., 44 (1995), 1307–1335 | DOI | MR

[50] L. D. Kudryavtsev, S. M. Nikol'skii, “Spaces of differentiable functions of several variables and imbedding theorems”, Analysis, v. III, Encyclopedia of Math. Sciences, 26, Spaces of differentiable functions, Springer, Heidelberg, 1990, 4–140 | MR

[51] P. G. Lemarié-Rieusset, “The Navier-Stokes equations in the critical Morrey–Campanato space”, Rev. Mat. Iberoam., 23 (2007), 897–930 | DOI | MR | Zbl

[52] P. G. Lemarié-Rieusset, “Multipliers and Morrey spaces”, Potential Anal., 38 (2013), 741–752 | DOI | MR | Zbl

[53] Y. Liang, Y. Sawano, T. Ullrich, D. Yang, W. Yuan, “New characterizations of Besov–Triebel- Lizorkin–Hausdorff spaces including coorbits and wavelets”, JFAA, 18 (2012), 1067–1111 | MR | Zbl

[54] Y. Liang, Y. Sawano, T. Ullrich, D. Yang, W. Yuan, A new framework for generalized Besov-type and Triebel–Lizorkin-type spaces, Dissertationes Math., 489, Warszawa, 2013, 114 pp. | DOI | Zbl

[55] U. Luther, “Representation, interpolation, and reiteration theorems for generalized approximation spaces”, Ann. Math. Pura Appl., 182 (2003), 161–200 | DOI | MR | Zbl

[56] U. Luther, J. M. Almira, “Generalized approximation spaces and applications”, Math. Nachr., 263–264 (2004), 3–35 | MR | Zbl

[57] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces”, Harmonic Analysis, ICM 1990 Satellite Conf. Proc., ed. S. Igari, Springer, Tokyo, 1991, 183–189 | MR

[58] A. Mazzucato, “Decomposition of Besov-Morrey spaces”, Harmonic Analysis (Mount Holyoke, 2001), Contemporary Math., 320, 2003, 279–294 | DOI | MR | Zbl

[59] A. Mazzucato, “Function space theory and applications to non-linear PDE”, Trans. Amer. Math. Soc., 355 (2003), 1297–1369 | DOI | MR

[60] A. M. Najafov, “Some properties of functions from the intersection of Besov–Morrey type spaces with dominant mixed derivatives”, Proc. A. Razmadze Math. Inst., 139 (2005), 71–82 | MR | Zbl

[61] A. M. Najafov, “Embedding theorems in the Sobolev–Morrey-type spaces $S_{p,a,\varkappa,\tau}W(G)$ with dominating mixed derivatives”, Siberian Math. J., 47 (2006), 505–516 | DOI | MR

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

[63] E. Nakai, “Orlicz-Morrey spaces and the Hardy–Littlewood maximal function”, Studia Math., 188 (2008), 193–221 | DOI | MR | Zbl

[64] 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 (in Russian) | MR | Zbl

[65] Y. V. Netrusov, “Embedding theorems for traces of Besov spaces and Lizorkin–Triebel spaces”, Dokl. Akad. Nauk SSSR, 298 (1988), 1326–1330 | MR | Zbl

[66] Y. V. Netrusov, “Metric estimates of the capacities of sets in Besov spaces”, Trudy Mat. Inst. Steklov, 190, 1989, 159–185 | MR

[67] S. M. Nikol'skii, Approximation of functions of several variables and imbedding theorems, Springer, Berlin, 1975 | MR

[68] F. Oru, Role des oscillations dans quelques problémes d'analyse non-linéaire, Doctorat de Ecole Normale Supérieure de Cachan, 1998

[69] Y. Peetre, “On the theory of $\mathcal{L}_{p,\lambda}$ spaces”, J. Funct. Anal., 4 (1969), 71–87 | DOI | MR | Zbl

[70] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press, Durham, 1976 | MR | Zbl

[71] J. Peetre, G. Sparr, “Interpolation of normed abelian groups”, Ann. Mat. Pura Appl., 92 (1972), 217–262 | DOI | MR | Zbl

[72] A. Pietsch, “Approximation spaces”, JAT, 32 (1980), 115–134 | DOI | MR

[73] J. Ross, “A Morrey–Nikol'skii inequality”, Proc. AMS, 78 (1980), 97–102 | MR | Zbl

[74] A. Ruiz, L. Vega, “Corrigenda to “Unique continuation for Schrödinger operators” and a remark on interpolation of Morrey spaces”, Publ. Mat., 39 (1995), 405–411 | DOI | MR | Zbl

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

[76] V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. London Math. Soc., 60 (1999), 237–257 | DOI | MR | Zbl

[77] V. S. Rychkov, “Littlewood–Paley theory and function spaces with $A_p^{loc}$ weights”, Math. Nachr., 224 (2001), 145–180 | 3.0.CO;2-2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[78] Y. Sawano, “A vector-valued sharp maximal inequality on Morrey spaces with non-doubling measures”, Georgian Math. J., 13:1 (2006), 153–172 | MR | Zbl

[79] Y. Sawano, “Wavelet characterization of Besov–Morrey and Triebel–Lizorkin–Morrey spaces”, Funct. Approx. Comment. Math., 38 (2008), 93–107 | DOI | MR | Zbl

[80] Y. Sawano, “Generalized Morrey spaces for non-doubling measures”, Nonlinear Differential Equations Appl., 15 (2008), 413–425 | DOI | MR | Zbl

[81] Y. Sawano, “Morrey spaces with non-doubling measures, II”, Journal of the Indonesian Mathematical Society, 14:2 (2008), 121–152 | MR | Zbl

[82] 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

[83] Y. Sawano, S. Sugano, H. Tanaka, “Identification of the image of Morrey spaces by the fractional integral operators”, Proc. A. Razmadze Math. Inst., 149 (2009), 87–93 | MR | Zbl

[84] Y. Sawano, S. Sugano, H. Tanaka, “Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces”, Trans. AMS, 363 (2011), 6481–6503 | DOI | MR | Zbl

[85] Y. Sawano, H. Tanaka, “Morrey spaces for non-doubling measures”, Acta Math. Sin. (Engl. Ser.), 21:6 (2005), 1535–1544 | DOI | MR | Zbl

[86] Y. Sawano, H. Tanaka, “Equivalent norms for the Morrey spaces with non-doubling measures”, Far East J. Math. Sci. (FJMS), 22:3 (2006), 387–404 | MR | Zbl

[87] Y. Sawano, H. Tanaka, “Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces for non-doubling measures”, Math. Nachr., 282 (2009), 1788–1810 | DOI | MR | Zbl

[88] Y. Sawano, H. Tanaka, “Predual spaces of Morrey spaces with non-doubling measures”, Tokyo J. Math., 32:2 (2009), 471–486 | DOI | MR | Zbl

[89] 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

[90] H.-J. Schmeißer, W. Sickel, Traces, Gagliardo–Nirenberg inequalities and Sobolev type embeddings for vector-valued function spaces, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/24/01, Jena, 2001, 58 pp.

[91] W. Sickel, “On pointwise multipliers for $F_{p,q}^s(\mathbb{R}^n)$ in case $\sigma_{p,q}/p$”, Annali Mat. Pura Appl., 176 (1999), 209–250 | DOI | MR | Zbl

[92] W. Sickel, “Pointwise multipliers for Lizorkin–Triebel spaces”, Operator theory: Advances and Appl., 110 (1999), 295–321 | MR | Zbl

[93] W. Sickel, “Smoothness spaces related to Morrey spaces — a survey, I”, Eurasian Math. J., 3 (2012), 104–143 | MR

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

[95] S. Spanne, “Sur l'interpolation entre les espace $\mathcal{L}_k^{p,\Phi}$”, Annali della Scuola Normale Superiore di Pisa, 20 (1966), 625–648 | MR | Zbl

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

[97] H. Triebel, Interpolation theory, function spaces, differential operators, North Holland, Amsterdam, 1978 | MR | Zbl

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

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

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

[101] H. Triebel, Function Spaces and wavelets on domains, EMS Publishing House, Zürich, 2008 | MR

[102] H. Triebel, Bases in function spaces, sampling, discrepancy, numerical integration, EMS Publ. House, Zürich, 2010 | MR | Zbl

[103] H. Triebel, Local function spaces, Lecture, given at the 2nd. International workshop on Interpolation theory, function spaces and related topics (Santiago de Compostela, July, 2011) http://www.mat.ucm.es/congresos/c2iwit/index.htm

[104] H. Triebel, Local function spaces, heat and Navier-Stokes equations, EMS Publ. House, Zürich, 2013 (to appear) | Zbl

[105] 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

[106] 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

[107] D. Yang, W. Yuan, “Dual properties of Triebel–Lizorkin-type spaces and their applications”, ZAA, 30 (2011), 29–58 | MR | Zbl

[108] 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

[109] W. Yuan, W. Sickel, D. Yang, “On the coincidence of certain approaches to smoothness spaces related to Morrey spaces”, Math. Nachr. (to appear)

[110] C. T. Zorko, “Morrey space”, Proc. AMS, 98 (1986), 586–592 | DOI | MR | Zbl