Science China Mathematics

, Volume 58, Issue 9, pp 1835–1908 | Cite as

Interpolation of Morrey-Campanato and related smoothness spaces

Articles

Abstract

We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.

Keywords

Morrey space Campanato space Besov-type space Triebel-Lizorkin-type space real and complex interpolation ±-method of interpolation Peetre-Gagliardo interpolation Calderón product 

MSC(2010)

46B70 46E35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams D R, Xiao J. Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ Math J, 2004, 53: 1629–1663MathSciNetMATHGoogle Scholar
  2. 2.
    Adams D R, Xiao J. Morrey potentials and harmonic maps. Comm Math Phys, 2011, 308: 439–456MathSciNetMATHGoogle Scholar
  3. 3.
    Adams D R, Xiao J. Morrey spaces in harmonic analysis. Ark Mat, 2012, 50: 201–230MathSciNetGoogle Scholar
  4. 4.
    Adams D R, Xiao J. Regularity of Morrey commutators. Trans Amer Math Soc, 2012, 364: 4801–4818MathSciNetMATHGoogle Scholar
  5. 5.
    Bennett C, Sharpley R. Interpolation of Operators. Boston: Academic Press, 1988MATHGoogle Scholar
  6. 6.
    Berezhnoi E I. Banach spaces, concave functions and interpolation of linear operators (in Russian). Funktsional Anal i Prilozhen, 1980, 14: 62–63MathSciNetGoogle Scholar
  7. 7.
    Bergh J, Löfström J. Interpolation Spaces: An Introduction. New York: Springer-Verlag, 1976MATHGoogle Scholar
  8. 8.
    Blasco O, Ruiz A, Vega L. Non interpolation in Morrey-Campanato and block spaces. Ann Sc Norm Super Pisa Cl Sci (4), 1999, 28: 31–40MathSciNetMATHGoogle Scholar
  9. 9.
    Bourdaud G. Remarques sur certains sous-espaces de BMO(Rn) et de bmo(Rn). Ann Inst Fourier, 2002, 52: 1187–1218MathSciNetMATHGoogle Scholar
  10. 10.
    Bownik M. Duality and interpolation of anisotropic Triebel-Lizorkin spaces. Math Z, 2008, 259: 131–169MathSciNetMATHGoogle Scholar
  11. 11.
    Brudnyĭ Yu A. Sobolev spaces and their relatives: Local polynomial approximation approach. In: Sobolev Spaces in Mathematics, vol. II. New York: Springer, 2009, 31–68Google Scholar
  12. 12.
    Brudnyĭ Yu A, Kruglyak N Ya. Interpolation Functors and Interpolation Spaces. Amsterdam: North Holland, 1991MATHGoogle Scholar
  13. 13.
    Calderón A P. Intermediate spaces and interpolation, the complex method. Studia Math, 1964, 24: 113–190MathSciNetMATHGoogle Scholar
  14. 14.
    Campanato S. Proprieta di inclusione per spazi di Morrey. Ric Mat, 1963, 12: 67–86MathSciNetMATHGoogle Scholar
  15. 15.
    Campanato S. Proprieta di hölderianita di alcune classi di funzioni. Ann Sc Norm Super Pisa, 1963, 17: 175–188MathSciNetGoogle Scholar
  16. 16.
    Campanato S. Proprieta di una famiglia di spazi funzionali. Ann Sc Norm Super Pisa, 1964, 18: 137–160MathSciNetMATHGoogle Scholar
  17. 17.
    Campanato S. Teoremi di interpolazione per transformazioni che applicano L p in C k,α. Ann Sc Norm Super Pisa, 1964, 18: 345–360MathSciNetMATHGoogle Scholar
  18. 18.
    Campanato S, Murthy M K V. Una generalizzazione del teorema di Riesz-Thorin. Ann Sc Norm Super Pisa (3), 1965, 19: 87–100MathSciNetMATHGoogle Scholar
  19. 19.
    Cobos F, Peetre J, Persson L E. On the connection between real and complex interpolation of quasi-Banach spaces. Bull Sci Math, 1998, 122: 17–37MathSciNetMATHGoogle Scholar
  20. 20.
    Dafni G, Xiao J. Some new tent spaces and duality theorems for fractional Carleson measures and Q α(Rn). J Funct Anal, 2004, 208: 377–422MathSciNetGoogle Scholar
  21. 21.
    Dchumakeva G T. A criterion for the imbedding of the Sobolev-Morrey class W p,Φl in the space C. Mat Zametki, 1985, 37: 399–406MathSciNetGoogle Scholar
  22. 22.
    El Baraka A. Function spaces of BMO and Campanato type. Electron J Differ Equ Conf, 2002, 9: 109–115MathSciNetGoogle Scholar
  23. 23.
    El Baraka A. An embedding theorem for Campanato spaces. Electron J Differential Equations, 2002, 66: 1–17MathSciNetGoogle Scholar
  24. 24.
    El Baraka A. Littlewood-Paley characterization for Campanato spaces. J Funct Spaces Appl, 2006, 4: 193–220MathSciNetMATHGoogle Scholar
  25. 25.
    Essén M, Janson S, Peng L, et al. Q spaces of several real variables. Indiana Univ Math J, 2000, 49: 575–615MathSciNetMATHGoogle Scholar
  26. 26.
    Frazier M, Jawerth B. A discrete transform and decompositions of distribution spaces. J Funct Anal, 1990, 93: 34–170MathSciNetMATHGoogle Scholar
  27. 27.
    Fu X, Lin H, Yang D, et al. Hardy spaces H p over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58: 309–388MathSciNetGoogle Scholar
  28. 28.
    Gagliardo E. Caratterizzazione costruttiva di tutti gli spazi di interpolazione tra spazi di Banach. Symposia Mathematica, 1968, 2: 95–106Google Scholar
  29. 29.
    Gustavsson J. On interpolation of weighted L p-spaces and Ovchinnikov’s theorem. Studia Math, 1982, 72: 237–251MathSciNetGoogle Scholar
  30. 30.
    Gustavsson J, Peetre J. Interpolation of Orlicz spaces. Studia Math, 1977, 60: 33–59MathSciNetMATHGoogle Scholar
  31. 31.
    Haroske D D, Skrzypczak L. Continuous embeddings of Besov-Morrey function spaces. Acta Math Sin Engl Ser, 2012, 28: 1307–1328MathSciNetMATHGoogle Scholar
  32. 32.
    Haroske D D, Skrzypczak L. Embeddings of Besov-Morrey spaces on bounded domains. Studia Math, 2013, 218: 119–144MathSciNetMATHGoogle Scholar
  33. 33.
    Hernández E, Weiss G. A First Course on Wavelets. Studies in Advanced Mathematics. Boca Raton: CRC Press, 1996Google Scholar
  34. 34.
    Janson S. Minimal and maximal methods of interpolation. J Funct Anal, 1981, 44: 50–73MathSciNetMATHGoogle Scholar
  35. 35.
    John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math, 1961, 14: 415–426MathSciNetMATHGoogle Scholar
  36. 36.
    Kahane J P, Lemarié-Rieuseut P G. Fourier Series and Wavelets. New York: Gordon and Breach Publ, 1995Google Scholar
  37. 37.
    Kalton N. Analytic functions in non-locally convex lattices. Studia Math, 1986, 83: 275–303MathSciNetGoogle Scholar
  38. 38.
    Kalton N. Plurisubharmonic functions on quasi-Banach spaces. Studia Math, 1986, 84: 297–324MathSciNetMATHGoogle Scholar
  39. 39.
    Kalton N, Mayboroda S, Mitrea M. Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations. Contemp Math, 2007, 445: 121–177MathSciNetGoogle Scholar
  40. 40.
    Kalton N, Mitrea M. Stability results on interpolation scales of quasi-Banach spaces and applications. Trans Amer Math Soc, 1998, 350: 3903–3922MathSciNetMATHGoogle Scholar
  41. 41.
    Kozono H, Yamazaki M. Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm Partial Differential Equations, 1994, 19: 959–1014MathSciNetMATHGoogle Scholar
  42. 42.
    Kreĭn S G, Petunin Yu I, Semënov E M. Interpolation of Linear Operators. Providence: Amer Math Soc, 1982Google Scholar
  43. 43.
    Kufner A, John O, Fučcik S. Function Spaces. Prague: Academia, 1977MATHGoogle Scholar
  44. 44.
    Lemarié-Rieusset P G. The Navier-Stokes equations in the critical Morrey-Campanato space. Rev Mat Iberoamericana, 2007, 23: 897–930MATHGoogle Scholar
  45. 45.
    Lemarié-Rieusset P G. The role of Morrey spaces in the study of Navier-Stokes and Euler equations. Eurasian Math J, 2012, 3: 62–93MathSciNetMATHGoogle Scholar
  46. 46.
    Lemarié-Rieusset P G. Multipliers and Morrey spaces. Potential Anal, 2013, 38: 741–752MathSciNetMATHGoogle Scholar
  47. 47.
    Lemarié-Rieusset P G. Sobolev multipliers, maximal functions and parabolic equations with a quadratic nonlinearity. Preprint, 2013, http://www.maths.univ-evry.fr/prepubli/387.pdfGoogle Scholar
  48. 48.
    Lemarié-Rieusset P G. Erratum to “Multipliers and Morrey spaces”. Potential Anal, 2014, 41: 1359–1362MathSciNetGoogle Scholar
  49. 49.
    Li P, Xiao J, Yang Q. Global mild solutions of fractional Navier-Stokes equations with small initial data in critical Besov-Q spaces. Electron J Differential Equations, 2014, 185: 37ppMathSciNetGoogle Scholar
  50. 50.
    Liang Y, Sawano Y, Ullrich T, et al. New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets. J Fourier Anal Appl, 2012, 18: 1067–1111MathSciNetMATHGoogle Scholar
  51. 51.
    Liang Y, Yang D, Yuan W, et al. A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math (Rozprawy Mat), 2013, 489: 1–114MathSciNetGoogle Scholar
  52. 52.
    Lin H, Yang D. Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci China Math, 2014, 57: 123–144MathSciNetGoogle Scholar
  53. 53.
    Long R, Yang L. BMO functions in spaces of homogeneous type. Sci China Ser A, 1984, 27: 695–708MathSciNetMATHGoogle Scholar
  54. 54.
    LozanovskiĭG Ja. A remark on a certain interpolation theorem of Calderón (in Russian). Funktsional Anal i Prilozhen, 1972, 6: 89–90Google Scholar
  55. 55.
    LozanovskiĭG Ja. On some Banach lattices IV. Sibirsk Mat Zh, 1973, 14: 97–108Google Scholar
  56. 56.
    Lu Y, Yang D, Yuan W. Interpolation of Morrey spaces on metric measure spaces. Canad Math Bull, 2014, 57: 598–608MathSciNetMATHGoogle Scholar
  57. 57.
    Lunardi A. Interpolation Theory. Pisa: Edizioni Normale, 2009MATHGoogle Scholar
  58. 58.
    Maligranda L. Orlicz Spaces and Interpolation (Seminars in Mathematics 5). Campinas: Departamento de Matemática, Universidade Estadual, 1989Google Scholar
  59. 59.
    Mazzucato A. Decomposition of Besov-Morrey spaces. Contemp Math, 2003, 320: 279–294MathSciNetGoogle Scholar
  60. 60.
    Mazzucato A. Besov-Morrey spaces: Function space theory and applications to non-linear PDE. Trans Amer Math Soc, 2003, 355: 1297–1369MathSciNetMATHGoogle Scholar
  61. 61.
    Mendez O, Mitrea M. The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations. J Fourier Anal Appl, 2000, 6, 503–531MathSciNetMATHGoogle Scholar
  62. 62.
    Meyer Y. Wavelets and Operators. Cambridge: Cambridge University Press, 1992MATHGoogle Scholar
  63. 63.
    Nakai E, Sawano Y. Orlicz-Hardy spaces and their duals. Sci China Math, 2014, 57: 903–962MathSciNetGoogle Scholar
  64. 64.
    Nilsson P. Interpolation of Banach lattices. Studia Math, 1985, 82: 135–154MathSciNetMATHGoogle Scholar
  65. 65.
    Ovchinnikov V I. The method of orbits in interpolation theory. Math Rep, 1984, 1: 349–515MathSciNetGoogle Scholar
  66. 66.
    Peetre J. On the theory of L p,λ spaces. J Funct Anal, 1969, 4: 71–87MathSciNetMATHGoogle Scholar
  67. 67.
    Peetre J. Sur l’utilisation des suites inconditionellement sommables dans la théorie des espaces dínterpolation. Rend Sem Mat Univ Padova, 1971, 46: 173–190MathSciNetGoogle Scholar
  68. 68.
    Peetre J. New Thoughts on Besov Spaces. Durham: Duke University Press, 1976MATHGoogle Scholar
  69. 69.
    Pick L, Kufner A, John O, et al. Function Spaces, vol. 1. Berlin: Walter de Gruyter & Co, 2012Google Scholar
  70. 70.
    Rafeiro H, Samko N, Samko S. Morrey-Campanato spaces: An overview. Oper Theory Adv Appl, 2013, 228: 293–323MathSciNetGoogle Scholar
  71. 71.
    Rosenthal M. Local means, wavelet bases and wavelet isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Math Nachr, 2013, 286: 59–87MathSciNetMATHGoogle Scholar
  72. 72.
    Ruiz A, Vega L. Corrigenda to “Unique continuation for Schrödinger operators with potential in Morrey spaces” and a remark on interpolation of Morrey spaces. Publ Mat, 1995, 3: 405–411MathSciNetGoogle Scholar
  73. 73.
    Runst T, Sickel W. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Berlin: Walter de Gruyter & Co, 1996MATHGoogle Scholar
  74. 74.
    Rychkov V S. On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J London Math Soc (2), 1999, 60: 237–257MathSciNetGoogle Scholar
  75. 75.
    Sawano Y. Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct Approx Comment Math, 2008, 38: 93–107MathSciNetMATHGoogle Scholar
  76. 76.
    Sawano Y. A note on Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Acta Math Sin Engl Ser, 2009, 25: 1223–1242MathSciNetMATHGoogle Scholar
  77. 77.
    Sawano Y. Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains. Math Nachr, 2010, 283: 1–32MathSciNetGoogle Scholar
  78. 78.
    Sawano Y, Tanaka H. Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math Z, 2007, 257: 871–904MathSciNetMATHGoogle Scholar
  79. 79.
    Sawano Y, Tanaka H. Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures. Math Nachr, 2009, 282: 1788–1810MathSciNetMATHGoogle Scholar
  80. 80.
    Sawano Y, Tanaka H. The Fatou property of block spaces. ArXiv:1404.2688, 2014Google Scholar
  81. 81.
    Sawano Y, Yang D, Yuan W. New applications of Besov-type and Triebel-Lizorkin-type spaces. J Math Anal Appl, 2010, 363: 73–85MathSciNetMATHGoogle Scholar
  82. 82.
    Shestakov V A. Interpolation of linear operators in spaces of measurable functions (in Russian). Funktsional Anal i Prilozhen, 1974, 8: 91–92Google Scholar
  83. 83.
    Shestakov V A. Complex interpolation in Banach spaces of measurable functions (in Russian). Vestnik Leningrad Univ, 1974, 19: 64–68Google Scholar
  84. 84.
    Shestakov V A. Transformations of Banach ideal spaces and interpolation of linear operators (in Russian). Bull Acad Polon Sci, 1981, 29: 569–577MathSciNetMATHGoogle Scholar
  85. 85.
    Sickel W. Smoothness spaces related to Morrey spaces — a survey. I. Eurasian Math J, 2012, 3: 110–149MathSciNetGoogle Scholar
  86. 86.
    Sickel W. Smoothness spaces related to Morrey spaces — a survey. II. Eurasian Math J, 2013, 4: 82–124MathSciNetMATHGoogle Scholar
  87. 87.
    Sickel W, Skrzypczak L, Vybíral J. Complex interpolation of weighted Besov- and Lizorkin-Triebel spaces (extended version). ArXiv:1212.1614, 2012Google Scholar
  88. 88.
    Sickel W, Skrzypczak L, Vybíral J. Complex interpolation of weighted Besov- and Lizorkin-Triebel spaces. Acta Math Sin Engl Ser, 2014, 30: 1297–1323MathSciNetMATHGoogle Scholar
  89. 89.
    Spanne S. Sur línterpolation entres les espaces L k(p,Φ). Ann Sc Norm Super Pisa, 1966, 20: 625–648MathSciNetMATHGoogle Scholar
  90. 90.
    Stampacchia G. L(p,λ)-spaces and interpolation. Comm Pure Appl Math, 1964, 17: 293–306MathSciNetMATHGoogle Scholar
  91. 91.
    Tan C, Li J. Littlewood-Paley theory on metric spaces with non doubling measures and its applications. Sci China Math, 2015, 58: 983–1004MathSciNetGoogle Scholar
  92. 92.
    Tang L, Xu J. Some properties of Morrey type Besov-Triebel spaces. Math Nachr, 2005, 278: 904–914MathSciNetMATHGoogle Scholar
  93. 93.
    Taylor M. Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm Partial Differential Equations, 1992, 17: 1407–1456MathSciNetMATHGoogle Scholar
  94. 94.
    Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam-New York: North-Holland Publishing, 1978Google Scholar
  95. 95.
    Triebel H. Complex interpolation and Fourier multipliers for the spaces B p,qs and F p,qs of Besov-Hardy-Sobolev type: The case 0 < p ≤ ∞, 0 < q ≤ ∞. Math Z, 1981, 176: 495–510MathSciNetGoogle Scholar
  96. 96.
    Triebel H. Theory of Function Spaces. Basel: Birkhäuser Verlag, 1983Google Scholar
  97. 97.
    Triebel H. Theory of Function Spaces II. Basel: Birkhäuser Verlag, 1992MATHGoogle Scholar
  98. 98.
    Triebel H. Theory of Function Spaces III. Basel: Birkhäuser Verlag, 2006MATHGoogle Scholar
  99. 99.
    Triebel H. Function Spaces and Wavelets on Domains. Zürich: European Mathematical Society, 2008Google Scholar
  100. 100.
    Triebel H. Local Function Spaces, Heat and Navier-Stokes Equations. Zürich: European Mathematical Society, 2013MATHGoogle Scholar
  101. 101.
    Triebel H. Hybrid Function Spaces, Heat and Navier-Stokes Equations. Zürich: European Mathematical Society, 2014Google Scholar
  102. 102.
    Turpin P. Convexités dans les espaces vectoriels topologiques généraux (in French). Dissertationes Math (Rozprawy Mat), 1974, 131: 221ppGoogle Scholar
  103. 103.
    Wojtaszczyk P. A Mathematical Introduction to Wavelets. Cambridge: Cambridge University Press, 1997MATHGoogle Scholar
  104. 104.
    Xiao J. Holomorphic Q Classes. Berlin: Springer, 2001MATHGoogle Scholar
  105. 105.
    Xiao J. Geometric Q p Functions. Basel: Birkhäuser Verlag, 2006MATHGoogle Scholar
  106. 106.
    Xu J. Decompositions of non-homogeneous Herz-type Besov and Triebel-Lizorkin spaces. Sci China Math, 2014, 57: 315–331MathSciNetMATHGoogle Scholar
  107. 107.
    Yang D, Yuan W. A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J Funct Anal, 2008, 255: 2760–2809MathSciNetMATHGoogle Scholar
  108. 108.
    Yang D, Yuan W. New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math Z, 2010, 265: 451–480MathSciNetMATHGoogle Scholar
  109. 109.
    Yang D, Yuan W. Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl Anal, 2013, 92: 549–561MathSciNetMATHGoogle Scholar
  110. 110.
    Yang D, Yuan W, Zhuo C. Fourier multipliers on Triebel-Lizorkin-type spaces. J Funct Spaces Appl, 2012, Article ID 431016Google Scholar
  111. 111.
    Yang D, Yuan W, Zhuo C. Complex interpolation on Besov-type and Triebel-Lizorkin-type spaces. Anal Appl, 2013, 11: 1350021Google Scholar
  112. 112.
    Yuan W. A note on complex interpolation and Calderón product of quasi-Banach spaces. ArXiv:1405.5735, 2014Google Scholar
  113. 113.
    Yuan W, Haroske D D, Skrzypczak L, et al. Embedding properties of Besov-type spaces. Appl Anal, 2015, 94: 318–340MathSciNetGoogle Scholar
  114. 114.
    Yuan W, Haroske D D, Skrzypczak L, et al. Embedding properties of weighted Besov-type spaces. Anal Appl (Singap), 2015, 13: 507–553MathSciNetGoogle Scholar
  115. 115.
    Yuan W, Sickel W, Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Berlin: Springer-Verlag, 2010Google Scholar
  116. 116.
    Yuan W, Sickel W, Yang D. On the coincidence of certain approaches to smoothness spaces related to Morrey spaces. Math Nachr, 2013, 286: 1571–1584MathSciNetMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingChina
  2. 2.Mathematisches InstitutFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations