Skip to main content
SpringerLink
Log in

Oscillations and differences in Triebel–Lizorkin–Morrey spaces

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

In this paper we are concerned with Triebel–Lizorkin–Morrey spaces \({\mathcal {E}}^{s}_{u,p,q}(\Omega )\) of positive smoothness s defined on (special or bounded) Lipschitz domains \(\Omega \subset {{\mathbb {R}}}^{d}\) as well as on \({{\mathbb {R}}}^{d}\). For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of \({\mathcal {E}}^{s}_{u,p,q}(\Omega )\) using differences of higher order. Special cases include standard Triebel–Lizorkin spaces \(F^s_{p,q} (\Omega )\) and hence classical \(L_p\)-Sobolev spaces \(H^s_p(\Omega )\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balci, AKh., Diening, L., Weimar, M.: Higher order Calderón–Zygmund estimates for the \(p\)-Laplace equation. J. Differ. Equ. 268, 590–635 (2020)

    Article  Google Scholar 

  2. Canuto, C., Nochetto, R.H., Stevenson, R., Verani, M.: Convergence and optimality of hp-AFEM. Numer. Math. 135, 1073–1119 (2017)

    Article  MathSciNet  Google Scholar 

  3. Christ, M., Seeger, A.: Necessary conditions for vector-valued operator inequalities in harmonic analysis. Proc. Lond. Math. Soc. 93(2), 447–473 (2006)

    Article  MathSciNet  Google Scholar 

  4. Cioica-Licht, P.A., Weimar, M.: On the limit regularity in Sobolev and Besov scales related to approximation theory. J. Fourier Anal. Appl. 26(1), 10 (2020)

    Article  MathSciNet  Google Scholar 

  5. Dahlke, S., Diening, L., Hartmann, C., Scharf, B., Weimar, M.: Besov regularity of solutions to the \(p\)-Poisson equation. Nonlinear Anal. 130, 298–329 (2016)

    Article  MathSciNet  Google Scholar 

  6. Dekel, S., Leviatan, D.: Whitney estimates for convex domains with applications to multivariate piecewise polynomial approximation. Found. Comput. Math. 4, 345–368 (2004)

    Article  MathSciNet  Google Scholar 

  7. Di Fazio, G., Hakim, D.I., Sawano, Y.: Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, vol. I. CRC Press, Boca Raton (2020)

    Google Scholar 

  8. Di Fazio, G., Hakim, D.I., Sawano, Y.: Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, vol. II. CRC Press, Boca Raton (2020)

    Google Scholar 

  9. Dorronsoro, J.R.: Poisson integrals of regular functions. Trans. Am. Math. Soc. 297, 669–685 (1986)

    Article  MathSciNet  Google Scholar 

  10. Drihem, D.: Characterizations of Besov-type and Triebel–Lizorkin-type spaces by differences. J. Funct. Spaces Appl. Article ID 328908, 24 (2012)

  11. DeVore, R.A., Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47(293), 115 (1984)

    MathSciNet  Google Scholar 

  12. Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Limiting embeddings of Besov-type and Triebel–Lizorkin-type spaces on domains and an extension operator. Ann. Mat. Pura Appl. 202(5), 2481–2516 (2023)

    Article  MathSciNet  Google Scholar 

  13. Haroske, D.D., Moura, S.D., Skrzypczak, L.: Smoothness Morrey spaces of regular distributions, and some unboundedness property. Nonlinear Anal. 139, 218–244 (2016)

    Article  MathSciNet  Google Scholar 

  14. Haroske, D.D., Moura, S.D., Skrzypczak, L.: Some embeddings of Morrey spaces with critical smoothness. J. Fourier Anal. Appl. 26(50), 31 (2020)

    MathSciNet  Google Scholar 

  15. Haroske, D.D., Schneider, C., Skrzypczak, L.: Morrey spaces on domains: different approaches and growth envelopes. J. Geom. Anal. 28, 817–841 (2018)

    Article  MathSciNet  Google Scholar 

  16. Haroske, D.D., Skrzypczak, L.: Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces, 23 pp (preprint) (2022). arXiv:2211.02594v1

  17. Hatano, N., Ikeda, M., Ishikawa, I., Sawano, Y.: Boundedness of composition operators on Morrey spaces and weak Morrey spaces. J. Inequal. Appl. 2021, 15 (2021)

    Article  MathSciNet  Google Scholar 

  18. Hedberg, L.I., Netrusov, Y.V.: An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation. Mem. Am. Math. Soc. 188(882), 97 (2007)

    MathSciNet  Google Scholar 

  19. Hovemann, M.: Besov–Morrey spaces and differences. Math. Rep. 23(73), 175–192 (2021)

    MathSciNet  Google Scholar 

  20. Hovemann, M.: Smoothness Morrey Spaces and Differences: Characterizations and Applications. Ph.D. Thesis, FSU Jena (2021)

  21. Hovemann, M.: Triebel–Lizorkin–Morrey spaces and differences. Math. Nachr. 295(4), 725–761 (2022)

    Article  MathSciNet  Google Scholar 

  22. Hovemann, M., Sickel, W.: Besov-type spaces and differences. Eurasian Math. J. 13(1), 25–56 (2020)

    Article  MathSciNet  Google Scholar 

  23. Kalyabin, G.A.: Theorems on extension, multipliers and diffeomorphisms for generalized Sobolev–Liouville classes in domains with a Lipschitz boundary. Trudy Mat. Inst. Steklov. 172, 173–186 (1985)

    MathSciNet  Google Scholar 

  24. Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel–Lizorkin-type spaces. Diss. Math. 489, 114 (2013)

    MathSciNet  Google Scholar 

  25. Lizorkin, P.I.: Operators connected with fractional derivatives and classes of differentiable functions. Trudy Mat. Inst. Steklov 117, 212–243 (1972)

    MathSciNet  Google Scholar 

  26. Lizorkin, P.I.: Properties of functions of the spaces \( \Lambda ^{r}_{p,\theta }\). Trudy Mat. Inst. Steklov 131, 158–181 (1974)

    MathSciNet  Google Scholar 

  27. Mazzucato, A.L.: Decomposition of Besov–Morrey spaces. In: Beckner, W., Nagel, A., Seeger, A., Smith, H.F. (eds.), Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), Contemporary Mathematics, vol. 320, pp. 279–294. Amer. Math. Soc, Providence (2003)

  28. Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1937)

    Article  MathSciNet  Google Scholar 

  29. Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains. J. Lond. Math. Soc. 60, 237–257 (1999)

    Article  MathSciNet  Google Scholar 

  30. Sawano, Y.: Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces on domains. Math. Nachr. 283, 1456–1487 (2010)

    Article  MathSciNet  Google Scholar 

  31. Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(6), 1535–1544 (2005)

    Article  MathSciNet  Google Scholar 

  32. Schott, T.: Function spaces with exponential weights I. Math. Nachr. 189, 221–242 (1998)

    Article  MathSciNet  Google Scholar 

  33. Seeger, A.: A note on Triebel–Lizorkin spaces. In: Ciesielski, Z. (ed.), Approximation and Function Spaces, Banach Center Publication, vol. 22, pp. 391–400. PWN-Polish Scientific Publishers, Warsaw (1989)

  34. Shvartsman, P.: Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of \({{\mathbb{R} }}^n\). Math. Nachr. 279, 1212–1241 (2006)

    Article  MathSciNet  Google Scholar 

  35. Sickel, W.: Smoothness spaces related to Morrey spaces: a survey, I. Eurasian Math. J. 3, 110–149 (2012)

    MathSciNet  Google Scholar 

  36. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    Google Scholar 

  37. Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278, 904–917 (2005)

    Article  MathSciNet  Google Scholar 

  38. Triebel, H.: Spaces of distributions of Besov type on Euclidean \(n\)-space. Duality interpolation. Ark. Mat. 11, 13–64 (1973)

    Article  MathSciNet  Google Scholar 

  39. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  40. Triebel, H.: Local approximation spaces. Z. Anal. Anwend. 8(3), 261–288 (1989)

    Article  MathSciNet  Google Scholar 

  41. Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)

    Book  Google Scholar 

  42. Triebel, H.: Theory of Function Spaces III. Birkhäuser, Basel (2006)

    Google Scholar 

  43. Triebel, H.: Hybrid Function Spaces, Heat and Navier–Stokes Equations, EMS Tracts in Mathematics, vol. 24. European Mathematical Society, Zürich (2014)

    Google Scholar 

  44. Yabuta, K.: Singular integral operators on Triebel–Lizorkin spaces. Bull. Fac. Sci. Ibaraki Univ. Ser. A 20, 9–17 (1988)

    Article  MathSciNet  Google Scholar 

  45. Yang, D., Yuan, W.: A new class of function spaces connecting Triebel–Lizorkin spaces and \(Q\) spaces. J. Funct. Anal. 255, 2760–2809 (2008)

    Article  MathSciNet  Google Scholar 

  46. Yang, D., Yuan, W.: New Besov-type spaces and Triebel–Lizorkin-type spaces including \(Q\) spaces. Math. Z. 265(2), 451–480 (2010)

    Article  MathSciNet  Google Scholar 

  47. Yao, L.: Some intrinsic characterizations of Besov–Triebel–Lizorkin–Morrey-type spaces on Lipschitz domains. J. Fourier Anal. Appl. 29, 21 (2023)

    Article  MathSciNet  Google Scholar 

  48. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)

  49. Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey–Campanato and related smoothness spaces. Sci. China Math. 58(9), 1835–1908 (2015)

    Article  MathSciNet  Google Scholar 

  50. Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin–Triebel–Morrey spaces on domains. Anal. Geom. Metr. Spaces 8, 268–304 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Winfried Sickel and Stephan Dahlke for several valuable discussions. Moreover, they like to thank the two anonymous reviewers for their constructive criticism.

Funding

Marc Hovemann has been supported by Deutsche Forschungsgemeinschaft (DFG), grant DA 360/24-1.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Philipps-University Marburg, Hans-Meerwein-Straße 6, 35043, Marburg, Germany

    Marc Hovemann

  2. Institute of Mathematics, Julius-Maximilians-Universität Würzburg (JMU), Emil-Fischer-Straße 30, 97074, Würzburg, Germany

    Markus Weimar

Authors
  1. Marc Hovemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Weimar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Hovemann.

Ethics declarations

Competing interests

The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Marc Hovemann has been supported by Deutsche Forschungsgemeinschaft (DFG), grant DA 360/24-1.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hovemann, M., Weimar, M. Oscillations and differences in Triebel–Lizorkin–Morrey spaces. Rev Mat Complut (2024). https://doi.org/10.1007/s13163-024-00487-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13163-024-00487-4

Keywords

Mathematics Subject Classification

Search

Navigation