Skip to main content
SpringerLink
Log in

Garsia-Rodemich Spaces: Bourgain-Brezis-Mironescu space, embeddings and rearrangement-invariant spaces

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We extend the construction of Garsia-Rodemich spaces in different directions. We show that the new space B, introduced by Bourgain, Brezis and Mironescu [6], can be described via a suitable scaling of the Garsia-Rodemich norms. As an application we give a new proof of the embeddings BMO ⊂ B ⊂ L(n’,∞). We then generalize the Garsia-Rodemich construction and introduce the GaRoX spaces associated with a rearrangement-invariant space X, in such a way that GaRoX = X, for a large class of rearrangement-invariant spaces. The underlying inequality for this new characterization of rearrangement-invariant spaces is an extension of the rearrangement inequalities of [17]. We introduce Gagliardo seminorms adapted to rearrangement-invariant spaces and use our generalized Garsia-Rodemich construction to prove Fractional Sobolev inequalities in this context.

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. L. Ambrosio, J. Bourgain, H. Brezis and A. Figalli, BMO type of norms related to the perimeter of sets, Comm. Pure Appl. Math. 69 (2016), 1062–1086.

    Article  MathSciNet  Google Scholar 

  2. S. Astashkin and M. Milman, Garsia-Rodemich spaces: Local maximal functions and interpolation, arXiv:1706.03045.

  3. W. Beckner, Embedding estimates and fractional smoothness, Int. Math. Res. Not. 2014 (2014), 390–417.

    Article  MathSciNet  Google Scholar 

  4. C. Bennett, R. DeVore and R. Sharpley, Weak -L ∞and BMO, Ann. of Math. 113 (1981), 601–611.

    Article  MathSciNet  Google Scholar 

  5. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, MA, 1988.

    MATH  Google Scholar 

  6. J. Bourgain, H. Brezis and P. Mironescu, A new function space and applications, J. Eur. Math. Soc. 17 (2015), 2083–2101.

    Article  MathSciNet  Google Scholar 

  7. J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, pp. 439–455.

    Google Scholar 

  8. D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. Math. J. 21 (1969), 1245–1254.

    Article  MathSciNet  Google Scholar 

  9. H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), 197–263.

    Article  MathSciNet  Google Scholar 

  10. H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), 773–789.

    Article  MathSciNet  Google Scholar 

  11. M. Cwikel, Y. Sagher and P. Shvartsmans, A new look at the John-Nirenberg and John-Strömberg theorems for BMO, J. Funct. Anal. 263 (2012), 129–167.

    Article  MathSciNet  Google Scholar 

  12. R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 3407–3430.

    Article  MathSciNet  Google Scholar 

  13. N. Fusco, V. Millot and M. Morino, Quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal. 261 (2011), 697–715.

    Article  MathSciNet  Google Scholar 

  14. A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973.

    MATH  Google Scholar 

  15. A.M. Garsia and E. Rodemich, Monotonicity of certain functional under rearrangements, Ann. Inst. Fourier (Grenoble) 24 (1974), 67–116.

    Article  MathSciNet  Google Scholar 

  16. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.

    Article  MathSciNet  Google Scholar 

  17. M. Milman, Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities, Ann. Acad. Sci. Fennicæ Math. 41 (2016), 491–501.

    Article  MathSciNet  Google Scholar 

  18. M. Milman, A note on self improvement of Poincaré-Sobolev inequalities via Garsia-Rodemich spaces, Pure Appl. Funct. Anal. 1 (2016), 429–439.

    MathSciNet  MATH  Google Scholar 

  19. M. Milman, BMO: Oscillations, self-improvement, Gagliardo coordinate spaces and reverse Hardy inequalities, in Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory, Springer-Verlag, Berlin, 2016, pp. 233–274.

    Google Scholar 

  20. R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129–142.

    Article  MathSciNet  Google Scholar 

  21. A. C. Ponce, Elliptic PDEs, Measures and Capacities: From the Poisson Equation to Nonlinear Thomas-Fermi Problems, European Mathematical Society, Zürich, 2016.

    Book  Google Scholar 

  22. A. C. Ponce and D. Spector, On formulae decoupling the total variation of BV functions, Nonlinear Anal. 154 (2017), 241–257.

    Article  MathSciNet  Google Scholar 

  23. E. Stein, Singular Integrals and the Differentiability of Functions, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  24. A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.

    MATH  Google Scholar 

  25. J. Xiao and Zh. Zhai, Fractional Sobolev, Moser-Trudinger, Morrey-Sobolev inequalities under Lorentz norms, J. Math. Sci. 166 (2010), 119–136.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Argentino de Matematica, Saavedra 15, 1083, Buenos Aires, Argentina

    Mario Milman

Authors
  1. Mario Milman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Milman.

Additional information

The author was partially supported by a grant from the Simons Foundation (#207929 to Mario Milman)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Milman, M. Garsia-Rodemich Spaces: Bourgain-Brezis-Mironescu space, embeddings and rearrangement-invariant spaces. JAMA 139, 121–141 (2019). https://doi.org/10.1007/s11854-019-0054-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-019-0054-2

Search

Navigation