Mathematische Annalen

, Volume 366, Issue 1–2, pp 695–720 | Cite as

Nonlinear commutators for the fractional p-Laplacian and applications

  • Armin SchikorraEmail author


We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded \(\frac{n}{s}\)-harmonic maps converge strongly outside at most finitely many points.

Mathematics Subject Classification

35D30 35B45 35J60 47G20 35S05 58E20 


  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230(4–6), 1859–1894 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blatt, S., Reiter, Ph., Schikorra, A.: Harmonic analysis meets critical knots (stationary points of the moebius energy are smooth). Trans. AMS (2014, accepted)Google Scholar
  4. 4.
    Bourgain, J., Brézis, H., Mironescu, P.: Another look at sobolev spaces. Optimal control and partial differential equations, pp. 439-455 (2001)Google Scholar
  5. 5.
    Brasco, L., Lindgren, E.: Higher sobolev regularity for the fractional \(p\)-Laplace equation in the superquadratic case (2015, preprint). arXiv:1508.01039
  6. 6.
    Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications (2015, preprint). arXiv:1504.08292
  7. 7.
    Da Lio, F.: Fractional harmonic maps into manifolds in odd dimension \(n {\>} 1\). Calc. Var. Partial Differ. Equ. 48(3–4), 421–445 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Da Lio, F.: Compactness and bubbles analysis for half-harmonic maps into spheres. Ann Inst. Henri Poincaré, Analyse non linèaire 32, 201–224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Da Lio, F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Da Lio, F., Rivière, T.: Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres. Anal. Partial Differ. Equ. 4(1), 149–190 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Di Castro, A., Kuusi, T., Palatucci, G.: Local behaviour of fractional \(p\)-minimizers (2014, preprint)Google Scholar
  12. 12.
    Di Castro, A., Kuusi, T., Palatucci, G.: Nonlocal harnack inequalities. J. Funct. Anal. 267, 1807–1836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, 2nd edn. Springer, New York (2009)Google Scholar
  15. 15.
    Iannizzotto, A., Mosconi, S., Squassina, M.: Global hölder regularity for the fractional p-laplacian. arXiv:1411.2956 (2014)
  16. 16.
    Iwaniec, T.: \(p\)-harmonic tensors and quasiregular mappings. Ann. Math. (2) 136(3), 589–624 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Iwaniec, T., Sbordone, C.: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143–161 (1994)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kuusi, T., Mingione, G., Sire, Y.: A fractional Gehring lemma, with applications to nonlocal equations. Rend. Lincei Mat. Appl. 25, 345–358 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337, 1317–1368 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kuusi, T., Mingione, G., Sire, Y.: Nonlocal self-improving properties. Anal. PDE 8, 57–114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schikorra, A.: Epsilon-regularity for systems involving non-local, antisymmetric operators (2012, preprint)Google Scholar
  22. 22.
    Schikorra, A.: Regularity of n/2-harmonic maps into spheres. J. Differ. Equ. 252, 1862–1911 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schikorra, A.: Integro-differential harmonic maps into spheres. Commun. Partial Differ Equ. 40(1), 506–539 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Triebel, H.: Theory of Function Spaces, Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  25. 25.
    Vazquez, J.-L.: The dirichlet problem for the fractional p-laplacian evolution equation (2015, preprint). arXiv:1506.00210

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Universität BaselBaselSwitzerland

Personalised recommendations