Nonlocal self-improving properties

Analysis and PDE

Volumn 8, Issue 1, 2015, Pages 57-114

  Kuusi, Tuomo ^a   Mingione, Giuseppe ^b   Sire, Yannick ^c  

^a AALTO UNIVERSITY   (Finland)

^b UNIVERSITY OF PARMA   (Italy)

^c AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Elliptic equations; Fractional differentiability; Nonlocal operators

Indexed keywords

EID: 84929758705     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2015.8.57     Document Type: Article

References (27)

1. R. F. Bass and M. Kassmann, "Hölder continuity of harmonic functions with respect to operators of variable order", Comm. Partial Differential Equations 30:7-9 (2005), 1249-1259.
2. R. F. Bass and H. Ren, "Meyers inequality and strong stability for stable-like operators", J. Funct. Anal. 265:1 (2013), 28-48.
3. C. Bjorland, L. Caffarelli, and A. Figalli, "Non-local gradient dependent operators", Adv. Math. 230:4-6 (2012), 1859-1894.
4. B. Bojarski and T. Iwaniec, "Analytical foundations of the theory of quasiconformal mappings in Rn", Ann. Acad. Sci. Fenn. Ser. A I Math. 8:2 (1983), 257-324.
5. X. Cabré and E. Cinti, "Sharp energy estimates for nonlinear fractional diffusion equations", Calc. Var. Partial Differential Equations 49:1-2 (2014), 233-269.
6. X. Cabré and J.-M. Roquejoffre, "The influence of fractional diffusion in Fisher-KPP equations", Comm. Math. Phys. 320:3 (2013), 679-722.
7. L. Caffarelli and L. Silvestre, "The Evans-Krylov theorem for nonlocal fully nonlinear equations", Ann. of Math. (2) 174:2 (2011), 1163-1187.
8. L. Caffarelli, C. H. Chan, and A. Vasseur, "Regularity theory for parabolic nonlinear integral operators", J. Amer. Math. Soc. 24:3 (2011), 849-869.
9. A. Di Castro, T. Kuusi, and G. Palatucci, "Local behavior of fractional p-minimizers", preprint, 2014, Available at http://cvgmt.sns.it/paper/2379/.
10. A. Di Castro, T. Kuusi, and G. Palatucci, "Nonlocal Harnack inequalities", J. Funct. Anal. 267:6 (2014), 1807-1836.
11. E. Di Nezza, G. Palatucci, and E. Valdinoci, "Hitchhiker's guide to the fractional Sobolev spaces", Bull. Sci. Math. 136:5 (2012), 521-573.
12. J. R. Dorronsoro, "A characterization of potential spaces", Proc. Amer. Math. Soc. 95:1 (1985), 21-31.
13. M. Felsinger and M. Kassmann, "Local regularity for parabolic nonlocal operators", Comm. Partial Differential Equations 38:9 (2013), 1539-1573.
14. N. Fusco and C. Sbordone, "Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions", Comm. Pure Appl. Math. 43:5 (1990), 673-683.
15. F. W. Gehring, "The L p-integrability of the partial derivatives of a quasiconformal mapping", Acta Math. 130 (1973), 265-277.
16. E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., River Edge, NJ, 2003.
17. T. Iwaniec, "The Gehring lemma", pp. 181-204 in Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), edited by P. Duren et al., Springer, New York, 1998.
18. M. Kassmann, "A priori estimates for integro-differential operators with measurable kernels", Calc. Var. Partial Differential Equations 34:1 (2009), 1-21.
19. J. Kinnunen and J. L. Lewis, "Higher integrability for parabolic systems of p-Laplacian type", Duke Math. J. 102:2 (2000), 253-271.
20. T. Kuusi and G. Mingione, "Universal potential estimates", J. Funct. Anal. 262:10 (2012), 4205-4269.
21. T. Kuusi, G. Mingione, and Y. Sire, "A fractional Gehring lemma, with applications to nonlocal equations", Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25:4 (2014), 345-358.
22. V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften 342, Springer, Heidelberg, 2011.
23. N. G. Meyers, "An L p-estimate for the gradient of solutions of second order elliptic divergence equations", Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 189-206.
24. N. G. Meyers and A. Elcrat, "Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions", Duke Math. J. 42 (1975), 121-136.
25. G. Mingione, "The singular set of solutions to non-differentiable elliptic systems", Arch. Ration. Mech. Anal. 166:4 (2003), 287-301.
26. E. M. Stein, "The characterization of functions arising as potentials", Bull. Amer. Math. Soc. 67 (1961), 102-104.
27. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.