Local 𝐋¹ estimates for elliptic systems of complex vector fields

Hounie, J.; Picon, T.

doi:10.1090/S0002-9939-2014-12219-2

Local $\mathbf {L}^{1}$ estimates for elliptic systems of complex vector fields
HTML articles powered by AMS MathViewer

by J. Hounie and T. Picon PDF

Proc. Amer. Math. Soc. 143 (2015), 1501-1514 Request permission

Abstract:

In this paper we present a strong local version of the Gagliardo-Nirenberg estimate that holds for elliptic systems of vector fields with smooth complex coefficients. We also consider $L^1$ estimates on forms analogous to those known in the case of the de Rham complex on $\mathbb {R}^N$.

References

Chérif Amrouche and Huy Hoang Nguyen, New estimates for the div, curl, grad operators and elliptic problems with $L^1$-data in the half-space, Appl. Math. Lett. 24 (2011), no. 5, 697–702. MR 2765145, DOI 10.1016/j.aml.2010.12.008
Jean Bourgain and Haïm Brezis, On the equation $\textrm {div}\, Y=f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003), no. 2, 393–426. MR 1949165, DOI 10.1090/S0894-0347-02-00411-3
Jean Bourgain and Haïm Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (2004), no. 7, 539–543 (English, with English and French summaries). MR 2057026, DOI 10.1016/j.crma.2003.12.031
Jean Bourgain and Haïm Brezis, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 277–315. MR 2293957, DOI 10.4171/JEMS/80
Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. MR 2397326, DOI 10.1017/CBO9780511543067
F. Bethuel, G. Orlandi, and D. Smets, Approximations with vorticity bounds for the Ginzburg-Landau functional, Commun. Contemp. Math. 6 (2004), no. 5, 803–832. MR 2100765, DOI 10.1142/S0219199704001537
Haïm Brezis and Jean Van Schaftingen, Circulation integrals and critical Sobolev spaces: problems of optimal constants, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 33–47. MR 2500488, DOI 10.1090/pspum/079/2500488
Haïm Brezis and Jean Van Schaftingen, Boundary estimates for elliptic systems with $L^1$-data, Calc. Var. Partial Differential Equations 30 (2007), no. 3, 369–388. MR 2332419, DOI 10.1007/s00526-007-0094-9
Sagun Chanillo and Po-Lam Yung, An improved Strichartz estimate for systems with divergence free data, Comm. Partial Differential Equations 37 (2012), no. 2, 225–233. MR 2876830, DOI 10.1080/03605302.2011.594475
Sagun Chanillo and Jean Van Schaftingen, Subelliptic Bourgain-Brezis estimates on groups, Math. Res. Lett. 16 (2009), no. 3, 487–501. MR 2511628, DOI 10.4310/MRL.2009.v16.n3.a9
Georges de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Publ. Inst. Math. Univ. Nancago, III, Hermann & Cie, Paris, 1955 (French). MR 0068889
David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
J. Hounie and T. Picon, Local Gagliardo-Nirenberg estimates for elliptic systems of vector fields, Math. Res. Lett. 18 (2011), no. 4, 791–804. MR 2831843, DOI 10.4310/MRL.2011.v18.n4.a16
Loredana Lanzani and Elias M. Stein, A note on div curl inequalities, Math. Res. Lett. 12 (2005), no. 1, 57–61. MR 2122730, DOI 10.4310/MRL.2005.v12.n1.a6
Vladimir Maz’ya, Bourgain-Brezis type inequality with explicit constants, Interpolation theory and applications, Contemp. Math., vol. 445, Amer. Math. Soc., Providence, RI, 2007, pp. 247–252. MR 2381898, DOI 10.1090/conm/445/08605
Vladimir Maz’ya, Estimates for differential operators of vector analysis involving $L^1$-norm, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 1, 221–240. MR 2578609, DOI 10.4171/JEMS/195
Irina Mitrea and Marius Mitrea, A remark on the regularity of the div-curl system, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1729–1733. MR 2470831, DOI 10.1090/S0002-9939-08-09671-8
Pierre Bousquet and Petru Mironescu, An elementary proof of an inequality of Maz’ya involving $L^1$ vector fields, Nonlinear elliptic partial differential equations, Contemp. Math., vol. 540, Amer. Math. Soc., Providence, RI, 2011, pp. 59–63. MR 2807409, DOI 10.1090/conm/540/10659
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 88770, DOI 10.2307/1970051
Donald Ornstein, A non-equality for differential operators in the $L_{1}$ norm, Arch. Rational Mech. Anal. 11 (1962), 40–49. MR 149331, DOI 10.1007/BF00253928
S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993), no. 4, 206–238 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 4, 841–867. MR 1246427
François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459
Jean Van Schaftingen, Estimates for $L^1$-vector fields, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 181–186 (English, with English and French summaries). MR 2078071, DOI 10.1016/j.crma.2004.05.013
Jean Van Schaftingen, Limiting fractional and Lorentz space estimates of differential forms, Proc. Amer. Math. Soc. 138 (2010), no. 1, 235–240. MR 2550188, DOI 10.1090/S0002-9939-09-10005-9
Jean Van Schaftingen, Estimates for $L^1$ vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 867–882. MR 2443922, DOI 10.4171/JEMS/133
Jean Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 877–921. MR 3085095, DOI 10.4171/JEMS/380
Po-Lam Yung, Sobolev inequalities for $(0,q)$ forms on CR manifolds of finite type, Math. Res. Lett. 17 (2010), no. 1, 177–196. MR 2592736, DOI 10.4310/MRL.2010.v17.n1.a14