MR3241906 Takashi ICHINOSE and Yoshimi SAITŌ Takashi ICHINOSE and Yoshimi SAITŌ Improved Sobolev Embedding Theorems for Vector-Valued Functions Funkcialaj Ekvacioj. Serio Internacia 57 2014 245--295 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-2/57_245.pdf http://www.ams.org/mathscinet-getitem?mr=MR3241906 The aim of this paper is to give an extension of the improved Sobolev embedding theorem for single-valued functions to the case of vector-valued functions which is involved with the three-dimensional massless Dirac operator together with the three- or two-dimensional Weyl-Dirac (or Pauli) operator, the Cauchy-Riemann operator and also the four-dimensional Euclidian Dirac operator. Sobolev inequality, Gagliardo-Nirenberg inequality, Improved Sobolev embedding theorem, Dirac-Sobolev inequality, Sobolev inequality for vector-valued functions, Dirac operator. 46E35, 46E40, 35Q40. 57-245 2014 Improved Sobolev Embedding Theorems for Vector-Valued Functions Takashi ICHINOSE and Yoshimi SAITŌ Takashi ICHINOSE and Yoshimi SAITŌ
1 Arai, A. Path integral representation of the index of Kähler-Dirac operators on an infinite-dimensional manifold J. Funct. Anal. 82 1989 330-369 MR0987298
2 Arai, A. A general class of infinite-dimensional Dirac operators and path integral representation of their index J. Funct. Anal. 105 1982 342-408 MR1160082
3 Balinsky, A., Evans, W. D., Saitō, Y. Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators J. Math. Phys. 49 2008 043524, 10 pp MR2412309
4 Balinsky, A., Evans, W. D., Umeda, T. The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$ Publ. Res. Inst. Math. Sci. 47 2011 791-801 MR2832806
5 Bethe, H. A., Salpeter, E. E. Quantum mechanics of one- and two-electron atoms Reprint of the 1957 original, A Plenum/Rosetta Edition, Plenum Publishing Corp., New York 1977 MR0441102
6 Bourgain, J.; Brezis, H. New estimates for the Laplacian, the div-curl, and related Hodge systems C. R. Math. Acad. Sci. Paris 338 2004 539-543 MR2057026
7 Cohen, A.; DeVore, R.; Meyer, Y.; Petrushev, P.; Xu, H. Nonlinear approximation and the space $BV({\Bbb R}^2)$ Amer. J. Math. 121 1999 587-628 MR1738406
8 Cohen, A.; Meyer, Y.; Oru,F. Improved Sobolev embedding theorem Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau 1998 Exp. No. XVI, 16 pp MR1660529
9 Fefferman, C. $L^p$ bounds for pseudo-differential operators Israel J. Math. 14 1973 413-417 MR0336453
10 Friedrich, T. Dirac Operators in Riemannian Geometry Graduate Studies in Mathematics, 25, American Mathematical Society, Providence, RI 2000 MR1777332
11 Good, R. H. Properties of the Dirac matrices Rev. Mod. Phys. 27 1955 187-211 MR0070467
12 Ichinose, T., Saitō, Y. Dirac-Sobolev spaces and Sobolev spaces Funkcial. Ekvac. 53 2010 291-310 MR2730625 2730625 13 Itzykson, C., Zuber, J. B. Quantum Field Theory International Series in Pure and Applied Physics. McGraw-Hill International Book Co., New York 1980 MR0585517
14 Lanzani, L.; Stein, E. M. A note on div curl inequalities Math. Res. Lett. 12 2005 57-61 MR2122730
15 Lawson, H. B.; Michelsohn, M-L. Spin Geometry Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ 1989 MR1031992
16 Ledoux, M. On improved Sobolev embedding theorems Math. Res. Lett. 10 2003 659-669 MR2024723
17 Lieb, E. H.; Loss, M. Analysis 2nd edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI 2001 MR1817225
18 Loss, M., Yau, H. T. Stability of Coulomb systems with magnetic fields, III. Zero energy bound states of the Pauli operators Comm. Math. Phys. 104 1986 283-290 MR0836005
19 Pauli, W. Contributions mathématiques à la théorie des matrices de Dirac Ann. Inst. H. Poncaré 6 1936 109-136 MR1508031
20 Stein, E. M. Singular Integrals and Differentiability Properties of Functions Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970 MR0290095 21 Triebel, H. Interpolation Theory, Function Spaces, Differential Operators 2nd edition, Johann Ambrosius Barth, Heidelberg 1995 MR1328645 22 Weisstein, Eric W., “Dirac Matrices”, From Wolfram MathWorld. http://mathworld.wolfram.com/DiracMatrices.html