Nirenberg's contributions to complex analysis

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May 2011, 30(2): 537-545. doi: 10.3934/dcds.2011.30.537

Nirenberg's contributions to complex analysis

Joseph J Kohn ^1,

1.	Princeton University, Mathematics Department, Princeton, NJ 08544, United States

Received September 2010 Published February 2011

Abstract
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This article is concerned with the fundamental contributions of Louis Nirenberg to complex analysis and their impact on the theory of partial differential equations. We explain some of his main results and sketch the developments that they engendered.

Keywords: pseudo convex, Partial differential equations, complex analysis, pseudo differential., subelliptic.

Mathematics Subject Classification: 32T25, 32T27, 32T40, 32W25, 35H20, 35N1.

Citation: Joseph J Kohn. Nirenberg's contributions to complex analysis. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 537-545. doi: 10.3934/dcds.2011.30.537

References:

[1]	T. Akahori, "A New Approach to the Local Embedding Theorem of CR-Structure for $n\ge4$,'' Memoirs of the A.M.S., Providence, RI, 1987.
[2]	S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math., 57 (1980), 283-289. doi: 10.1007/BF01418930.
[3]	J. Bokobza and A. Unterberger, Les operators de Calderón-Zygmund précisés, C. R. Acad. Sci. Paris, 259 (1964), 1612-1614.
[4]	L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, II: Complex Monge-Ampère, and uniformly elliptic equations, Comm. Pure Appl. Math., 38 (1985), 209-252. doi: 10.1002/cpa.3160380206.
[5]	D. Catlin, Necessary conditions for subellipticity of the $\bar\partial$-Neumann problem, Ann. of Math., 117 (1983), 147-171. doi: 10.2307/2006974.
[6]	D. Catlin, Subelliptic estimates for the $\bar\partial$-Neumann problem on pseudoconvex domains, Ann. of Math., 126 (1987), 131-191. doi: 10.2307/1971347.
[7]	S. S. Chern, H. I. Levine and L. Nirenberg, Intrinsic norms on a complex manifold, Global Anal., Univ. of Tokyo Press, (1969), 119-139.
[8]	S.-C. Chen and M.-C. Shaw, "Partial Differential Equations in Several Complex Variables," AMS/IP Stud. Adv. Math. 19, AMS Providence, R. I., 2001.
[9]	J. P. D'Angelo, Finite type conditions for real hypersurfaces, J. Diff. Geom., 14 (1979), 59-66.
[10]	C. L. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974), 1-65. doi: 10.1007/BF01406845.
[11]	G. Fichera, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem. Ser. 8, 5 (1956), 97-120.
[12]	K. Kodaira, L. Nirenberg and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math., 68 (1958), 450-459. doi: 10.2307/1970256.
[13]	J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds I and II, Ann. of Math. 78 (1963), 112-148; 79 (1964), 450-472.
[14]	J. J. Kohn, Boundary behavior of $\bar\partial$ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542.
[15]	J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math., 18 (1965), 443-492. doi: 10.1002/cpa.3160180305.
[16]	J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math., 18 (1965), 269-305. doi: 10.1002/cpa.3160180121.
[17]	J. J. Kohn and L. Nirenberg, Degenerate ellptic-parabolic equations of second order, Comm. Pure Appl. Math., 20 (1967), 797-782. doi: 10.1002/cpa.3160200410.
[18]	J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265-268. doi: 10.1007/BF01428194.
[19]	M. Kuranishi, Strongly psedoconvex CR structures over small balls, Part I, An a priori estimate, Part II, A regularity theorem, Part III, An embedding theorem, Ann. of Math., 115 (1982), 451-500; 116 (1982), 1-64; 116 (1982), 249-330.
[20]	P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gårding's inequality, Comm. Pure Appl. Math., 19 (1966), 473-492. doi: 10.1002/cpa.3160190409.
[21]	C. B. Morrey, The analytic embedding of abstract real analytic manifolds, Ann. of Math., 40 (1958), 62-70.
[22]	A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), 391-404. doi: 10.2307/1970051.
[23]	L. Nirenberg, "A Complex Frobenius Theorem," Proc. Conf. Analytic Functions, vol. 1, Institute for Advanced Study, Princeton (1957), 172-189.
[24]	L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys, 29 (1974), 251-262.
[25]	L. Nirenberg and D. C. Spencer, On rigidity of holomorphic imbeddings, Contributions to function theory, Tata Institute of Fundamental Research, Bombay, (1960), 133-137.
[26]	L. Nirenberg, S. Webster and P. Yang, Local boundary regularity of holomorphic mappings, Comm Pure and Appl. Math., 33 (1980), 305-338. doi: 10.1002/cpa.3160330306.
[27]	O. A. Oleinik, A boundary value problem for linear elliptic parabolic equations, Doklady Akad. Nauk. SSSR, 163 (1963), 577-580.
[28]	E. J. Straube, "Lectures on the $\mathcal L^2$-Sobolev Theory of the $\bar\partial$-Neumann Problem,'' LSI Lectures in Mathematics and Physics, European Mathematical Society, 2010. doi: 10.4171/076.

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References:

[1]	T. Akahori, "A New Approach to the Local Embedding Theorem of CR-Structure for $n\ge4$,'' Memoirs of the A.M.S., Providence, RI, 1987.
[2]	S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math., 57 (1980), 283-289. doi: 10.1007/BF01418930.
[3]	J. Bokobza and A. Unterberger, Les operators de Calderón-Zygmund précisés, C. R. Acad. Sci. Paris, 259 (1964), 1612-1614.
[4]	L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, II: Complex Monge-Ampère, and uniformly elliptic equations, Comm. Pure Appl. Math., 38 (1985), 209-252. doi: 10.1002/cpa.3160380206.
[5]	D. Catlin, Necessary conditions for subellipticity of the $\bar\partial$-Neumann problem, Ann. of Math., 117 (1983), 147-171. doi: 10.2307/2006974.
[6]	D. Catlin, Subelliptic estimates for the $\bar\partial$-Neumann problem on pseudoconvex domains, Ann. of Math., 126 (1987), 131-191. doi: 10.2307/1971347.
[7]	S. S. Chern, H. I. Levine and L. Nirenberg, Intrinsic norms on a complex manifold, Global Anal., Univ. of Tokyo Press, (1969), 119-139.
[8]	S.-C. Chen and M.-C. Shaw, "Partial Differential Equations in Several Complex Variables," AMS/IP Stud. Adv. Math. 19, AMS Providence, R. I., 2001.
[9]	J. P. D'Angelo, Finite type conditions for real hypersurfaces, J. Diff. Geom., 14 (1979), 59-66.
[10]	C. L. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974), 1-65. doi: 10.1007/BF01406845.
[11]	G. Fichera, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem. Ser. 8, 5 (1956), 97-120.
[12]	K. Kodaira, L. Nirenberg and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math., 68 (1958), 450-459. doi: 10.2307/1970256.
[13]	J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds I and II, Ann. of Math. 78 (1963), 112-148; 79 (1964), 450-472.
[14]	J. J. Kohn, Boundary behavior of $\bar\partial$ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542.
[15]	J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math., 18 (1965), 443-492. doi: 10.1002/cpa.3160180305.
[16]	J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math., 18 (1965), 269-305. doi: 10.1002/cpa.3160180121.
[17]	J. J. Kohn and L. Nirenberg, Degenerate ellptic-parabolic equations of second order, Comm. Pure Appl. Math., 20 (1967), 797-782. doi: 10.1002/cpa.3160200410.
[18]	J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265-268. doi: 10.1007/BF01428194.
[19]	M. Kuranishi, Strongly psedoconvex CR structures over small balls, Part I, An a priori estimate, Part II, A regularity theorem, Part III, An embedding theorem, Ann. of Math., 115 (1982), 451-500; 116 (1982), 1-64; 116 (1982), 249-330.
[20]	P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gårding's inequality, Comm. Pure Appl. Math., 19 (1966), 473-492. doi: 10.1002/cpa.3160190409.
[21]	C. B. Morrey, The analytic embedding of abstract real analytic manifolds, Ann. of Math., 40 (1958), 62-70.
[22]	A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), 391-404. doi: 10.2307/1970051.
[23]	L. Nirenberg, "A Complex Frobenius Theorem," Proc. Conf. Analytic Functions, vol. 1, Institute for Advanced Study, Princeton (1957), 172-189.
[24]	L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys, 29 (1974), 251-262.
[25]	L. Nirenberg and D. C. Spencer, On rigidity of holomorphic imbeddings, Contributions to function theory, Tata Institute of Fundamental Research, Bombay, (1960), 133-137.
[26]	L. Nirenberg, S. Webster and P. Yang, Local boundary regularity of holomorphic mappings, Comm Pure and Appl. Math., 33 (1980), 305-338. doi: 10.1002/cpa.3160330306.
[27]	O. A. Oleinik, A boundary value problem for linear elliptic parabolic equations, Doklady Akad. Nauk. SSSR, 163 (1963), 577-580.
[28]	E. J. Straube, "Lectures on the $\mathcal L^2$-Sobolev Theory of the $\bar\partial$-Neumann Problem,'' LSI Lectures in Mathematics and Physics, European Mathematical Society, 2010. doi: 10.4171/076.

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