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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

J. Bokobza and A. Unterberger, Les operators de Calderón-Zygmund précisés, C. R. Acad. Sci. Paris, 259 (1964), 1612-1614.  Google Scholar

[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.  Google Scholar

[5]

D. Catlin, Necessary conditions for subellipticity of the $\bar\partial$-Neumann problem, Ann. of Math., 117 (1983), 147-171. doi: 10.2307/2006974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. P. D'Angelo, Finite type conditions for real hypersurfaces, J. Diff. Geom., 14 (1979), 59-66.  Google Scholar

[10]

C. L. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974), 1-65. doi: 10.1007/BF01406845.  Google Scholar

[11]

G. Fichera, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem. Ser. 8, 5 (1956), 97-120.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. J. Kohn, Boundary behavior of $\bar\partial$ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. B. Morrey, The analytic embedding of abstract real analytic manifolds, Ann. of Math., 40 (1958), 62-70.  Google Scholar

[22]

A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), 391-404. doi: 10.2307/1970051.  Google Scholar

[23]

L. Nirenberg, "A Complex Frobenius Theorem," Proc. Conf. Analytic Functions, vol. 1, Institute for Advanced Study, Princeton (1957), 172-189. Google Scholar

[24]

L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys, 29 (1974), 251-262. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

O. A. Oleinik, A boundary value problem for linear elliptic parabolic equations, Doklady Akad. Nauk. SSSR, 163 (1963), 577-580. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

J. Bokobza and A. Unterberger, Les operators de Calderón-Zygmund précisés, C. R. Acad. Sci. Paris, 259 (1964), 1612-1614.  Google Scholar

[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.  Google Scholar

[5]

D. Catlin, Necessary conditions for subellipticity of the $\bar\partial$-Neumann problem, Ann. of Math., 117 (1983), 147-171. doi: 10.2307/2006974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. P. D'Angelo, Finite type conditions for real hypersurfaces, J. Diff. Geom., 14 (1979), 59-66.  Google Scholar

[10]

C. L. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974), 1-65. doi: 10.1007/BF01406845.  Google Scholar

[11]

G. Fichera, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem. Ser. 8, 5 (1956), 97-120.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. J. Kohn, Boundary behavior of $\bar\partial$ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. B. Morrey, The analytic embedding of abstract real analytic manifolds, Ann. of Math., 40 (1958), 62-70.  Google Scholar

[22]

A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), 391-404. doi: 10.2307/1970051.  Google Scholar

[23]

L. Nirenberg, "A Complex Frobenius Theorem," Proc. Conf. Analytic Functions, vol. 1, Institute for Advanced Study, Princeton (1957), 172-189. Google Scholar

[24]

L. Nirenberg, On a question of Hans Lewy, Russian Math. Surveys, 29 (1974), 251-262. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

O. A. Oleinik, A boundary value problem for linear elliptic parabolic equations, Doklady Akad. Nauk. SSSR, 163 (1963), 577-580. Google Scholar

[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.  Google Scholar

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