# American Institute of Mathematical Sciences

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

## Nirenberg's contributions to complex analysis

 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.
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
 [1] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [2] Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure & Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627 [3] Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291 [4] JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042 [5] Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 [6] Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481 [7] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [8] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [9] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [10] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [11] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [12] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [13] Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 [14] Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021019 [15] Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763 [16] Keizo Hasegawa. Complex moduli and pseudo-Kahler structures on three-dimensional compact complex solvmanifolds. Electronic Research Announcements, 2007, 14: 30-34. doi: 10.3934/era.2007.14.30 [17] Rui Zhang, Yong-Kui Chang, G. M. N'Guérékata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525 [18] Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056 [19] Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240 [20] Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

2019 Impact Factor: 1.338