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S.: Spectral aspects of a class of differential operators, preprint 2001\nHedenmalm, H., Korenblum, B., Zhu, K.: Beurling type invaeiant subspaces of the Bergman spaces. J. London Math. Soc., 53(2), 601–614 (1996)\nGuo, K.: Defect operators for submodules of H 2d . J, reine angew. Math., 573, 181–209 (2004)\nGuo, K.: Defect operators, defect functions and defect indices for analytic submoduls. J. Func. Anal., 213, 380–411 (2004)\nGuo, K., Zheng, D.: Invariant subspaces, quasi-invariant subspaces and Hankel operators. J. Funct. Anal., 187, 308–342 (2001)\nJiang, C., Jin, Y., Wang, Z.: About operator weighted shift. Acta Mathematica Sinica, English Series, 23(8), 1385–1390 (2007)\nBourdon, P.: Similarity of parts to the whole for certain multiplication operators, Proc. Amer. Math. Soc. V. 99, N. 3, March 1987\nZhu, K.: Restriction of the Bergman shift to an invariant subspace. Quart. J. Math. Orford, 48(2), 519–532 (1997)\nGuo, K.: Homogeneous quasi-invariant subspaces of the Fock space. J. Aust. Math. Soc., 75, 399–407 (2003)\nRichter, S.: Unitary equivalence of invariant subspaces of Bergman and Dirichlet spaces. Pacific J. Math., 133, 151–156 (1988)",{"EN":233},"In this paper, we study the relation between the ordered reproducing Hilbert space and its reproducing kernel. 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Ann. of Math., 144, 453–496 (1996)\nCaffarelli, L., Nirenberg, L., Spruck, J.: Dirichlet problem for nonlinear second order elliptic equations I, Monge-Ampère equations. Comm. Pure Appl. Math., 37, 369–402 (1984)\nCaffarelli, L., Nirenberg, L., Spruck, J.: Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math., 155, 261–301 (1985)\nCaffarelli, L., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations IV: Starshaped compact Weigarten hypersurfaces. In: Current Topics in Partial Differential Equations, Y. Ohya, K. Kasahara and N. Shimakura (eds.), Kinokunize, Tokyo, 1–26 (1985)\nCaffarelli, L., Nirenberg, L., Spruck, J.: Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces. Comm. Pure Appl. Math., 41, 47–70 (1988)\nChang, S. Y., Gursky, M., Yang, P.: An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math., 155, 709–787 (2002)\nChen, C. Q.: The interior gradient estimate of Hessian quotient equations. J. Differential Equations, 259, 1014–1023 (2015)\nChen, C. Q., Zhang, D. K.: The Neumann problem of Hessian quotient equations. Bulletin of Mathematical Sciences, DOI: https:\u002F\u002Fdoi.org\u002F10.1142\u002FS1664360720500186 (2020)\nChen, S.: Boundary value problems for some fully nonlinear elliptic equations. Cal. Var. Partial Differential Equations, 30, 1–15 (2007)\nChen, Y. Z., Wu L. C.: Second Order Elliptic Equations and Elliptic Systems. Amer. Math. Soc., Providence, RI, 1998\nChou, K. S., Wang, X. J.: A variation theory of the Hessian equation. Comm. Pure Appl. Math., 54(9), 1029–1064 (2001)\nDinew, S., Kolodziej, S.: Liouville and Calabi-Yau type theorems for complex Hessian equations. American Journal of Mathematics, 139, 403–415 (2017)\nEscobar, J.: The Yamabe problem on manifolds with boundary. J. Diff. Geom., 35, 21–84 (1992)\nEvans, L. C.: Classical solutions of fully nonlinear convex second-order elliptic equations. Comm. Pure Appl. Math., 35, 333–363 (1982)\nGilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, x+401 pp. ISBN: 3-540-08007-4, 1977\nGerhardt, C.: Global regularity of the solutions to the capillary problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3, 151–176 (1976)\nGuan, B.: Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J., 163, 1491–1524 (2014)\nGuan, B., Guan, P. F.: Convex hypersurfaces of prescribed curvature. Ann. of Math., 156, 655–674 (2002)\nGuan, P. F., Ma, X. N.: The Christoffel-Minkowski problem I: convexity of solutions of a Hessian equations. Invent. Math., 151, 553–577 (2003)\nHou, Z. L., Ma, X. N., Wu, D. M.: A second order estimate for complex Hessian equations on a compact Kähler manifold. Math. Res. Lett., 17, 547–561 (2010)\nHuisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math., 183, 45–70 (1999)\nIvochkina, N.: Solutions of the Dirichlet problem for certain equations of Monge-Ampère type (in Russian). Mat. Sb., 128, 403–415 (1985)\nJiang, F. D., Trudinger, N. S.: Oblique boundary value problems for augmented Hessian equations I. Bulletin of Mathematical Sciences, 8, 353–411 (2018)\nJiang, F. D., Trudinger, N. S.: Oblique boundary value problems for augmented Hessian equations II. Nonlinear Analysis: Theory, Methods & Applications, 154, 148–173 (2017)\nJin, Q. N., Li, A. B., Li, Y. Y.: Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary. Cal. Var. Partial Differential Equations, 28, 509–543 (2007)\nKrylov, N. V.: Boundedly inhomogeneous elliptic and parabolic equations. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 46, 487–523 (1982)\nKrylov, N. V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 47, 75–108 (1983)\nKrylov, N. V.: Degenerate nonlinear elliptic equations. (Russian) Mat. Sb. (N.S.), 120(162), 311–330 (1983)\nLi, Y. Y.: Some existence results for fully nonlinear elliptic equations of Monge-Ampère type. Comm. Pure Appl. Math., 43, 233–271 (1990)\nLi, Y. Y.: Interior gradient estimates for solutions of certain fully nonlinear elliptic equations. J. Differential Equations, 90(1), 172–185 (1991)\nLi, Y. Y., Luc, N.: A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary. Adv. Math., 251, 87–110 (2014)\nLieberman, G.: Second Order Parabolic Differential Equations, World Scientific Publishing Co., 1996\nLieberman, G.: Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing, 2013\nLieberman, G., Trudinger, N.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Amer. Math. Soc., 295, 509–546 (1986)\nLions, P. L., Trudinger, N., Urbas, J.: The Neumann problem for equations of Monge-Ampère type. Comm. Pure Appl. Math., 39, 539–563 (1986)\nMa, X. N., Qiu, G. H.: The Neumann problem for Hessian equations. Communications in Mathematical Physics, 366, 1–28 (2019)\nMa, X. N., Trudinger, N., Wang, X. J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal., 177, 151–183 (2005)\nMa, X. N., Wang, P. H., Wei, W.: Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains. Journal of Functional Analysis, 274, 252–277 (2018)\nMa, X. N., Xu, J. J.: Gradient estimates of mean curvature equations with Neumann boundary condition. Advances in Mathematics, 290, 1010–1039 (2016)\nPogorelov, A. V.: The Minkowski Multidimensional Problem, John Wiley, 1978\nQiu, G. H., Xia, C.: Classical Neumann Problems for Hessian Equations and Alexandrov-Fenchel’s Inequalities. Int. Math. Res. Not., 20, 6285–6303 (2019)\nSchnürer, O. C., Smoczyk, K.: Neumann and second boundary value problems for Hessian and Gauss curvature flows. Ann. Inst Henri Poincarè-Anal. Non Lin., 20, 1043–1073 (2003)\nSchnürer, O. C., Schwetlick, H. R.: Translating solutions for Gauss curvature flows with Neumann boundary conditions. Pacific Journal of Mathematics, 213, 89–109 (2004)\nSheng, W. M., Trudinger, N., Wang, X. J.: Convex hypersurfaces of prescribed Weingarten curvatures. Comm. Anal. Geom., 12, 213–232 (2004)\nSimon, L., Spruck, J.: Existence and regularity of a capillary surface with prescribed contact angle. Arch. Rational Mech. Anal., 61, 19–34 (1976)\nSpruck, J.: Geometric aspects of the theory of fully nonlinear elliptic equations. Clay Mathematics Proceedings, 2, 283–309 (2005)\nTrudinger, N. S.: On degenerate fully nonlinear elliptic equations in balls. Bulletin of the Australian Math. Soc., 35, 299–307 (1987)\nTrudinger, N. S.: On the Dirichlet problem for Hessian equations. Acta Math., 175, 151–164 (1995)\nUral’tseva, N.: Solvability of the capillary problem. Vestnik Leningrad. Univ. No. 19 (1973), 54–64, No. 1 (1975), 143–149 [Russian]. English Translation in Vestnik Leningrad Univ. Math., 6, 363–375 (1979); 8, 151–158 (1980)\nUrbas, J.: Nonlinear oblique boundary value problems for Hessian equations in two dimensions. Ann. Inst Henri Poincarè-Anal. 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C., Zalcman, L.: Normal families and shared values. Bull. London Math. Soc., 32, 325–331 (2000)\nZhang, G. M., Sun, W., Pang, X. C.: On the normality of certain king of holomorphic functions. Chin. Ann. Math. Ser. A, 26(6), 765–770 (2005)\nChang, J. M., Fang, M. L., Zalcman, L.: Normal families of holomorphic functions. Illinois Math. J., 48(1), 319–337 (2004)\nLucas, F.: Géométrie des polynômes. J. École Polytech., 46(1), 1–33 (1879)\nMarden, M.: Geometry of Polynomials, American Mathematical Society, Providence, Rhode Island, 1966\nPang, X. C.: Bloch’s principle and normal criterion. Sci. China Ser. A, 32, 782–791 (1989)\nZalcman, L.: Normal families: new perspectives. Bull. Amer. Math. Soc. (N.S.), 35, 215–230 (1998)\nNevo, S.: Applications of Zalcman’s lemma to Q m-normal families. Analysis, 21, 289–325 (2001)\nNevo, S., Pang, X. C., Zalcman, L.: Quasinormality and meromorphic functions with multiple zeros. J. Anal. Math., 101, 1–23 (2007)\nZalcman, L.: A heuristic principle in complex function theory. Amer. Math. Monthly, 82, 813–817 (1975)\nClunie, J., Hayman, W. K.: The spherical derivative of integral and meromorphic functions. Comment Math. Helvet., 40, 117–148 (1966)\nLehto, O.: The spherical derivative of a meromorphic function in the neighborhood of an isolated singularity. Comment Math. Helvet., 33, 196–205 (1959)\nPang, X. C., Zalcman, L.: Normal families of meromorphic functions with multiple zeros and poles. Israel J. Math., 136, 1–9 (2003)",{"EN":759},"Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f\n                        (k)(z)| ≤ c, where c is a constant. 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Ann. Inst. Fourier (Grenoble), 56(6), 1633–1662 (2006)\nBerndtsson, B.: The openness conjecture for plurisubharmonic functions, arXiv:1305.5781\nBerndtsson, B., Lempert, L.: A proof of the Ohsawa—Takegoshi theorem with sharp estimates. J. Math. Soc. Japan, 68(4), 1461–1472 (2016)\nBlocki, Z.: Suita conjecture and the Ohsawa—Takegoshi extension theorem. Invent. Math., 193(1), 149–158 (2013)\nDemailly, J. P.: Transcendental proof of a generalized Kawamata—Viehweg vanishing theorem. In: Geometrical and Algebraical Aspects in Several Complex variables (Cetraro, 1989), Sem. Conf., Vol. 8, EditEl, Rende, 1991, 81–94\nDemailly, J. P.: Complex analytic and differential geometry, electronically accessible at https:\u002F\u002Fwww-fourier.ujf-grenoble.fr\u002F∼demailly\u002Fmanuscripts\u002Fagbook.pdf\nDemailly, J. P.: Multiplier ideal sheaves and analytic methods in algebraic geometry, In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes, Vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 1–148\nDemailly, J. P.: Analytic methods in algebraic geometry, International Press, Somerville, MA; Higher Education Press, Beijing, 2012\nDemailly, J. P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler—Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4), 34(4), 525–556 (2001)\nGuan, Q. A.: A sharp effectiveness result of Demailly’s strong openness conjecture. Adv. Math., 348, 51–80 (2019)\nGuan, Q. A., Zhou, X. Y.: Strong openness conjecture and related problems for plurisubharmonic functions. arXiv:1401.7158\nGuan, Q. A., Zhou, X. Y.: A proof of Demailly’s strong openness conjecture. Ann. of Math. (2), 182(2), 605–616 (2015)\nGuan, Q. A., Zhou, X. Y.: Effectiveness of Demailly’s strong openness conjecture and related problems. Invent. Math., 202(2), 635–676 (2015)\nGuan, Q. A., Zhou, X. Y.: A solution of an L2 extension problem with an optimal estimate and applications. Ann. of Math. (2), 181(3), 1139–1208 (2015)\nHörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd Ed., North-Holland Publishing Co., Amsterdam, 1990\nJonsson, M., Mustaţă, M.: An algebraic approach to the openness conjecture of Demailly and Kollár. J. Inst. Math. Jussieu, 13(1), 119–144 (2014)\nMaitani, F., Yamaguchi, H.: Variation of Bergman metrics on Riemann surfaces. Math. Ann., 330(3), 477–489 (2004)\nNadel, A.: Multiplier ideal sheaves and Kähler—Einstein metrics of positive scalar curvature. Ann. of Math. (2), 132(3), 549–596 (1990)\nOhsawa, T.: On the extension of L2 holomorphic functions. V. Effects of generalization. Nagoya Math. J., 161, 1–21 (2001)\nOhsawa, T.: Analysis of Several Complex Variables, American Mathematical Society, Providence, RI, 2002\nOhsawa, T.: L2 approaches in several complex variables. Development of Oka—Cartan Theory by L2 Estimates for the \\(\\bar \\partial \\) Operator, Springer, Tokyo, 2015\nOhsawa, T.: L2 approaches in several complex variables. Towards the Oka—Cartan Theory with Precise Bounds, Springer, Tokyo, 2018\nRudin, W.: Functional analysis, McGraw-Hill Book Co., New York—Düsseldorf—Johannesburg, 1973\nSiu, Y. T.: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China Ser. A, 48(suppl.), 1–31 (2005)\nTian, G.: On Köhler—Einstein metrics on certain Kähler manifolds with C1(M) > 0. Invent. Math., 89(2), 225–246 (1987)",{"EN":909},"In this note, we present an L2 extension approach to the effectiveness result of strong openness property of multiplier ideal sheaves.",{"EN":911},"L2 Extension and Effectiveness of Strong Openness Property",{"VOID":913},"10.1007\u002Fs10114-022-1220-5","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10114-022-1220-5",[916,931],{"id":917,"sortIndex":115,"researcher":20,"roles":918,"affiliations":919,"properties":928},"fc80e22b-9919-44b2-817e-2abf2df5d3a5",[246],[920],{"id":20,"sortIndex":21,"affiliation":921,"properties":20},{"id":922,"createTime":923,"updateTime":923,"relativeEntities":924,"slug":20,"properties":925,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"108bb899-583b-40ad-9955-38dfe1aec038","2024-01-11T00:11:12.463+00:00",[],{"title":926},{"VI":927},"School of Mathematical Sciences, Peking University, Beijing, P.R. 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Comm. Partial Differential Equations, 16, 1223–1253 (1991)\nChen, W. X., Li, C. M.: Classification of solutions some nonlinear elliptic equation. Duke Math. J., 63, 615–622 (1991)\nChen, W. X., Li, C. M.: Qualitative properties of solutions to some nonlinear elliptic equations in ℝ2. Duke Math. J., 71, 427–439 (1993)\nCheng, K. S., Lin, C. S.: On the asymptotic behavior of solutions of the conformal Gaussian curvature equation in ℝ2. Math. Ann., 308, 119–139 (1997)\nDunne, G.: Self-dual Chern-Simons Theories, Lecture Notes in Physics, New series, Monographs, m36, Springer, New York, 1995\nFordy, A. P., Wood, J. C.: Harmonic Maps and Integrable Systems, Aspects of Mathematics, E23. Friedr. Vieweg Sohn, Braunschweig, 1994\nGuest, M. A.: Harmonic Maps, Loops Groups, and Integrabl Systems, London Mathematical Society Student Texts, 38, Cambridge University Press, Cambridge, 1997\nJost, J., Wang, G.: Analytic aspects of the Toda system. I. A Moser-Trudinger inequality. Comm. Pure Appl. Math., 54, 1289–1319 (2001)\nJost, J., Wang, G.: Classification of solutions of a Toda system in ℝ2. Int. Math. Res. Not., 2002, 277–290 (2002)\nKao, H. C., Lee, K.: Self-dual SU(3) Chern-Simons Higgs systems. Phys. Rev. D, 50(3), 6626–6632 (1994)",{"EN":989},"In this paper, we establish a priori estimates to the generalized second order Toda system \n                  \n                    \n                  \n                  $$\\left\\{ \\begin{gathered}\n   - \\Delta u_1 (x) = 2R_1 (x)e^{u_1 }  - R_2 e^{u_2 } , \\hfill \\\\\n   - \\Delta u_2 (x) =  - R_1 (x)e^{u_1 }  + 2R_2 e^{u_2 }  \\hfill \\\\ \n\\end{gathered}  \\right.$$\n                 in ℝ2, and discuss the convergence and asymptotic behavior of its solutions, where R\n                \n                  i\n                (x), i = 1, 2, is bounded function in ℝ2. Consequently, we prove that all the solutions satisfy an identity, which is somewhat a generalization of the well-known Kazdan-Warner condition.",{"EN":991},"A remark on the generalized second order Toda system",{"VOID":993},"10.1007\u002Fs10114-013-2080-9","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10114-013-2080-9",[996],{"id":997,"sortIndex":21,"researcher":20,"roles":998,"affiliations":999,"properties":1008},"3117c286-2c94-44aa-8f5d-a26e88f7a3c4",[246],[1000],{"id":20,"sortIndex":21,"affiliation":1001,"properties":20},{"id":1002,"createTime":1003,"updateTime":1003,"relativeEntities":1004,"slug":20,"properties":1005,"entityType":49,"verifyStatus":19,"verifyTime":20,"verifyNote":20,"syncStatus":19,"languages":20,"translateLanguages":20,"viewCount":21},"4e086e4c-4f9d-4cf2-8c3f-8aac65dacba3","2024-01-15T02:15:32.628+00:00",[],{"title":1006},{"VI":1007},"School of Mathematics and Statistics, Tianshui Normal University, Tianshui, P.R. 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