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  • 1. Hedenmalm, Håkan
    et al.
    Jakobsson, S.
    Shimorin, Serguei
    KTH, Superseded Departments, Mathematics.
    A biharmonic maximum principle for hyperbolic surfaces2002In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 550, p. 25-75Article in journal (Refereed)
  • 2. Hedenmalm, Håkan
    et al.
    Shimorin, Serguei
    KTH, Superseded Departments, Mathematics.
    Hele-Shaw flow on hyperbolic surfaces2002In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 81, no 3, p. 187-222Article in journal (Refereed)
    Abstract [en]

    Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele-Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele-Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.

  • 3.
    Hedenmalm, Håkan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the universal integral means spectrum of conformal mappings near the origin2007In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 135, no 7, p. 2249-2255Article in journal (Refereed)
    Abstract [en]

    We improve the local estimate near the origin of the integral means spectrum for conformal mappings obtained in our paper from 2005. We also study some algebraic aspects of higher order forms associated with the given conformal mapping.

  • 4.
    Hedenmalm, Håkan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weighted Bergman spaces and the integral means spectrum of conformal mappings2005In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 127, no 2, p. 341-393Article in journal (Refereed)
    Abstract [en]

    The classical theory of conformal mappings involves best possible pointwise estimates of the derivative, thus supplying a measure of the extremal expansion/contraction possible for a conformal mapping. It is natural to consider also the integral means of \phi'\(t) along circles \z\ = r, where phi is the conformal mapping in question and t is a real parameter (0 < r < 1 if phi is defined in the unit disk, while 1 < r < +infinity if phi is defined in the exterior disk). The extremal growth rate as r --> 1 of the integral means which follows from the classical pointwise estimates is by far too fast. Better estimates were found by Clunie, Makarov, Pommerenke, Bertilsson, Shimorin, and others. Here we introduce a new method-based on area-type estimates-which discards as little as possible of the information supplied by the area methods. The result is a considerable improvement in the estimates of the integral means spectrum known tip to this point.

  • 5.
    Hedenmalm, Håkan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sola, Alan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Norm expansion along a zero variety2008In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 254, p. 1601-1625Article in journal (Refereed)
    Abstract [en]

    The reproducing kernel function of a weighted Bergman space over domains in C-d is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d = 2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z(1) = z(2) as subvariety) and the ball (with z(2) = 0 as subvariety), as well as for a weighted Bargmann-Fock space over C-2 (with the diagonal z(1) = z(2) as subvariety).

  • 6.
    Shimorin, Serguei
    KTH, Superseded Departments, Mathematics.
    A multiplier estimate of the Schwarzian derivative of univalent functions2003In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 30, p. 1623-1633Article in journal (Refereed)
  • 7.
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Branching points area theorems for univalent functions2007In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 18, no 1, p. 141-181Article in journal (Refereed)
    Abstract [en]

    Area theorems of a new type are obtained by considering branching point compositions with univalent functions. Such theorems can be formulated both in the form of integral estimates and in the form of Grunsky and Goluzin-type inequalities.

  • 8.
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Commutant lifting and factorization of reproducing kernels2005In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 224, no 1, p. 134-159Article in journal (Refereed)
    Abstract [en]

    A general version of the commutant lifting theorem for operators between different spaces is proved. It includes as special cases the lifting theorems of Ball-Trent-Vinnikov and Volberg-Treil. A multivariable variant of the Volberg-Treil theorem is obtained as a corollary. A certain factorization property of reproducing kernels is shown to be a sufficient condition for the lifting. Another factorization property is shown to be a necessary condition.

  • 9. Shimorin, Serguei
    Complete Nevanlinna-Pick property of Dirichlet-type spaces2002In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 191, no 2, p. 276-296Article in journal (Refereed)
    Abstract [en]

    We prove that all Dirichlet-type spaces of functions analytic in the unit disk whose derivatives are square area integrable with superharmonic weights have complete Nevanlinna-Pick reproducing kernels. As a corollary, we obtain a commutant lifting theorem for cyclic analytic two-isometries.

  • 10.
    Shimorin, Serguei
    KTH, Superseded Departments, Mathematics.
    On Beurling-type theorems in weighted l(2) and Bergman spaces2003In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 131, no 6, p. 1777-1787Article in journal (Refereed)
    Abstract [en]

    We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted l(2) and Bergman spaces.

  • 11.
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Trigonometric obstacle problem and weak factorization2006In: Bergman Spaces and Related Topics in Complex Analysis, Proceedings / [ed] Borichev, A; Hedenmalm, H; Zhu, K, PROVIDENCE, RI: AMER MATHEMATICAL SOC , 2006, Vol. 404, p. 187-197Conference paper (Refereed)
    Abstract [en]

    Using an argument related to an obstacle problem for trigonometric polynomials, we prove that any real valued polynomial is the difference of two positive polynomials of the same degree with the control of L-1-norm. This fact has an application to weak factorization of polynomials.

  • 12.
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Two applications of the Bergman spaces techniques2010In: FIVE LECTURES IN COMPLEX ANALYSIS / [ed] Contreras MD; DiazMadrigal S, 2010, Vol. 525, p. 141-161Conference paper (Refereed)
    Abstract [en]

    I am often asked by different people why the theory of Bergman spaces is so interesting that it attracts such an attention of researchers. My answer to this question is short: diversity makes theory so beautiful and interesting. First, it is diversity of methods used in the theory, sometimes coming from very unexpected areas of mathematics. Second, it is diversity of applications. The aim of this mini-course is to present two of such applications, one to an old problem in potential theory, the Hadamard conjecture on biharmonic Green functions, and another to estimates of conformal mappings. In both cases the results are obtained by an interplay of different techniques, Bergman spaces playing an important role.

    These lectures cannot be considered as an introduction to the theory of Bergman spaces. Only those aspects of this theory which are needed for our applications will be touched upon. An interested reader is referred to recent books [21] and [9].

  • 13.
    Shimorin, Serguei
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Weighted composition operators associated with conformal mappings2005In: Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume / [ed] Ebenfelt, P; Gaustafsson, B; Khavinson, D; Putinar, M, 2005, Vol. 156, p. 217-237Conference paper (Refereed)
    Abstract [en]

    For conformal self-maps rho of the unit disk, we study weighted composition operators f -> f circle rho center dot (rho')(b). We are interested in their boundedness, compactness and contractivity properties as operators acting in weighted Djrbashian-Bergman spaces.

  • 14.
    Shimorin, Serguei
    KTH, Superseded Departments, Mathematics.
    Wold-type decompositions and wandering subspaces for operators close to isometries2001In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 531, p. 147-189Article in journal (Refereed)
    Abstract [en]

    For operators T satisfying certain inequalities we obtain a weak analog of the classical Wold-Kolmogorov decomposition theorem, representing T as a direct sum of a unitary operator and a shift operator acting in some Hilbert space of analytic functions. The concept of a dual operator is introduced, which reflects relationships between shift operators acting in two Hilbert spaces of analytic functions dual to each other with respect to the Cauchy pairing. Among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels.

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