| 1. |
- Aleman, Alexandru, et al.
(författare)
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Weak products of complete pick spaces
- 2021
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 70:1, s. 325-352
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Tidskriftsartikel (refereegranskat)abstract
- Let H be the Drury-Arveson or Dirichlet space of the unit ball of Cd. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑∞n=1 fngn, where ∑∞n=1 ||fn|| ||gn|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → closH ☉H N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of Cd provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕∞n=1 H induces a bounded row multiplication operator ⊕∞n=1 H → H . For the Drury-Arveson space Hd2 this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.
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| 2. |
- Bergman, Alex
(författare)
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On cyclicity in de Branges-Rovnyak spaces
- 2023
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Ingår i: Indiana University Mathematics Journal. - 0022-2518.
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Tidskriftsartikel (refereegranskat)abstract
- We study the problem of characterizing the cyclic vec-tors in de Branges-Rovnyak spaces. Based on a description of theinvariant subspaces we show that the difficulty lies entirely in un-derstanding the subspace (aH2)⊥ and give a complete function the-oretic description of the cyclic vectors in the case dim(aH2)⊥ < ∞.Incidentally, this implies analogous results for certain generalizedDirichlet spaces D(μ). Most of our attention is directed to the in-finite case where we relate the cyclicity problem to describing theexposed points of H1 and provide several sufficient conditions. Anecessary condition based on the Aleksandrov-Clark measures of bis also presented.
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| 3. |
- Charlier, Christophe, et al.
(författare)
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The "Good" Boussinesq Equation : a Riemann-Hilbert Approach
- 2022
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518 .- 1943-5258. ; 71:4, s. 1505-1562
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Tidskriftsartikel (refereegranskat)abstract
- We develop an inverse scattering transform formalism for the "good " Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a 3 x 3 matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.
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| 4. |
- Gardiner, Stephen J., et al.
(författare)
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Isoperimetric Inequalities for Bergman Analytic Content
- 2020
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Ingår i: Indiana University Mathematics Journal. - : INDIANA UNIV MATH JOURNAL. - 0022-2518 .- 1943-5258. ; 69:4, s. 1231-1249
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Tidskriftsartikel (refereegranskat)abstract
- The Bergman p-analytic content (1 <= p < infinity) of a planar domain Omega measures the L-p (Omega)-distance between (z) over bar and the Bergman space A(p) (Omega) of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman p-analytic content in terms of the St. Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.
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| 5. |
- Kim, I., et al.
(författare)
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Plurisubharmonicity and Geodesic Convexity of Energy Function on Teichmuller Space
- 2022
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 71:1, s. 1-36
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Tidskriftsartikel (refereegranskat)abstract
- Let pi : X -> T be Teichmuller curve over Teichmuller space T, such that the fiber X-z = pi(-1) (z) is exactly the Riemann surface given by the complex structure z is an element of T. For a fixed Riemannian manifold M and a continuous map u(0): M -> X-z0, let E(z) denote the energy function of the harmonic map u(z) : M -> X-z homotopic to u(0), z is an element of T. We obtain the first and the second variations of the energy function E(z), and show that logE(z) is strictly plurisubharmonic on Teichmuller space, and that both E(z) and logE(z) are plurisubharmonic exhausting functions. We also obtain a precise formula on the second variation of E- (1/2) if dim M = 1. In particular, we get the formula of Axelsson-Schumacher on the second variation of the geodesic length function. We give also a simple and corrected proof for the theorem of Yamada, the convexity of energy function E(t) along Weil-Petersson geodesics. As an application we show that E(t)(c) is also strictly convex for c > 5/6 and convex for c = 5/6 along Weil-Petersson geodesics. We also re-prove a Kerckhoff's theorem, which is a positive answer to the Nielsen realization problem.
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| 6. |
- Lang, Lionel, et al.
(författare)
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On the Number of Intersection Points of the Contour of an Amoeba with a Line
- 2021
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518 .- 1943-5258. ; 70:4, s. 1335-1353
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Tidskriftsartikel (refereegranskat)abstract
- In this note, we investigate the maximal number of intersection points of a line with the contour of a hypersurface amoeba in R-n. We define the latter number to be the R-degree of the contour. We also investigate the R-degree of related sets such as the boundary of an amoeba and the amoeba of the real part of a hypersurface defined over R. For all these objects, we provide bounds for the respective R-degrees.
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| 7. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Chern Forms of Hermitian Metrics with Analytic Singularities on Vector Bundles
- 2022
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 71:5, s. 153-189
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Tidskriftsartikel (refereegranskat)abstract
- We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular Hermitian metric h with analytic singularities on a holomorphic vector bundle E. The currents are constructed as pushforwards of generalized Monge-Ampere products on the projectivization of E. The Chern and Segre currents represent the Chern and Segre classes of E, respectively, and coincide with the Chern and Segre forms of E and h where h is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined.
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| 8. |
- Nilsson, Mårten
(författare)
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Plurisubharmonic functions with discontinuous boundary behavior
- 2023
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Ingår i: Indiana University Mathematics Journal. - 0022-2518.
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Tidskriftsartikel (refereegranskat)abstract
- We study the Dirichlet problem for the complex Monge–Amp`ereoperator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complexMonge–Amp`ere measures. This result is optimal in the unit disk, as boundaryfunctions with b-pluripolar discontinuity then coincide with functions that arecontinuous almost everywhere. We also show that neither of these propertiesof the boundary function – being continuous almost everywhere or having discontinuities forming a b-pluripolar set – are necessary conditions in order toestablish uniqueness and continuity of the solution in higher dimensions. Inparticular, there are situations where it is enough to prescribe the boundarybehavior at a set of arbitrarily small Lebesgue measure.
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