Alexander Borichev

A Blaschke-type condition and its application to complex Jacobi matrices

We obtain a Blaschke-type necessary condition on zeros of analytic functions on the unit disc with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.

Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

In the theory of commutative Banach algebras with unit, an el- ement generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space H2 on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of func- tions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskĭi, Shields, Korenblum, Brown, and Frankfurt. Department of Mathematics, University of Bordeaux I, 351, cours de la Liberation, 33405 Talence, France E-mail address: borichev@math.u-bordeaux.fr Department of Mathematics, Lund University, Box 118, 22100 Lund, Sweden E-mail address: haakan@maths.lth.se

Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Given martingales W and Z such that W is differentially subordinate to Z, Burkholder obtained the sharp martingale inequality E|W |p ≤ (p∗− 1)pE|Z|p, where p∗ = max{p, p p−1}. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if p ≥ 2 and W is a conformal martingale differentially subordinate to any martingale Z, then E|W |p ≤ [(p − p)/2]p/2E|Z|p. In this paper, we establish that if p ≥ 2, Z is conformal, and W is any martingale subordinate to Z, then E|W |p ≤ [ √ 2(1−zp)/zp]E|Z|, where zp is the smallest positive zero of a certain solution of the Laguerre ODE. We also prove the sharpness of this estimate, and an analogous one in the dual case for 1 < p < 2. Finally, we give an application of our results. Previous estimates on the L norm of the Beurling–Ahlfors transform give at best ‖B‖p . √ 2 p as p → ∞. We improve this to ‖B‖p . 1.3922 p as p→∞.

Riesz bases of reproducing kernels in Fock-type spaces

In a scale of Fock spaces

Weighted integrability of polyharmonic functions

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Large Bergman spaces: Invertibility, cyclicity, and subspaces of arbitrary index

with radial weights

Systems, Approximation, Singular Integral Operators, and Related Topics

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On Burkholder function for orthogonal martingales and zeros of Legendre polynomials

we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if

Pseudo-holomorphic functions at the critical exponent

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Distortion Growth for Iterations of Diffeomorphisms of the Interval

grows at most like