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• 1. Abkar, A.
A Riesz representation formula for super-biharmonic functions2001In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 26, no 2, p. 305-324Article in journal (Refereed)

Let u be a real-valued function defined on the unit disk D. We call u super-biharmonic provided that u is locally integrable and the bi-laplacian Delta (2)u is a positive distribution on D. In this paper, we shall establish a representation formula for super-biharmonic functions, This formula can be regarded as an analogue of the Poisson-Jensen representation formula for subharmonic functions.

• 2. Abuzyarova, Natalia
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Branch point area methods in conformal mapping2006In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 99, p. 177-198Article in journal (Refereed)

The classical estimate of Bieberbach that vertical bar a(2)vertical bar <= 2 for a given univalent function phi(z) = z + a(2)z(2) +... in the class S leads to the best possible pointwise estimates of the ratio phi''(z)/phi'(z) for phi is an element of S, first obtained by K oe be and Bieberbach. For the corresponding class E of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of psi(z)/psi'(z) for psi is an element of Sigma. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K oe be-Bieberbach estimate, is firmly rooted in area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.

• 3. Aleman, A.
KTH, Superseded Departments, Mathematics. KTH, Superseded Departments, Mathematics.
Real zero polynomials and Pólya-Schur type theorems2004In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 94, p. 49-60Article in journal (Refereed)
• 4.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Berezin Transform in Polynomial Bergman Spaces2010In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 63, no 12, p. 1533-1584Article in journal (Refereed)

Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.

• 5. Ameur, Yacin
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
FLUCTUATIONS OF EIGENVALUES OF RANDOM NORMAL MATRICES2011In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 159, no 1, p. 31-81Article in journal (Refereed)

In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane-the "droplet." We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.

• 6. Ameur, Yacin
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Random normal matrices and ward identities2015In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, no 3, p. 1157-1201Article in journal (Refereed)

We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

• 7. Baranov, Anton
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Boundary properties of Green functions in the plane2008In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 145, no 1, p. 1-24Article in journal (Refereed)

We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 article of Jones and Makarov [11]. The main tools are an integral identity as well as a uniform Sobolev embedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case where p = 2 of a more general Grunsky identily for L-p-spaces.

• 8. Borichev, A.
KTH, Superseded Departments, Mathematics.
Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index2004In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 207, no 1, p. 111-160Article in journal (Refereed)

In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate non-trivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.

• 9. Borichev, Alexander
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Weighted integrability of polyharmonic functions2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, p. 464-505Article in journal (Refereed)

To address the uniqueness issues associated with the Dirichlet problem for the N-harmonic equation on the unit disk D in the plane, we investigate the L-P integrability of N-harmonic functions with respect to the standard weights (1 vertical bar z vertical bar(2))(alpha). The question at hand is the following. If u solves Delta(N)u = 0 in D, where Delta stands for the Laplacian, and integral(D)vertical bar u(Z)vertical bar(p)(1 - vertical bar z vertical bar(2))(alpha)dA(z) < +infinity, must then u(z) 0? Here, N is a positive integer, alpha is real, and 0 < p < +infinity; dA is the usual area element. The answer will, generally speaking, depend on the triple (N, p, alpha). The most interesting case is 0 < p < 1. For a given N, we find an explicit critical curve p bar right arrow beta(N, p) - a piecewise affine function - such that for alpha > beta(N, p) there exist nontrivial functions u with Delta Nu = 0 of the given integrability, while for alpha <= beta(N, p), only u(z) 0 is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in PHN, alpha p (D) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions - the cellular decomposition - which decomposes the polyharmonic weighted LP space in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the (p, alpha) plane into cells. The above uniqueness for the Dirichlet problem may be considered for any elliptic operator of order 2N. However, the above-mentioned critical integrability curve will depend rather strongly on the given elliptic operator, even in the constant coefficient case, for N > 1.

• 10. Canto-Martín, Francisco
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Perron-Frobenius operators and the Klein-Gordon equation2014In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 16, no 1, p. 31-66Article in journal (Refereed)

For a smooth curve Gamma and a set Lambda in the plane R-2, let AC(Gamma; Lambda) be the space of finite Borel measures in the plane supported on Gamma, absolutely continuous with respect to arc length and whose Fourier transform vanishes on Lambda. Following [12], we say that (Gamma, Lambda) is a Heisenberg uniqueness pair if AC(Gamma; Lambda) = {0}. In the context of a hyperbola Gamma, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Gamma of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Gamma; Lambda) when it is nonzero. We will fix the curve Gamma to be the hyperbola x(1)x(2) = 1, and the set Lambda = Lambda(alpha,beta) to be the lattice-cross Lambda(alpha,beta) = (alpha Zeta x {0}) boolean OR ({0} x beta Z), where alpha, beta are positive reals. We will also consider Gamma(+), the branch of x(1)x(2) = 1 where x(1) > 0. In [12], it is shown that AC(Gamma; Lambda(alpha,beta)) = {0} if and only if alpha beta <= 1. Here, we show that for alpha beta > 1, we get a rather drastic "phase transition": AC(Gamma; Lambda(alpha,beta)) is infinite-dimensional whenever alpha beta > 1. It is shown in [13] that AC(Gamma(+); Lambda(alpha,beta)) = {0} if and only if alpha beta < 4. Moreover, at the edge alpha beta = 4, the behavior is more exotic: the space AC(Gamma(+); Lambda(alpha,beta)) is one-dimensional. Here, we show that the dimension of AC(Gamma(+); Lambda(alpha,beta)) is infinite whenever alpha beta > 4. Dynamical systems, and more specifically Perron-Frobenius operators, play a prominent role in the presentation.

• 11. Goden, Julia
KTH, Superseded Departments, Mathematics.
The composition operators on the space of Dirichlet series with square summable coefficients1999In: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 46, p. 313-329Article in journal (Refereed)
• 12.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Asymptotic expansion of polyanalytic Bergman kernels2014In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 12, p. 4667-4731Article in journal (Refereed)

We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if (partial derivative) over bar (q) f = 0 for the given positive integer q. Polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels.

We obtain asymptotic formulae in the bulk for the q-analytic Bergman kernel in the setting of the power weights e(-2mQ), as the positive real parameter m tends to infinity. This is only known previously for q = 1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjostrand, which is based on ideas from microlocal analysis.

We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality.

• 13.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Polyanalytic Ginibre Ensembles2013In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 153, no 1, p. 10-47Article in journal (Refereed)

For integers n,q=1,2,3,aEuro broken vertical bar aEuro parts per thousand, let Pol (n,q) denote the -linear space of polynomials in z and , of degree a parts per thousand currency signn-1 in z and of degree a parts per thousand currency signq-1 in . We supply Pol (n,q) with the inner product structure of the resulting Hilbert space is denoted by Pol (m,n,q) . Here, it is assumed that m is a positive real. We let K (m,n,q) denote the reproducing kernel of Pol (m,n,q) , and study the associated determinantal process, in the limit as m,n ->+a while n=m+O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble-the eigenvalue process of random (normal) matrices with Gaussian weight. There is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first Landau levels have been filled. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk . We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m (-1/2); the typical distance is the same both for interior and for boundary points of . This amounts to obtaining the asymptotical behavior of the generating kernel K (m,n,q) . Following (Ameur et al. in Commun. Pure Appl. Math. 63(12):1533-1584, 2010), the asymptotics of the K (m,n,q) are rather conveniently expressed in terms of the Berezin measure (and density) For interior points |z|< 1, we obtain that in the weak-star sense, where delta (z) denotes the unit point mass at z. Moreover, if we blow up to the scale of m (-1/2) around z, we get convergence to a measure which is Gaussian for q=1, but exhibits more complicated Fresnel zone behavior for q > 1. In contrast, for exterior points |z|> 1, we have instead that , where is the harmonic measure at z with respect to the exterior disk . For boundary points, |z|=1, the Berezin measure converges to the unit point mass at z, as with interior points, but the blow-up to the scale m (-1/2) exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q (1/2) m (-1/2).

• 14.
KTH, Superseded Departments, Mathematics.
A Beurling-Rudin theorem for H^\infty1987In: Illinois Journal of Mathematics, ISSN 0019-2082, E-ISSN 1945-6581, Vol. 31, p. 629-644Article in journal (Refereed)
• 15.
KTH, Superseded Departments, Mathematics.
A comparison between the closed modular ideals in L^1(w) and l^1(w)1986In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 58, p. 275-300Article in journal (Refereed)
• 16.
KTH, Superseded Departments, Mathematics.
A computation of Green functions for the weighted biharmonic Green functions \Delta|z|^{-2\alpha}\Delta1994In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 75, p. 51-78Article in journal (Refereed)
• 17.
KTH, Superseded Departments, Mathematics.
A factoring theorem for a weighted Bergman space1993In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 4, p. 163-174Article in journal (Refereed)
• 18.
KTH, Superseded Departments, Mathematics.
A factoring theorem for the Bergman space1994In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 26, p. 113-126Article in journal (Refereed)
• 19.
KTH, Superseded Departments, Mathematics.
A factorization theorem for square area-integrable analytic functions1991In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 422, p. 45-68Article in journal (Refereed)
• 20.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
A note on Poincare metric related quantities in conformal mapping2004In: Complex variables, theory and application, ISSN 0278-1077, Vol. 49, no 7-9, p. 549-554Article in journal (Refereed)
• 21.
KTH, Superseded Departments, Mathematics.
An invariant subspace of the Bergman space having the codimension two property1993In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 443, p. 1-9Article in journal (Refereed)
• 22. Hedenmalm, Håkan
An off-diagonal estimate of Bergman kernels2000In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 79, no 2, p. 163-172Article in journal (Refereed)

For weights on the unit disk that are logarithmically subharmonic and reproducing for the origin, an estimate from above and below is obtained for the Bergman kernel associated with the weight. In particular, that kernel is zero free, and bounded from above by twice the unweighted Bergman kernel.

• 23.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Saint Petersburg State University,Russian Federation.
Bloch functions and asymptotic tail variance2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 313, p. 947-990Article in journal (Refereed)

Let P denote the Bergman projection on the unit disk D, P mu(z) := integral(D) mu(w)/(1-z (w) over bar)(2) dA(w), z is an element of D, where dA is normalized area measure. We prove that if vertical bar mu(z)vertical bar <= 1 on D, then the integral I(mu()a,r) := integral(2 pi)(0) exp {a r(4)vertical bar P mu(re(i theta))vertical bar(2)/log 1/1-r(2)}d theta/2 pi, 0 < r < 1, has the bound I-mu(a,r) <= C(a) := 10(1-a)(-3/2) for 0 < a < 1, irrespective of the choice of the function mu. Moreover, for a > 1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection P-mu (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0 < a < pi(2)/64 = 0.1542.... We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0 < k < 1. The estimate reads, for t is an element of C and 0 < k < 1, B(k,t) <= {1/4 k(2)vertical bar t vertical bar(2)(1 + 7k)(2), for vertical bar t vertical bar <= 2/k(1 + 7k)(2), k vertical bar t vertical bar - 1/(1 + 7k)(2), for vertical bar t vertical bar >= 2/k(1 + 7k)(2), which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with vertical bar t vertical bar <= 2/k, we should have B(k, t) = 1/4k(2)t(2).

• 24.
KTH, Superseded Departments, Mathematics.
Boundary value problems for weighted biharmonic operators1997In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 8, p. 661-674Article in journal (Refereed)
• 25.
KTH, Superseded Departments, Mathematics.
Bounded analytic functions and closed ideals1987In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 48, p. 142-166Article in journal (Refereed)
• 26.
KTH, Superseded Departments, Mathematics.
Closed ideals in the ball algebra1989In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 21, p. 469-474Article in journal (Refereed)
• 27.
KTH, Superseded Departments, Mathematics.
Closed ideals in the bidisc algebra1990In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 28, p. 111-117Article in journal (Refereed)
• 28.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Codebreakers: Arne Beurling and the Swedish crypto program during World War II2006In: The Mathematical intelligencer, ISSN 0343-6993, E-ISSN 1866-7414, Vol. 28, no 1, p. 57-59Article, book review (Other academic)
• 29.
KTH, Superseded Departments, Mathematics.
Comparisons of ideal structures in algebras of analytic functions of several complex variables1988In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 63, p. 305-314Article in journal (Refereed)
• 30.
KTH, Superseded Departments, Mathematics.
Cyclicity in Bergman-type spaces1995In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 5, p. 253-262Article in journal (Refereed)
• 31.
KTH, Superseded Departments, Mathematics.
Dirichlet series and functional analysis2004In: Legacy Of Niels Henrik Abel / [ed] Laudal, OA; Piene, R, BERLIN: SPRINGER-VERLAG BERLIN , 2004, p. 673-684Conference paper (Refereed)
• 32.
KTH, Superseded Departments, Mathematics.
Formal power series and nearly analytic functions1991In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 57, p. 61-70Article in journal (Refereed)
• 33.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Gösta Mittag-Leffler : A Man of Conviction by Arild Stubhaug2012In: The Mathematical intelligencer, ISSN 0343-6993, E-ISSN 1866-7414, Vol. 34, no 1, p. 63-64Article, book review (Other academic)
• 34.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Heisenberg's uncertainty principle in the sense of Beurling2012In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 118, no 2, p. 691-702Article in journal (Refereed)

We shed new light on Heisenberg's uncertainty principle in the sense of Beurling, by offering a fundamentally different proof which allows us to weaken the assumptions rather substantially. The new formulation is pretty much optimal, as can be seen from examples. Our arguments involve Fourier and Mellin transforms. We also introduce a version which applies to two given functions. Finally, we show how our approach applies in the higher dimensional setting.

• 35.
KTH, Superseded Departments, Mathematics.
Interpolating sequences and invariant subspaces of given index in the Bergman spaces1996In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 477, p. 13-30Article in journal (Refereed)
• 36.
KTH, Superseded Departments, Mathematics.
Maximal invariant subspaces in the Bergman space1998In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 36, p. 97-101Article in journal (Refereed)
• 37.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On Hormander's solution of the -equation. I2015In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 281, no 1-2, p. 349-355Article in journal (Refereed)

We explain how Hormander's classical solution of the -equation in the plane with a weight which permits growth near infinity carries over to the rather opposite situation when we ask for decay near infinity. Here, however, a natural condition on the datum needs to be imposed. The condition is not only natural but also necessary to have the result at least in the Fock weight case. The norm identity which leads to the estimate is related to general area-type results in the theory of conformal mappings.

• 38.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the asymptotic free boundary for the American put option problem2006In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 314, no 1, p. 345-362Article in journal (Refereed)

In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifespan, by analyzing binomial trees. It is considerably more challenging to make the decision for American put options with long lifespan. In order to provide a satisfactory analysis, we look at the corresponding free boundary problem, and show that the free boundary-which is the curve that separates the two decisions, to exercise or not to-has an asymptotic expansion, where the coefficient of the main term is expressed as an integral in terms of the free boundary. This raises the perspective that one could use numerical simulation to approximate the integral and thus get an effective way to make correct decisions for long life options.

• 39.
KTH, Superseded Departments, Mathematics.
On the primary ideal structure at infinity for analytic Beurling algebras1985In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 23, p. 129-158Article in journal (Refereed)
• 40.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the uniqueness theorem of Holmgren2015In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 281, no 1-2, p. 357-378Article in journal (Refereed)

We review the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the existence and uniqueness issues. Here, we consider a global uniqueness problem of Holmgren's type. Perhaps surprisingly, we obtain a connection with the theory of quadrature identities, which demonstrates that rather subtle algebraic properties of the curve come into play. For instance, if is the interior domain of an ellipse, and I is a proper arc of the ellipse , then there exists a nontrivial biharmonic function u in which is three-flat on I (i.e., all partial derivatives of u of order vanish on I) if and only if the ellipse is a circle. Another instance of the same phenomenon is that if is bounded and simply connected with -smooth Jordan curve boundary, and if the arc is nowhere real-analytic, then we have local uniqueness already with sub-Cauchy data: if a function is biharmonic in for some planar neighborhood of I, and is three-flat on I, then it vanishes identically on , provided that is connected. Finally, we consider a three-dimensional setting, and analyze it partially using analogues of the square of the standard Cauchy-Riemann operator. In a special case when the domain is of periodized cylindrical type, we find a connection with the massive Laplacian [the Helmholz operator with imaginary wave number] and the theory of generalized analytic (or pseudoanalytic) functions of Bers and Vekua.

• 41.
KTH, Superseded Departments, Mathematics.
Outer functions in function algebras on the bidisc1988In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 306, p. 697-714Article in journal (Refereed)
• 42.
KTH, Superseded Departments, Mathematics.
Outer functions of several complex variables1988In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 80, p. 9-15Article in journal (Refereed)
• 43.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Planar Beurling transform and Grunsky inequalities2008In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 33, no 2, p. 585-596Article in journal (Refereed)

In recent work with Baranov, it was explained how to view the classical Grunsky inequalities in terms of an operator identity, involving a transferred Beurling operator induced by the conformal mapping. The main property used is the fact that the Beurling operator is unitary on L-2(C). As the Beurling operator is also bounded oil L-p(C) for 1 < p < infinity (with so far unknown norm), all analogous operator identity was found which produces a generalization of the Grunsky inequalities to the L-p setting. Here, we consider weighted Hilbert spaces L-theta(2)(C) with weight, vertical bar z vertical bar(2 theta), for 0 <= theta <= 1, and find that the Beurling operator perturbed by adding a Cauchy-type operator acts unitarily on L-0(2) (C). After transferring to the unit disk D with the conformal mapping, we find a generalization of the Grunsky inequalities ill the setting of the space L-theta(2) (D); this generalization seems to be essentially known, but the formulation is new. As a special case, the generalization of the Grunsky inequalities contains the Prawitz theorem used in a recent paper with Shirnorin. We also mention an application to quasiconformal maps.

• 44.
KTH, Superseded Departments, Mathematics.
Spectral properties of invariant subspaces in the Bergman space1993In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 116, p. 441-448Article in journal (Refereed)
• 45.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Beurling operator for the hyperbolic plane2012In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 37, no 1, p. 3-18Article in journal (Refereed)

We find a Beurling operator for the hyperbolic plane, and obtain an L-2 norm identity for it, as well as two-sided L-p estimates.

• 46.
KTH, Superseded Departments, Mathematics.
The dual of a Bergman space on simply connected domains2002In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 88, p. 311-335Article in journal (Refereed)
• 47.
KTH, Superseded Departments, Mathematics.
Thin interpolating sequences and three algebras of bounded functions1987In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 99, p. 489-495Article in journal (Refereed)
• 48.
KTH, Superseded Departments, Mathematics.
Translates of functions of two variables1989In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 58, p. 251-297Article in journal (Refereed)
• 49.
KTH, Superseded Departments, Mathematics.
Invariant subspaces in quasi-Banach spaces of analytic functions2001In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 12, p. 83-100Article in journal (Refereed)
• 50.
KTH, Superseded Departments, Mathematics.
Invariant subspaces on multiply connected domains1998In: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 42, p. 521-557Article in journal (Refereed)
12 1 - 50 of 84
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