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Hedenmalm, H. (2020). BLOCH FUNCTIONS, ASYMPTOTIC VARIANCE, AND GEOMETRIC ZERO PACKING. American Journal of Mathematics, 142(1), 267-321
Open this publication in new window or tab >>BLOCH FUNCTIONS, ASYMPTOTIC VARIANCE, AND GEOMETRIC ZERO PACKING
2020 (English)In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 142, no 1, p. 267-321Article in journal (Refereed) Published
Abstract [en]

Motivated by a problem in quasiconformal mapping, we introduce a problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity. The problem will be referred to as geometric zero packing, and is somewhat analogous to studying Fekete point configurations. The associated quantity is a density, denoted pc in the planar case, and pH in the case of the hyperbolic plane. We refer to these densities as discrepancy densities for planar and hyperbolic zero packing, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures. The universal asymptotic variance Sigma(2) associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: Sigma(2) = 1- rho H. We obtain the estimates 3.2 x 10(-5) < rho H <= 0.12087, where the upper estimate is derived from the estimate from below on Sigma(2) obtained by Astala, Ivrii, Perala, and Prause, and the estimate from below is more delicate. In particular, it follows that Sigma(2) < 1, which in combination with the work of ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached. Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics B(k, t) similar to 1/4 Sigma(2)vertical bar t vertical bar(2) for small t and k, the conjectured formula B(k, t) = 1/4 k(2)vertical bar t vertical bar(2) is not true. As for the actual numerical values of the discrepancy density rho(C), we obtain the estimate from above rho(C) <= 0.061203 ... by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The value of pH is expected to be somewhat close to that of rho(C).

Place, publisher, year, edition, pages
JOHNS HOPKINS UNIV PRESS, 2020
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-267501 (URN)000508886200008 ()2-s2.0-85078752683 (Scopus ID)
Note

QC 20200407

Available from: 2020-04-07 Created: 2020-04-07 Last updated: 2020-05-25Bibliographically approved
Hedenmalm, H. & Wennman, A. (2020). Off-Spectral Analysis of Bergman Kernels. Communications in Mathematical Physics, 373(3), 1049-1083
Open this publication in new window or tab >>Off-Spectral Analysis of Bergman Kernels
2020 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 373, no 3, p. 1049-1083Article in journal (Refereed) Published
Abstract [en]

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behavior is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying weights on the complex plane. We obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off-spectral point, which we also refer to as the root function at the point in question. The expansion is valid in the entire off-spectral component containing the root point, and protrudes into the spectrum as well. This allows us to obtain error function transition behavior of the density of states along the smooth interface. Previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum, which contrasts with our non-local expansions.

Place, publisher, year, edition, pages
Springer, 2020
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-267844 (URN)10.1007/s00220-019-03667-2 (DOI)000518630600007 ()2-s2.0-85078012453 (Scopus ID)
Note

QC 20200427

Available from: 2020-02-26 Created: 2020-02-26 Last updated: 2020-04-27Bibliographically approved
Hedenmalm, H. & Wennman, A. (2018). A critical topology for L^p Carleman classes with 0<p<1. Mathematische Annalen, 371(3-4), 1803-1844
Open this publication in new window or tab >>A critical topology for L^p Carleman classes with 0<p<1
2018 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 371, no 3-4, p. 1803-1844Article in journal (Refereed) Published
Abstract [en]

In this paper, we explore a sharp phase transition phenomenon which occurs for (Formula presented.)-Carleman classes with exponents (Formula presented.). These classes are defined as for the standard Carleman classes, only the (Formula presented.)-bounds are replaced by corresponding (Formula presented.)-bounds. We study the quasinorms (Formula presented.)for some weight sequence (Formula presented.) of positive real numbers, and consider as the corresponding (Formula presented.)-Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the (Formula presented.)-Carleman class. A particular degenerate instance is when (Formula presented.) for (Formula presented.) and (Formula presented.) for (Formula presented.). This would give the (Formula presented.)-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these (Formula presented.)-Sobolev spaces are highly degenerate for (Formula presented.). Indeed, the canonical map (Formula presented.) fails to be injective, and there is even an isomorphism (Formula presented.)corresponding to the canonical map (Formula presented.) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general (Formula presented.)-Carleman classes defined by a weight sequence (Formula presented.). If (Formula presented.) has some regularity properties, and if the given collection of test functions is what we call (Formula presented.)-tame, then we find that there is a sharp boundary, defined in terms of the weight (Formula presented.): on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of (Formula presented.)-smooth functions and the canonical map (Formula presented.) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the (Formula presented.) setting, with (Formula presented.).

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-228086 (URN)10.1007/s00208-018-1654-3 (DOI)000439931600025 ()2-s2.0-85042074601 (Scopus ID)
Funder
Swedish Research Council, 2016-04912
Note

QC 20180518

Available from: 2018-05-17 Created: 2018-05-17 Last updated: 2019-05-22Bibliographically approved
Hedenmalm, H., Stolyarov, D. M., Vasyunin, V. I. & Zatitskiy, P. B. (2018). Sharpening Holder's inequality. Journal of Functional Analysis, 275(5), 1280-1319
Open this publication in new window or tab >>Sharpening Holder's inequality
2018 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 275, no 5, p. 1280-1319Article in journal (Refereed) Published
Abstract [en]

We strengthen Holder's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of the Pythagorean theorem for the L-p-spaces. Our treatment of the subject matter is based on Bellman functions of four variables.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2018
Keywords
Holder's inequality, Sharpening, Pythagorean theorem
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-232378 (URN)10.1016/j.jfa.2018.05.003 (DOI)000437388300008 ()2-s2.0-85047084973 (Scopus ID)
Note

QC 20180726

Available from: 2018-07-26 Created: 2018-07-26 Last updated: 2018-07-26Bibliographically approved
Hedenmalm, H. (2015). On Hormander's solution of the -equation. I. Mathematische Zeitschrift, 281(1-2), 349-355
Open this publication in new window or tab >>On Hormander's solution of the -equation. I
2015 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 281, no 1-2, p. 349-355Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
(partial derivative)over-bar-Equation, Weighted estimates (partial derivative)over-bar-equation, Hormander's theorem, Growing weight, Existence, Uniqueness of the solution
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-173413 (URN)10.1007/s00209-015-1487-7 (DOI)000359830400014 ()2-s2.0-84939575144 (Scopus ID)
Note

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
Hedenmalm, H. (2015). On the uniqueness theorem of Holmgren. Mathematische Zeitschrift, 281(1-2), 357-378
Open this publication in new window or tab >>On the uniqueness theorem of Holmgren
2015 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 281, no 1-2, p. 357-378Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Cauchy problem, Dirichlet problem, Holmgren's uniqueness theorem
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-173414 (URN)10.1007/s00209-015-1488-6 (DOI)000359830400015 ()2-s2.0-84939575710 (Scopus ID)
Note

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
Ameur, Y., Hedenmalm, H. & Makarov, N. (2015). Random normal matrices and ward identities. Annals of Probability, 43(3), 1157-1201
Open this publication in new window or tab >>Random normal matrices and ward identities
2015 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, no 3, p. 1157-1201Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Random normal matrix, eigenvalues, Ginibre ensemble, Ward identity, loop equation, Gaussian free field
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-172251 (URN)10.1214/13-AOP885 (DOI)000354665200007 ()2-s2.0-84929252808 (Scopus ID)
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council
Note

QC 20150821

Available from: 2015-08-21 Created: 2015-08-14 Last updated: 2017-12-04Bibliographically approved
Haimi, A. & Hedenmalm, H. (2014). Asymptotic expansion of polyanalytic Bergman kernels. Journal of Functional Analysis, 267(12), 4667-4731
Open this publication in new window or tab >>Asymptotic expansion of polyanalytic Bergman kernels
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 12, p. 4667-4731Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Polyanalytic functions, Bergman kernel, Asymptotic expansion, Bulk universality
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-122142 (URN)10.1016/j.jfa.2014.09.002 (DOI)000346226500003 ()2-s2.0-84921969223 (Scopus ID)
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2012-3122
Note

QC 20150220. Updated from submitted to published.

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2017-12-06Bibliographically approved
Canto-Martín, F., Hedenmalm, H. & Montes-Rodríguez, A. (2014). Perron-Frobenius operators and the Klein-Gordon equation. Journal of the European Mathematical Society (Print), 16(1), 31-66
Open this publication in new window or tab >>Perron-Frobenius operators and the Klein-Gordon equation
2014 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 16, no 1, p. 31-66Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Trigonometric system, inversion, Perron-Frobenius operator, Koopman operator, invariant measure, Klein-Gordon equation, ergodic theory
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-139512 (URN)10.4171/JEMS/427 (DOI)000328255500002 ()2-s2.0-84890361563 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20140114

Available from: 2014-01-14 Created: 2014-01-14 Last updated: 2017-12-06Bibliographically approved
Hedenmalm, H. & Nieminen, P. J. (2014). The Gaussian free field and Hadamard's variational formula. Probability theory and related fields, 159(1-2), 61-73
Open this publication in new window or tab >>The Gaussian free field and Hadamard's variational formula
2014 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 159, no 1-2, p. 61-73Article in journal (Refereed) Published
Abstract [en]

We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator-called the Hadamard operator-associated with a given flow of growing domains.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-147034 (URN)10.1007/s00440-013-0501-4 (DOI)000335900500002 ()2-s2.0-84900473387 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20140624

Available from: 2014-06-24 Created: 2014-06-23 Last updated: 2017-12-05Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-4971-7147

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