## Håkan Hedenmalm's scientific interests and long term projects

References are to items in the
list of publications.

** ONGOING INTERESTS **

**(Quasi)conformal mapping.** An analytic function which maps any two points
to two separate points is said to be *univalent*. For such a univalent
function (also known as a conformal mapping), defined on the open unit disk,
which maps onto some region of the complex plane, we have the well-known
classical pointwise estimates of Koebe-Bieberbach. It is, however, natural
to ask for estimates *in the mean*. This led Nikolai Makarov to
introduce the * integral means spectral function*. It is considered the
main remaining theme in conformal mapping to find optimal estimates of
this function. In the paper [49] with Shimorin, we develop a general method
to tackle this problem. Another approach based on Marcinkiewicz-Zygmund
integrals was explored in the work [58] with Baranov.
More recently, quasiconformal mappings with small dilatation were studied
in [74] and the related paper *Bloch functions, asymptotic variance, and
geometric zero packing* (AJM, to appear). There, an unexpected connection
with the mathematical physics of superconductivity by Nobel prize laureate
Abrikosov was found. Also, a well-known conjecture by Astala was shown to
be incorrect, but in the positive sense that Astala's question has an even
more interesting answer than expected.

**Random normal matrices.**
The eigenvalues of a random normal nXn matrix form a system of n points in
the complex plane. The joint probability density is propertional to the
modulus of all the distances times a localizing potential. This point process
is known to be determinantal, governed by a so-called correlation kernel.
The correlation kernel is essentially the reproducing kernel for a space of
of polynomials of degree at most n-1 supplied with a Fock-type inner product.
The asymptotic behavior of the correlation kernel then gives the basic
behavior of the random point process. This direction is represented by
papers [59], [61], [65], as well as by forthcoming work with Aron Wennman.

**Polyanalytic determinantal processes.**
We define a point process as in the case of the random normal matrices,
declared to be determinantal with correlation kernel associated with a space
of polynomials of both the variable and its complex conjugate. This connects
with the mathematical physics of so-called higher Landau levels.
This direction is represented by
papers [66], [70], as well as by forthcoming work with Antti Haimi and Aron
Wennman.

** PAST INTERESTS** (could be resumed anytime)

**Wiener-type tauberian theorems.** My first scientific paper,
[1], concerns Wiener's famous tauberian theorem [N. Wiener, Tauberian
theorems, Ann. of Math. 33 (1932), 1--100], in the context of a weight;
the weight has the property that the Fourier transform maps the associated
weighted L^1 space to a space of holomorphic functions on a strip.
The result generalizes earlier work of Korenblum (1958).
Not much is known about similar matters in higher dimensions, at least not
when analyticity appears. The paper [10] was an attempt to alter
that: it partially solves a problem of B. Ya. Levin concerning L^1
functions on the quarter-plane. However, even in the one-variable case, the
situation was not fully understood. With Borichev, the main remaining
difficulty -- not having a sufficiently good grasp of the regularity of
Fourier transforms of functions in weighted L^p spaces -- was resolved,
and the result was a Wiener tauberian theorem for the half-line even for
quasi-analytic weights (in the sense of Beurling) [24]. Later, with Ben
Natan, Benyamini, and Weit [25, 29], the technique was extended to
obtain the non-commutative Wiener tauberian theorem asked for by Ehrenpreis
and Mautner in [Some properties of the Fourier transform on semi simple Lie
groups, Ann. of Math. 61 (1955), 406--439].

**Transfer themes. ** In the study of closed ideals in commutative
Banach algebras, or, more generally, invariant subspaces in Banach spaces,
it is of general interest to transfer information concerning the structure in
one instance to that of another. For instance, if we have a space of
holomorphic functions on an annulus, and associated spaces of holomorphic
functions on the two disks obtained by filling in the hole at the origin or
the hole at infinity, what are the relations between the invariant subspaces,
where on the annulus we have biinvariant subspaces, and on the two disks
forward and backward invariant ones, respectively? These issues were resolved
in [2, 3, 5, 6, 7, 34, 39]. For example, in [5], it was shown that the
closed ideals of H^\infty on the unit disk can be understood in terms of the
closed ideals of the bigger uniform algebra H^\infty+C.

**Several complex variables.** In a series of papers,
[8, 9, 89-1, 11, 13], I tried to study the closed ideals of
well-known algebras of holomorphic functions of several complex variables.
For instance, if *f* is a function in the bidisk algebra, which
vanishes only at the point (1,1) in the closed bidisk, then *f*
generates the closed ideal of all functions vanishing at (1,1) if and
only if the functions *f*(·,1) and *f*(1,·),
which belong to the disk algebra, are both outer. This generalizes a
classical theorem of Carleman (1926) to the two-dimensional setting.

**Bergman spaces.** In 1991, I found a factorization theorem for
the Bergman space of area-summable holomorphic functions in the unit disk
space [15], which is analogous to the classical inner-outer factorization
of Hardy space functions. The main difference is that the inner
functions have such a complicated nonlinear structure in the Bergman space
setting. Further results were obtained in
[16, 17, 21, 95-3, 26, 33]. Inner functions are intimately
connected with the general structure of invariant subspaces also in the
Bergman space, as discovered by Aleman, Richter, Sundberg [Beurling's
theorem for the Bergman space, Acta Math. 177 (1996), 275--310]. Spectral
notions for the shift operator on quotient Bergman spaces were introduced
in [20]. The mysterious appearance of invariant subspaces in the
Bergman space of index two or more was explained in concrete terms in the
papers [19, 28]. With Borichev, the well-known conjecture
of Shapiro, Shields, and Korenblum -- claiming that invertible elements of
the Bergman space are necessarily cyclic (weakly invertible) -- was
disproved with rather intricate counterexamples [26, 32]. These were
later extended with Borichev and Volberg to large Bergman spaces [45].

**Biharmonic operators.** The structure theorem of Aleman-Richter-Sundberg
for the Bergman space invariant subspaces relies on properties of the
biharmonic Green function; the most important one of them is that it is
positive on the bidisk. The latter property is also used for the
factorization theory. Questions concerning the relationship between inner
functions and corresponding invariant subspaces automatically lead to the
issue whether the Green function for the operator $\Delta\omega^{-1}\Delta$ is
positive, provided that $\omega$ is a weight on the unit disk with
$\log\omega$ subharmonic and such that the weight times area measure is
reproducing for the origin on harmonic functions. The Green function for the
bilaplacian models the clamped plate, and it also occurs in the theory of
creeping flow. The introduction of the weight means that we alter the
geometry of the disk: $\log\omega$ subharmonic means that the geometry is
hyperbolic. My first paper on the matter was [22], and later a kind
of concavity property of Green functions with respect to the weight was
found in [30]. The above-mentioned problem for reproducing
logarithmically subharmonic weights was finally solved in a joint paper
with Jakobsson and Shimorin [35, 42]. The solution involves
finding the Hele-Shaw flow analogue of the classical theorem of Hadamard
(1898) which says that the exponential mapping from the tangent plane to
the surface is a global diffeomorphism provided that the surface is
hyperbolic, complete, and simply connected.

** Riemannian geometry of surfaces. **
The paper with Shimorin [41] on Hele-Shaw flow on curved surfaces suggests
that there are connections between complex analysis and Riemannian geometry
that are yet to be uncovered. For instance, with my student Yolanda Perdomo,
we considered the problem of finding a smooth surface with given curvature
form having minimal area under a normalizing condition [46]. This
differential geometry problem leads to deep complex analysis questions
regarding the possible appearance of extraneous zeros in the Bergman kernel
function.

**Dirichlet series.** With Lindqvist and Seip [31, 36], the Hilbert
space of square summable Dirichlet series is studied with respect to the
pointwise multipliers. This space is analogous to the classical Hardy space
$H^2$ for Taylor series. With Gordon, a complete description of the
composition operators was obtained [99-3].
The analogue of Carleson's convergence theorem for Fourier series is obtained
for Dirichlet series with Saksman in [03-1]. The main motivation for
studying Dirichlet series is their connection with analytic number theory.