Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.Comment: 26 page

On meromorphic functions whose image has finite spherical area

In this paper, we study meromorphic functions on a domain $\Omega \subset \mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $\Omega$ union a tree of spheres. In the second part, we show that a set $E \subset \Omega$ is removable if and only if it is negligible for extremal distance.Comment: 21 page

The Geometry of the Weil-Petersson Metric in Complex Dynamics

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. We show that the Weil-Petersson metric is incomplete and study its metric completion. Our work parallels known results for the Teichmuller space of a punctured torus.Mathematic

Prescribing inner parts of derivatives of inner functions

Let ℐ be the set of inner functions whose derivative lies in the Nevanlinna class. We show that up to a post-composition with a Möbius transformation, an inner function F ∈ ℐ is uniquely determined by the inner part of its derivative. We also characterize inner functions which can be represented as Inn F′ for some F ∈ ℐ in terms of the associated singular measure, namely, it must live on a countable union of Beurling–Carleson sets. This answers a question raised by K. Dyakonov

Prescribing inner parts of derivatives of inner functions

Let ℐ be the set of inner functions whose derivative lies in the Nevanlinna class. We show that up to a post-composition with a Möbius transformation, an inner function F ∈ ℐ is uniquely determined by the inner part of its derivative. We also characterize inner functions which can be represented as Inn F′ for some F ∈ ℐ in terms of the associated singular measure, namely, it must live on a countable union of Beurling–Carleson sets. This answers a question raised by K. Dyakonov

Beurling-Carleson sets, inner functions and a semi-linear equation

Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure $\mu$ on the unit circle, let $S_\mu$ denote the singular inner function with singular measure $\mu$. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle such as the membership of $S'_\mu$ in the Nevanlinna class $\mathcal N$, area conditions on level sets of $S_\mu$ and wepability. It was known that each of these properties hold for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that $\mu$ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of $S'_\mu$ in the Hardy space $H^p$, membership of $S_\mu$ in the Besov space $B^p$ and $(1-p)$-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of $\alpha$-Beurling-Carleson sets are in bijection with nearly-maximal solutions of $\Delta u = u^p \cdot \chi_{u > 0}$ when $p > 3$ and $\alpha = \frac{p-3}{p-1}$.Comment: 49 page