Volume 85, Nº 3 (2021)

We obtain a lower bound for the rate of convergence of multipoint Pade approximants of functions holomorphicallyextendable from a compact set to a union of domains whose boundaries possess a symmetry property. Thebound obtained matches a known upper bound for the same quantity.

The structure of proper holomorphic maps with multiplicity higher than one from bounded Reinhardt domains in $\mathbb C^2$ onto two-dimensional complex manifolds is described.

We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly ellipticequations with constant complex coefficients on compact sets in $\mathbb{R}^2$ (the particular case of harmonicapproximations is not distinguished).The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of aLaurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations).The proof uses an improvement of Vitushkin's scheme, special geometric constructions, and methods of the theory of singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operatorson $\mathbb{R}^2$, the problem considered is technically more difficult than the analogous problemfor $\mathbb{R}^d$, $d>2$.

In this paper we consider non-local energies defined on probability measures in the plane, given by a convolutioninteraction term plus a quadratic confinement. The interaction kernel is $-\log|z|+\alpha x^2/|z|^2$, $z=x+iy$, with $-1<\alpha<1$. This kernel is anisotropic except for the Coulomb case $\alpha=0$. We present a short compact proofof the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domainenclosed by an ellipse with horizontal semi-axis $\sqrt{1-\alpha}$ and vertical semi-axis $\sqrt{1+\alpha}$.Letting $\alpha \to 1^-$, we find that the semicircle law on the vertical axis is the unique minimizer of the correspondingenergy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote thefirst sections of this paper to presenting some well-known background material in the simplest way possible, so thatreaders unfamiliar with the subject find the proofs accessible.

The famous Jacobian conjecture (JC) remains open even for dimension $2$. In this paper we study it by extending theclass of polynomial mappings to quasi-polynomial ones. We show that any possible non-invertible polynomialmapping with non-zero constant Jacobian can be transformed into a special reduced form by a sequence ofelementary transformations.

We show that the model surface of a germ of a holomorphically homogeneousCR-manifold is holomorphically homogeneous. We also obtain restrictions on the multiplicitiesin the Bloom–Graham type of a germ of a holomorphically homogeneous CR-manifold.

We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.

We define a new class of functions of variable smoothness that areanalytic in the unit disc and continuous in the closed disc. We construct the theoryof the Nevanlinna outer-inner factorization, taking into account the influence of theinner factor on the outer function, for functions of the new class.