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Leonid V. Kovalev

Publications and source records attributed to Leonid V. Kovalev.

At least 19 recordsLinked to original sources
arXiv2026

Symmetric products of absolute Lipschitz retracts

We prove that the Qth symmetric product of an absolute $λ$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)λ$-Lipschitz retract. This answers a question of De Pauw about the target spaces of Almgren's Q-valued maps. The main tool is a reconstruction theorem for bounded products of metric stars with finitely many rays, modulo a finite permutation group: these quotients are absolute 3-Lipschitz retracts, independently of the group and the number of coordinates.

math.MG↗
arXiv2026

Schwarz-contractivity of the Poisson operator

We say that the Poisson operator $P_r$ is Schwarz-contractive from a space $X$ to a space $Y$ if $\|P_r\|_{X\to Y}\le r$ holds for all $0<r<1$. The classical Schwarz lemma is the case $H^\infty_0\to H^\infty$, the subscript indicating zero mean. We are concerned with the non-holomorphic case $L^\infty_0 \to L^p$. For real functions Schwarz-contractivity holds up to the sharp exponent $p_{\mathbb R}=4.109\ldots$. For complex functions we prove it for $p\le 3$ and conjecture that the critical exponent is $4$.

math.CV↗
arXiv2026

Hardy-type norms of matrices

We show that the function that assigns to each square matrix $A$ the geometric mean of $|Ax|$ over all unit vectors $x$ is a norm. The same holds for $L^p$ means with $0<p<1$. Some properties of these norms are proved, and others are conjectured.

math.FA↗
arXiv2023

Anti-absorbing ternary operations on metric spaces

The existence of a median-type ternary operation on a metric space is known to have a number of implications for the geometry of the space. For such operations, if two of the three arguments coincide, they also coincide with the output of the operation. We consider ternary operations with the opposite property: if two of the arguments coincide, the output is equal to the third one. The existence of such an operation is a necessary condition for the space to be an absolute retract.

math.MG↗
arXiv2022

Lipschitz means and mixers on metric spaces

The standard arithmetic measures of center, the mean and median, have natural topological counterparts which have been widely used in continuum theory. In the context of metric spaces it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.

math.MG↗
arXiv2021

Lipschitz clustering in metric spaces

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones.

math.MG↗
arXiv2021

Extreme values of the derivative of Blaschke products and hypergeometric polynomials

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.

math.CV↗
arXiv2021

Continuity of logarithmic capacity

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.

math.CV↗
arXiv2020

Growth rate of Lipschitz constants for retractions between finite subset spaces

For any metric space $X$, finite subset spaces of $X$ provide a sequence of isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence of Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry of $X$ in a subtle way. Such retractions are known to exist when $X$ is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence $\{r_n\}$ can be uniformly Lipschitz. We give a negative answer by proving that $\operatorname{Lip}(r_n)$ must grow with $n$ when $X$ is a normed space or an Hadamard space.

math.MG↗
arXiv2020

Circle embeddings with restrictions on Fourier coefficients

This paper continues the investigation of the relation between the geometry of a circle embedding and the values of its Fourier coefficients. First, we answer a question of Kovalev and Yang concerning the support of the Fourier transform of a starlike embedding. An important special case of circle embeddings are homeomorphisms of the circle onto itself. Under a one-sided bound on the Fourier support, such homeomorphisms are rational functions related to Blaschke products. We study the structure of rational circle homeomorphisms and show that they form a connected set in the uniform topology.

math.CV↗
arXiv2020

Near-isometric duality of Hardy norms with applications to harmonic mappings

Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.

math.CV↗
arXiv2019

Algebraic structure of the range of a trigonometric polynomial

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.

math.CV↗
arXiv2018

Fourier series of circle embeddings

We study the Fourier series of circle homeomorphisms and circle embeddings, with the emphasis on Blaschke product approximation and the vanishing of Fourier coefficients. The analytic properties of the Fourier series are related to the geometry of the circle embeddings, and have implications for the curvature of minimal surfaces.

math.CV↗
arXiv2018

Optimal extension of Lipschitz embeddings in the plane

We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli. Furthermore, every Lipschitz embedding of the circle extends to a Lipschitz homeomorphism of the plane, again with a linear bound on the constant.

math.CV↗
arXiv2017

Removable sets for intrinsic metric and for holomorphic functions

We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.

math.CV↗