Asymptotic soliton-like solutions to the Benjamin–Bona–Mahony equation with variable coefficients and a strong singularity

The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.

Belonging of Laplace-Stieltjes integrals to convergence classes

For positive continuous functions $\alpha$ and $\beta$ increasing to $+\infty$ on $[x_0,+\infty)$ and the Laplace--Stieltjes integral $I(\sigma)=\int\limits_{0}^{\infty}f(x)e^{x\sigma}dF(x),\,\sigma\in{\Bbb R}$, a generalized convergence $\alpha\beta$-class is defined by the condition $$\int\limits_{\sigma_0}^{\infty}\dfrac{\alpha(\ln\,I(\sigma))}{\beta(\sigma)}d\sigma<+\infty.$$ Under certain conditions on the functions $\alpha$, $\beta$, $f$, and $F$, it is proved that the integral $I$ belongs to the generalized convergence $\alpha\beta$-class if and only if $\int\limits_{x_0}^{\infty}\alpha'(x)\beta_1 \left(\dfrac1{x}\ln\dfrac1{f(x)}\right)<+\infty,\,\beta_1(x)= \int\limits_{x}^{+\infty}\dfrac{d\sigma}{\beta(\sigma)}$. For a positive, convex on $(-\infty,\,+\infty)$ function $\Phi$ and the integral $I$, a convergence $\Phi$-class is defined by the condition $\int\limits_{\sigma_0}^{\infty}\dfrac{\Phi'(\sigma)\ln\,I(\sigma)}{\Phi^2(\sigma)}d\sigma<+\infty$, and it is proved that under certain conditions on $\Phi$, $f$ and $F$, the integral $I$ belongs to the convergence $\Phi$-class if and only if $\int\limits_{x_0}^{\infty}\dfrac{dx}{\Phi'\left(({1/x)\ln\,(1/f(x))}\right)}<+\infty$. Conditions are also found for the integral of the Laplace--Stieltjes type $\int\limits_{0}^{\infty} f(x)g(x\sigma)dF(x)$ to belong to the generalized convergence $\alpha\beta$-class if and only if the function $g$ belongs to this class.

Fourier transforms on weighted amalgam type spaces

We introduce weighted amalgam-type spaces and analyze their relations with some known spaces. Integrability results for the Fourier transform of a function with the derivative from one of those spaces are proved. The obtained results are applied to the integrability of trigonometric series with the sequence of coefficients of bounded variation.

Teichmuller space theory and classical problems of geometric function theory

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions.

Conformality, classical and generalized

We discuss several topics: the concept of conformal mapping of Riemannian and pseudo-Riemannian manifolds, conformal rigidity of higher-dimensional domains, and conformal flexibility of two-dimensional domains of Euclidian and Minkowski planes. We present an extension of the concept of conformal mapping proposed by M. Gromov and recall an open problem related to it.

On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is H\"{o}lder continuous in the ''weak'' (logarithmic) sense with the exponent power $\alpha=1/2.$ Moreover, it belongs to the class $W^{1, 2}_{\rm loc}.$ Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.

An alternative capacity in metric measure spaces

A new condenser capacity $\CMp(E,G)$ is introduced as an alternative to the classical Dirichlet capacity in a metric measure space $X$. For $p>1$, it coincides with the $M_p$-modulus of the curve family $\Gamma(E,G)$ joining $\partial G$ to an arbitrary set $E \subset G$ and, for $p = 1$, it lies between $AM_1(\Gamma(E,G))$ and $M_1(\Gamma(E,G))$. Moreover, the $\CMp(E,G)$-capacity has good measure theoretic regularity properties with respect to the set $E$. The $\CMp(E,G)$-capacity uses Lipschitz functions and their upper gradients. The doubling property of the measure $\mu$ and Poincar\'e inequalities in $X$ are not needed.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with $\BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $\BMO$-spaces, we prove that $\BMO$-quasiconformal mappings generate bounded composition operators from Hardy--Sobolev spaces to Sobolev spaces.

A note on meromorphic functions with finite order and of bounded l-index

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$