Related papers: On some extremal problems in spaces of harmonic fu…
We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…
Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…
In this paper we consider an extremal problem in geometry. Let $\lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $\Gamma$. We give full characterization of the extremal behavior of the function…
Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the $k$-Hessian operator acting on $\Phi^{k}_{0,\mathrm{rad}}(B)$, the space of radially symmetric $k$-admissible functions on the unit ball…
In this paper we prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n.
We study some Hardy-type inequalities involving a general norm in $R^n$ and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.
In this paper, we show that the extremal length functions on Teichm\"uller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions.…
In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial…
In this paper, we investigate the extremal functions for anisotropic Trudinger-Moser inequalities. Our method uses convex symmetrization, the continuity of the supremum function, together with the relation between the supremums of the…
With this paper, we begin a series of studies of extremal problems for estimating distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave…
Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of…
We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable…
In this paper, we present the solution to Kolmogorov's problem for the classes of multiply monotone and completely monotone functions together with its connections to the Markov moment problem, Hermite-Birkhoff interpolation problem, and…
In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.
A multidimensional extremal problem in the idempotent algebra setting is considered which consists in minimizing a nonlinear functional defined on a finite-dimensional semimodule over an idempotent semifield. The problem integrates two…
We provide infinitely many solutions of a Dirichlet problem on balls.
We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several…
We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the…
We consider the problem of extending the integrals of motion of soliton equations to the space of all finite-gap solutions. We consider the critical points of these integrals on the moduli space of Riemann surfaces with marked points and…
In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are related to the…