Reverse estimates in logarithmic Bloch spaces

We obtain sharp reverse estimates for the logarithmic Bloch spaces on the unit disk. As an application, we study composition operators with values in the space BMOA.     

Sharp weighted logarithmic bound for maximal operators

The paper contains the study of sharp weighted logarithmic estimates for maximal operators on probability spaces equipped with a tree-like structure. These inequalities can be regarded as LlogL versions of the classical estimates of Fefferman and Stein. The proof exploits the existence of a certain special function, enjoying appropriate majorization and concavity conditions.     

Log-Sobolev inequalities and regions with exterior exponential cusps

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log-Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy-type inequality. Our main application is to show that domains which have exterior exponential cusps, and hence have no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusps are not too sharp. Finally, we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp.     

An analog of the Schwarz lemma for locally quasiconformal automorphisms of the unit disk

We obtain sharp estimates for the module of functions in the classes of normalized locally quasiconformal authomorphisms of the unit disk with given majorants of M. A. Lavrent’ev’s characteristic. The estimates are analogs of Schwarz’s lemma and A. Mori’s theoremand they imply the classical growth theorems for quasiconformal authomorphisms of the disk. In the classes we also prove sharp estimates of the conformal radius and the radius of covering disk. The main results are obtained by methods of extremal lengths and symmetrization.     

Products of polynomials in uniform norms

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment . Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.

     

Boundary Distortion Estimates for Holomorphic Maps

We establish some estimates of the angular derivatives from below for holomorphic self-maps of the unit disk ${\mathbb {D}}$ at one and two fixed points of the unit circle provided there is no fixed point inside ${\mathbb {D}}$ . The results complement Cowen–Pommerenke and Anderson–Vasil’ev type estimates in the case of univalent functions. We use the method of extremal length and a semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to obtain a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.     

A Sharp Version of Mahler's Inequality for Products of Polynomials

In this note we give some sharp estimates for norms of polynomialsvia the products of norms of their linear terms. Different convexnorms on the unit disc are considered. 1991 Mathematics SubjectClassification 30C10, 11C08.     

Pointwise Remez-Type Inequalities in the Unit Disk

We obtain a sharp upper bound on exponentials of logarithmic potentials at any point of the unit disk D, provided that the area of the subset of \overline{D}, where such exponentials are at most 1, is known.     

Distortion theorems for univalent meromorphic functions on an annulus

We apply the capacity and symmetrization methods to distortion theorems for analytic functions in an annulus. We show that the classical Teichmüller estimate for the capacity of a doubly-connected domain yields a series of the already known and new inequalities for univalent functions. In particular, we supplement the results of Grötzsch, Duren, and Huckemann. Using the dissymmetrization of condensers we establish sharp estimates for local distortion and the distortion of level curves in n ≥ 2 symmetric directions. In terms of Robin functions we give an analog of the Nehari inequality: some general distortion theorem for several points taking into account the boundary behavior of the function and describing the cases of equalities. As a corollary we give analogs of some inequalities of Solynin, Pommerenke, and Vasil’ev that were obtained previously for univalent and bounded functions in a disk. We prove a distortion theorem that involves the Schwarzian derivatives at symmetric points on the unit circle.     

Differential inequalities in linear- and affine-invariant families of harmonic mappings

In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

某类双调和映射的Landau型定理

令f(z)=h(z)+g(z)为开单位圆盘U内的调和映射.本文首先建立了有界调和映射的精确系数估计,其次得到了满足有界性条件|h(z)|+|g(z)|≤M的正规化调和映射的精确系数估计.作为其应用,得到了双调和映射的一些Landau型定理.这些结果改进和推广了早期作者的相关结果.     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Bloch Functions in Several Complex Variables, I

Bloch functions on the unit disk are those analytic functionsf for which the quantity |f'(z)|(1 – |z|²) is bounded(for z in the disk). In this paper, Bloch functions on boundedhomogeneous domains in complex m-space are studied. The mainresult shows that many of the equivalent definitions of Blochfunctions on the unit disk are also equivalent in the generalsetting. Current Address: 39 Trinity College, Dublin 2, Ireland.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

Differential Inequalities and Starlike Functions

We develop sufficient conditions for various types of starlikeness for functions analytic in the unit disk. The problems that we consider are more general than similar problems considered by previous authors. Furthermore we obtain stronger estimates than in earlier works and in several cases sharp ones.     

Talagrand's T_2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction-Diffusion equations are provided.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H ^k,∞; k ∈ ?* of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L.Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S.Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces H ^k,∞, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.