Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Hölder Continuity of Random Processes

For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by

Hölder Regularity of μ-Similar Functions

Given a positive measure μ, d contractions on [0,1] and a function g on ℝ, we are interested in function series F that we call “μ-similar functions” associated with μ, g and the contractions. These series F are defined as infinite sums of rescaled and translated copies of the function g, the dilation and translations depending on the choice of the contractions. The class of μ-similar functions F intersects the classes of self-similar and quasi-self-similar functions, but the heterogeneity we introduce in the location of the copies of g make the class much larger. We study the convergence and the global and local regularity properties of the μ-similar functions. We also try to relate the multifractal properties of μ to those of F and to develop a multifractal formalism (based on oscillation methods) associated with F.     

Oscillation of Functions in the Hölder Class

Potential Analysis - We study the size of the set of points where the α-divided difference of a function in the Hölder class Λα is bounded below by a fixed positive constant....     

Hölder Continuity of Roots of Complex and p-adic Polynomials

Let f be a polynomial with coefficients in an algebraically closed, valued field. We show a refinement of the principle of continuity of roots, namely, that each root α of f is locally Hölder continuous of order 1/μ as a function of the coefficients of f, where μ is the root multiplicity of α. This is derived as a consequence of a principle that could be called continuity of factors, namely, that if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are locally Lipschitz continuous as functions of the coefficients of f. The proofs are elementary and of an algebraic nature.     

Hölder Continuity of Solutions of Nondivergent Degenerate Second-Order Elliptic Equations

One considers a degenerate nondivergent second-order linear elliptic equation. In the model case, the matrix of its higher-order coefficients is diagonal and its elements are powers of the moduli of the independent variables. A sufficient condition on the power exponents is obtained which ensures interior a priori Hölder estimates of the solutions.     

On Hölder Continuity of Quasiconformal Maps with VMO Dilatation

If the dilatation tensor or the matrix dilatation of a quasiconformal mapping $ f!:! Gto {bf R} ^ n $ belongs to the space VMO of functions with vanishing mean oscillation, then f is locally Hölder continuous with every exponent f < 1.     

About the Uniform Hölder Continuity of Generalized Riemann Function

In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by

$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$

using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.

     

Integral Estimates of Derivatives of Rational Functions in Hölder Domains

Doklady Mathematics - Given a bounded rational function of degree n in a Hölder domain in the complex plane, it is shown that the area integral of the modulus of its derivative is bounded by a...     

Hölder Continuity of Homeomorphisms with Controlled Growth of Their Spherical Means

We continue the study of homeomorphisms preserving integrally controlled weighted p-module of the ring domains. It was established earlier that under appropriate growth condition for the spherical mean of the weight such mappings are locally Hölder continuous with respect to logarithms of the distances. In this paper, we consider much more general growth conditions and derive the differentiability almost everywhere, local Lipschitz and Hölder continuity. The sharpness of these results is illustrated by several examples. The distortion estimates for measures under such mappings are also established.     

Hölder Continuity of the Solution Set of the Ky Fan Inequality

This paper is concerned with the Hölder continuity of the perturbed solution set to a convex Ky Fan Inequality. We establish some new sufficient conditions for the uniqueness and Hölder continuity of the solution set of the Ky Fan Inequality both in the given space and in its image space by perturbing the objective function and the feasible set. Our methods and results are different from the corresponding ones in the literature. Some examples are given to analyze the obtained results.     

Large Deviations of the Ergodic Averages: From Hölder Continuity to Continuity Almost Everywhere

For many dynamical systems that are popular in applications, estimates are known for the decay of large deviations of the ergodic averages in the case of Hölder continuous averaging functions. In the present article, we show that these estimates are valid with the same asymptotics in the case of bounded almost everywhere continuous functions. Using this fact, we obtain, in the case of such functions, estimates for the rate of convergence in Birkhoff’s ergodic theorem and for the distribution of the time of return to a subset of the phase space.     

Approximation of Functions from the Hölder Class on a Half-Axis

Let (0 < < 1) be the Hölder class on the semi-axis [0,). We characterize the class by rates of approximation of its functions by entire functions of order 1/2 belonging to a special class (similarly to the classical JacksonBernstein theorem). Bibliography: 4 titles.     

Hölder Continuity of Solutions to Parametric Weak Generalized Ky Fan Inequality

In this paper, by using a scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric weak generalized Ky Fan Inequality in the case where the solution mapping is a general set-valued one. The result is different from the recent ones in the literature.     

Hölder Continuity of the Green Function and Markov Brothers’ Inequality

Let V _E be the pluricomplex Green function associated with a compact subset E of \(\mathbb{C}^{N}\) . The well-known Hölder continuity property of E means that there exist constants B>0,γ∈(0,1] such that V _E (z)≤B?dist(z,E) ^γ . The main result of this paper says that this condition is equivalent to a Vladimir Markov-type inequality; i.e., ∥D ^α P∥ _E ≤M ^|α|(degP) ^m|α|(|α|!)^1?m ∥P∥ _E , where m,M>0 are independent of the polynomial P of N variables. We give some applications of this equivalence, e.g., for convex bodies in \(\mathbb{R}^{N}\) , for uniformly polynomially cuspidal sets and for some disconnect compact sets.     

Some Further Generalizations of Ostrowski Inequality for Hölder Functions and Functions with Bounded Derivatives

Some generalizations of Ostrowski inequality for Hölder functions and functions with bounded derivatives are given.     

Hölder Continuity and Optimal Control for Nonsmooth Elliptic Problems

The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.