Intrinsic Hölder Continuity of Harmonic Functions

We consider the stochastic differential equation (SDE) of the form

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$

where \(\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d\) is globally Lipschitz continuous and L={L(t):t≥0} is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let \((\mathcal {P}_{t})_{t\ge 0}\) be the Markovian semigroup associated to X defined by \(\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]\), t≥0, \(x\in {\mathbb {R}}^{d}\), \(f\in \mathcal {B}_{b}({\mathbb {R}}^{d})\). Let B be a pseudo–differential operator characterized by its symbol q. Fix \(\rho \in \mathbb {R}\). In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that

$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$

     

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.     

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

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 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.     

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.

     

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.     

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.     

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 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.     

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.     

On the Approximation of Functions of the Hölder Class by Triharmonic Poisson Integrals

We determine the exact value of the upper bound for the deviation of the triharmonic Poisson integral from functions of the Hölder class.     

On the Approximation of Functions of the Hölder Class by Biharmonic Poisson Integrals

We determine the exact value of the upper bound of the deviation of biharmonic Poisson integrals from functions of the Hölder class.     

Estimate of the Hölder Exponent Based on the $$epsilon$$ -Complexity of Continuous Functions

Mathematical Notes -     

Hölder Continuity for the Fokas–Olver–Rosenau–Qiao Equation

It has been shown that the Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation is well-posed for initial data \(u_0\in H^s\) , \(s>5/2\) , with its data-to-solution map \(u_0\mapsto u\) being continuous but not uniformly continuous. This work further investigates the continuity properties of the solution map and shows that it is Hölder continuous in the \(H^r\) topology when \(0\le r . The Hölder exponent is given explicitly and depends on both \(s\) and \(r\) .     

On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces

This paper is devoted to the compactness of the hypercomplex commutator S _γ M _a ? M _a S _γ, where S _γ is the Cauchy singular integral operator (in the Douglis sense), a is a Hölder continuous hypercomplex function and M _a is the multiplication operator given by M _a f = a f. We extend a known compactness sufficient condition for the commutator of the Cauchy singular integral operator to the frame of the hypercomplex analysis, where γ is merely required to be an arbitrary regular closed Jordan curve.     

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.