Directional Hölder Metric Regularity

This paper studies the first-order behavior of the value function of a parametric optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Fréchet subdifferential of the value function of a parametric minimization problem, we derive a formula for computing the Fréchet subdifferential of the value function to a parametric optimal control problem. The obtained results improve and extend some previous results.     

Large Deviations of Solutions of Hyperbolic SPDE's in the Hölder Norm

In this paper, we prove the large deviations principle for solutions of a hyperbolic stochastic partial differential equation, in the Hölder topology of index for all 0 < . This result generalizes those in [5] and [10] to the Hölder norm, and the result in [3] for solutions of a class fo stochastic differential equations involving a two-parameter Wiener process. These solutions are obtained by small perturbations of the noise.     

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 Estimates for the ∂¯-Operator on Polycylinders

Hölder estimates for the ?¯-operator are obtained on polycylinders and applied to a number of problems-cohomology with bounds, the corona problem and approximation of Hölder functions by holomorphic functions.     

Metric properties of classes of Hölder surfaces on Carnot groups

Area formulas for classes of Hölder mappings of Carnot groups and the corresponding graph mappings are obtained. The calculation of a nonintrinsic measure is exemplified.     

Generalized Hölder Spaces of Holomorphic Functions in Domains in the Complex Plane

We study some nonstandard spaces of functions holomorphic in domains on the complex plain with certain smoothness conditions up to the boundary. The first type is the space of Hölder-type holomorphic functions with prescribed modulus of continuity \(\omega =\omega (h)\), and the second is the variable exponent holomorphic Hölder space with the modulus of continuity \(|h|^{\lambda (z)}\). We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.     

Estimates in Hölder Classes for the Solution of an Inhomogeneous Dirichlet Problem for a Singularly Perturbed Homogeneous Convection–Diffusion Equation

Computational Mathematics and Mathematical Physics - An inhomogeneous Dirichlet boundary value problem for a singularly perturbed homogeneous convection–diffusion equation with constant...     

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 Continuity of the Unique Solution to Quasiequilibrium Problems in Metric Spaces

We extend and sharpen the result in Ait Mansour and Scrimali (J. Glob. Optim. 40:175–184, 2008) on the Hölder continuity of the solution sets of quasivariational inequalities in Hilbert spaces to the case of quasiequilibrium problems in metric spaces. Applications to some important problems are also provided.     

Hölder Estimates for the Regular Component of the Solution to a Singularly Perturbed Convection–Diffusion Equation

In a half-plane, a homogeneous Dirichlet boundary value problem for an inhomogeneous singularly perturbed convection–diffusion equation with constant coefficients and convection directed orthogonally away from the boundary of the half-plane is considered. Assuming that the right-hand side of the equation belongs to the space C^λ, 0 < λ < 1, and the solution is bounded at infinity, an unimprovable estimate of the solution is obtained in a corresponding Hölder norm (anisotropic with respect to a small parameter).     

Hölder Continuity for Solution Mappings of Parametric Non-convex Strong Generalized Ky Fan Inequalities

Abstract

This article focuses on a new approach to investigate the Hölder continuity for the solution mapping of a parametric non-convex strong generalized Ky Fan inequality. Based on a non-convex separation theorem, the union relation between the solution set of the parametric non-convex strong generalized Ky Fan inequality and the solution sets of a series of Ky Fan inequalities, is established. Without density results and any information on the solution mapping, a sufficient condition for the Hölder continuity of the solution mapping to the parametric non-convex strong generalized Ky Fan inequality is given by using the key union relation. Our method does not impose any convexity, monotonicity, and the single-valuedness of the solution mapping.     

On the Cauchy Problem for Nondegenerate Parabolic Integro-Differential Equations in the Scale of Generalized Hölder Spaces

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of Hölder spaces of functions whose regularity is defined by a radially O-regularly varying Lévy measure. Existence and uniqueness and the estimates of the solution are derived.

     

On the Hölder Property of Trajectories in a Set of Full Wiener Measure on the Heisenberg Group

Mathematical Notes -     

Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Hölder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Hölder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.     

Hölder rigidity for matrices

It is proved that a (C ₁, C ₂)-Hölder valuation is (2, α)-equivalent to classical valuation on the set of matrices over a skew field and on the set of cubic matrices over a field. These results provide an extension of the Garcia theorem.     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

Hölder Conditions for Endomorphisms of Hyperbolic Groups

It is proved that an endomorphism φ of a hyperbolic group G satisfies a Hölder condition with respect to a visual metric if and only if φ is virtually injective and Gφ is a quasiconvex subgroup of G. If G is virtually free or torsion-free co-hopfian, then φ is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective.