Continuous families of Hölder functions that are not of bounded variation

This paper deals with three classes of functions of great importance in analysis and its applications. We construct a family of Hölder functions in the closed unit interval having two continuous parameters. Those functions are not of bounded variation for any pair of values of the Hölder constant and exponent. The construction depends on a change of variables given by a Lipschitz function with constant equal to 1. Several questions related to the concepts of genericity, surjectivity and deformability are posed at the end.     

A decomposition of functions with zero means on circles

It is well known that every Hölder continuous function on the unit circle is the sum of two functions such that one of these functions extends holomorphically into the unit disc and the other extends holomorphically into the complement of the unit disc. We prove that an analogue of this holds for Hölder continuous functions on an annulus A which have zero averages on all circles contained in A which surround the hole.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

Let {X(t)} _t∈? be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

Extending Lipschitz and Hölder maps between metric spaces

We introduce a stochastic generalization of Lipschitz retracts, and apply it to the extension problems of Lipschitz, Hölder, large-scale Lipschitz and large-scale Hölder maps into barycentric metric spaces. Our discussion gives an appropriate interpretation of a work of Lee and Naor.     

Polyanalytic Hardy decomposition of higher order Lipschitz functions

This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve Γ into a sum of two polyanalytic functions in each open domain defined by Γ. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on the boundary of a domain.     

The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups

The polynomial sub-Riemannian differentiability is established for the large classes of Hölder mappings in the sub-Riemannian sense, namely, the classes of smooth mappings, their graphs, and the graphs of Lipschitz mappings in the sub-Riemannian sense defined on nilpotent graded groups. We also describe some special bases that carry the sub-Riemannian structure of the preimage to the image.     

Nonlinear scalarizing functions in set optimization problems

In this paper, we obtain Hölder continuity of the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín (J. Math. Anal. Appl. 2007;325:1–18). Moreover, we introduce the nonlinear scalarizing function for u-type less order relation and establish continuity, convexity and Hölder continuity of the nonlinear scalarizing function for u-type less order relation. As applications, we firstly obtain Lipschitz continuity of solution mapping to the parametric equilibrium problems and then establish Lipschitz continuity of strongly approximate solution mappings for l-type less order relation, u-type less order relation and set less order relation to the parametric set optimization problems by using convexity and Hölder continuity of the nonlinear scalarizing functions.     

Stochastic Reaction-Diffusion Systems With Hölder Continuous Multiplicative Noise

We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Hölder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

Global optimization of Hölder functions

We propose a branch-and-bound framework for the global optimization of unconstrained Hölder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate Hölder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.     

An Uniparametric Secant–Type Method for Nonsmooth Generalized Equations

We are concerned with the problem of approximating a locally unique solution of a nonsmooth generalized equation in Banach spaces using an uniparametric Secant–type method. We provide a local convergence analysis under ω–conditioned divided difference which generalizes the usual Lipschitz continuous and Hölder continuous conditions used in [14].     

On the Local Convergence of Modified Homeier-Like Method in Banach Spaces

The aim of this article is to investigate the local convergence analysis of the multi-step Homeier-like approach in order to approximate the solution of nonlinear equations in Banach spaces, which fulfilled the Lipschitz as well as Hölder continuity condition. The Hölder condition is more relax than Lipschitz condition. Also, the existence and uniqueness theorem has been derived and found their error bounds. Numerical examples are available to appear the importance of theoretical discussions.     

Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions

In this paper, new results for Hölder continuity of the unique solution to a parametric generalized vector quasiequilibrium problem are established via nonlinear scalarization, with and without using the free-disposal condition. Especially, a new kind of monotonicity hypothesis is proposed. The globally Lipschitz property together with other useful properties of the well-known Gerstewitz nonlinear scalarization function are fully exploited for proving. Moreover, our approach does not impose any convexity condition on the considered model. The oriented distance function is also employed for studying Hölder continuity.     

Weakly hyperbolic equations with Lipschitz or Hölder continuous coefficients with respect to time

We consider second order weakly hyerbolic equations, with Lipschitz or Hölder continuous coefficients with respect to time, concerning the well posedness of the Cauchy problem and the propagation of the singularities in the framework of Gevrey classes.     

Nonlinear potentials of the cauchy-dirichlet problem for the integrodifferential Bellman equation

The Cauchy-Dirichlet problem for the integrodifferential Bellman equation, arising in the theory of contolled Ito processes, is investigated. Sufficient conditions are given, under which this problem has a viscosity solution Lipschitz continuous inx and Hölder continuous int with the exponent 1/2. The proof is based on the method of nonlinear potentials.     

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.     

Un teorema di confronto per funzioni armoniche in domini con frontiera hölderiana

We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Hölder continuous function. In particular we sketch a non probabilistic proof of a Harnack-type principle, due to Bañuelos, Bass and Burdzy.     

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.