The local Hölder function of a continuous function

This work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0.     

Multifractal Formalism for Selfsimilar Functions Expanded in Singular Basis

Selfsimilar functions can be written as the superposition of similar structures, at different scales, generated by a function g. Their expressions look like wavelet decompositions. In the case where g is regular, the multifractal formalism has been proved for the corresponding selfsimilar function, for Hölder exponents smaller than the regularity of g. In this paper, we show, in the case where g is the Schauder function (or the Haar function or a spline-type wavelet), that for larger Hölder exponents, the singularities of g can disturb the Hölder exponents of the associated selfsimilar function, modify the shape of the spectrum of singularities, and finally affect the validity of the multifractal formalism.     

Hölder Exponent for a Two-Parameter Lévy Process

We study the regularity of a two-parameter Lévy process in the neighbourhood of a fixed point and then we compute the Hölder exponent of such a process.     

Characterization of Pointwise Hölder Regularity

We study different characterizations of the pointwise Hölder spacesC^s(x₀), including rate of approximation by smooth functions and iterated differences. As an application of our results we study the class of functions that are Hölder exponents and prove that the Hölder exponent of a continuous function is the limit inferior of a sequence of continuous functions, thereby improving a theorem of S. Jaffard.     

On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

We study the Dirichlet problem for the parabolic equation u_t = Δu^m, m > 0, in a bounded, non-cylindrical and non-smooth domain Ω ^{N + 1}, N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent is critical as in the classical theory of the one-dimensional heat equation u_t = u_xx.     

Stability of Calderón inverse conductivity problem in the plane

It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable when the conductivities are Hölder continuous with any exponent α>0.     

A matrix reverse Hölder inequality

A matrix reverse Hölder inequality is given. This result is a counterpart to the concavity property of matrix weighted geometric means. It extends a scalar inequality due to Gheorghiu and contains several Kantorovich type inequalities.     

Berry–Esseen theorem and local limit theorem for non uniformly expanding maps

In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed , and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.     

Remarks on Hölder continuity for solutions of the p-Laplace evolution equations

We study the interior Hölder regularity problem for the gradient of solutions of the p-Laplace evolution equations with the external forces. Misawa gave some conditions for the Hölder continuity of the gradient of solutions. We show Hölder estimates of the solutions with weaker condition as for Misawa.     

Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type

Weighted norm inequalities with general weights are established for the maximal singular integral operators on spaces of homogeneous type, when the kernel satisfies a Hörmander regularity condition on one variable and a Hölder regularity condition on the other variable.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

The 1-d stochastic wave equation driven by a fractional Brownian sheet

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet.     

Reverse Hölder inequalities for singular parabolic equations near the boundary

We show that weak solutions to a singular parabolic partial differential equation globally belong to a higher Sobolev space than assumed a priori. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. The results extend to singular parabolic systems as well. Motivation for studying reverse Hölder inequalities comes partly from applications to regularity theory.     

Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient. In particular, the given estimate is independent of the regularity of the coefficients. As in the original paper by Krylov and Safonov, the proof relies on a probabilistic interpretation of the equation.     

Complexity penalized support estimation

We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate supports which consists of disconnected regions. We will prove that the estimator achieves minimax rates of convergence up to a logarithmic factor simultaneously over a scale of Hölder smoothness classes for the boundary of the support. The proof assumes a sharp boundary for the support.     

Hölder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems

In this paper, we establish Hölder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to Hörmander type vector fields by deriving a parabolic Poincaré inequality and the higher integrability of gradients of weak solutions.     

Self-Affine Fractal Functions and Wavelet Series

We consider functions represented by series ∑_g  G c_gψ(g ^− 1(x)) of wavelet-type, where G is a group generated by affine functions L₁,…,L_n and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point.     

Positive solutions of the generalized Emden-Fowler equation in Hölder spaces

A singular sublinear BVP related to the Emden-Fowler equation is considered. Existence, nonexistence, and regularity of positive solutions in Hölder spaces is obtained.     

Real harmonizable multifractional stable process and its local properties

A real harmonizable multifractional stable process is defined, its Hölder continuity and localizability are proved. The existence of local time is shown and its regularity is established.