Area formulas for classes of Hölder continuous mappings of Carnot groups

We prove area formulas for classes of the mappings that are Hölder continuous in the sub-Riemannian sense and defined on nilpotent graded groups. Moreover, in one of the model cases, we establish an area formula for calculating the initial measure and a measure close to it.     

Hölder regularity of optimal mappings in optimal transportation

It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C ^1,α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper.     

Regularity results for deep-water waves with Hölder continuous vorticity

In this paper we study the regularity properties of periodic deep-water waves travelling under the influence of gravity. The flow beneath the wave surface is assumed to be rotational and the vorticity function is taken to be uniformly Hölder continuous. Excluding the presence of stagnation points, we transform the problem on a fixed reference half-plane and we use Schauder estimates to prove that the streamlines and the free surface of such waves are real-analytic graphs.     

Exponential convergence for functional SDEs with Hölder continuous drift

Abstract

Applying Zvonkin’s transform, the exponential convergence in Wasserstein distance for a class of functional SDEs with Hölder continuous drift is obtained. This combining with log-Harnack inequality implies the same convergence in the sense of entropy, which also yields the convergence in total variation norm by Pinsker’s inequality.     

On Hölder calmness of solution mappings in parametric equilibrium problems

We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces.     

On the Perron-Frobenius-ruelle operator for rational maps on the riemann sphere and for Hölder continuous functions

Let      

Hölder mappings of Carnot groups and intrinsic bases

For classes of Hölder mappings of Carnot groups and the corresponding graph mappings, a method for constructing intrinsic bases is described, which makes it possible to push forward the Hausdorff dimension of preimages to images.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

Some exact equivalents for the Brownian motion in Hölder norm

Summary Some exact equivalents of small probabilities are given for the Wiener measure on spaces of Hölder paths. It turns out that most of them are easier to derive than their counterparts in the uniform norm because of a classical result of Z. Ciesielski which makes the Brownian motion on these spaces easy to handle. In particular we study the equivalents of the probability of B in a fixed ball, ofB in a small ball and we give applications to the speed of clustering in Strassen law.     

Hölder norm estimate for the Hilbert transform in Clifford analysis

Let Ω ? ? ⁿ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum (d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface.     

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.     

Eigenfunction statistics for Anderson model with Hölder continuous single site potential

We consider random Schrödinger operators on \(\ell ^{2}(\mathbb {Z}^{d})\) with α-Hölder continuous (0<α≤1) single site distribution. In localized regime, we study the distribution of eigenfunctions in space and energy simultaneously. In a certain scaling limit, we prove limit points are Poisson.     

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....     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Mathematische Zeitschrift - We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $$C^{k}$$ quasi-periodic $$SL(2,{\mathbb {R}})$$ cocycle is...     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

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.     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).