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Leindler [5] obtained certain estimates of the approximation of Fourier series by Nörlund-Voronoi means in the Hölder metric. Making use of the equality of two norms in the Hölder space and a theorem of Leindler-Meir-Totik [6], we improve these estimates.     

Weak $$A_\infty $$ Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type

In the Euclidean setting, the Fujii–Wilson-type \(A_\infty \) weights satisfy a reverse Hölder inequality (RHI), but in spaces of homogeneous type the best-known result has been that \(A_\infty \) weights satisfy only a weak reverse Hölder inequality. In this paper, we complement the results of Hytönen, Pérez and Rela and show that there exist both \(A_\infty \) weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property, but the self-improving property of the strong reverse Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.     

Regularity results for fully nonlinear parabolic integro-differential operators

In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov–Backelman–Pucci estimate with $0<\sigma <2$ . And we show a Harnack inequality, Hölder regularity, and $C^{1,\alpha }$ -regularity of the solutions by obtaining decay estimates of their level sets.     

Reverse Hölder Property for Strong Weights and General Measures

We present dimension-free reverse Hölder inequalities for strong \(A^*_p\) weights, \(1\le p < \infty \). We also provide a proof for the full range of local integrability of \(A_1^*\) weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For \(p=\infty \), we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed \(A_p^*-A_\infty ^*\) weighted estimates.     

Stability of quasiminimizers of the p-Dirichlet integral with varying p on metric spaces

We prove a stability result, with respect to the varying exponentp, for a family of quasiminimizers of the p-Dirichlet energyfunctional on a doubling metric measure space. In addition,we prove global higher integrability for upper gradients ofquasiminimizers with fixed boundary data, provided that theboundary data belong to a slightly better Newtonian space.     

Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ .     

A Family of Functions with Two Different Spectra of Singularities

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\) -exponent. We prove that in the general case the Hölder and \(p\) -exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.     

Free interpolation in Hölder classes in quasiconformal domains

The free interpolation problem for Hölder classes on a simply connected bounded domain whose boundary is a quasiconformal curve is studied. The necessary and sufficient conditions on a closed set $E \subset \overline G $ under which the whole Hölder space on E is generated by the restrictions of analytic functions of the Hölder class to the domain G are discussed. Bibliography: 7 titles.     

Schauder theory for Dirichlet elliptic operators in divergence form

Let A be a strongly elliptic operator of order 2m in divergence form with Hölder continuous coefficients of exponent ${\sigma \in (0,1)}$ defined in a uniformly C ^1+σ domain Ω of ${\mathbb{R}^n}$ . Regarding A as an operator from the Hölder space of order m +  σ associated with the Dirichlet data to the Hölder space of order ?m +  σ, we show that the inverse (A ? λ)^?1 exists for λ in a suitable angular region of the complex plane and estimate its operator norms. As an application, we give a regularity theorem for elliptic equations.     

Regularity of harmonic maps from polyhedra to CAT(1) spaces

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the \((n-2)\)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points \(x \in X^{(k)}\) with \(k \le n-2\), we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of \(x \in X\).     

Morrey-Sobolev Spaces on Metric Measure Spaces

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space (??, d, μ). The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if ?? supports a weak Poincaré inequality and the measure μ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors Q-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj?asz gradient, the authors also introduce the Haj?asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj?asz-Morrey-Sobolev space when μ is doubling and ?? supports a weak Poincaré inequality. In particular, on the Euclidean space \({\mathbb R}^n\) , the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj?asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when (??, d) is geometrically doubling and μ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when μ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj?asz-Morrey-Sobolev spaces.     

Hölder regularity for parabolic De Giorgi classes in metric measure spaces

We give a proof for the Hölder continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak (1, p)-Poincaré inequality and to satisfy the annular decay property.     

Hölder exponents of the Green functions of planar polynomial Julia sets

If the Green function gE of a compact set ${E \subset \mathbb{C}}$ is Hölder continuous, then the Hölder exponent of the set E is the supremum over all such α that $$|g_E(z)-g_E(w)|\leq M|z-w|^\alpha,\, z, w \in \mathbb{C}.$$ We give a lower bound for the Hölder exponent of the Julia sets of polynomials. In particular, we show that there exist totally disconnected planar sets with the Hölder exponent greater than 1/2 as well as fat continua with the boundary nowhere smooth and with the Hölder exponent as close to 1 as we wish.     

On Weak*-Convergence in \boldsymbol{H^1_L(\boldsymbol{\mathbb R}^d)}

Let L?=???Δ?+?V be a Schrödinger operator on $\mathbb R^d$ , d?≥?3, where V is a nonnegative function, $V\ne 0$ , and belongs to the reverse Hölder class RH _d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space $H^1_L(\mathbb R^d)$ .     

Local Hölder Continuity Property of the Densities of Solutions of SDEs with Singular Coefficients

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier–Printems (Bernoulli 16(2):343–360, 2010), where the density is shown to exist if both coefficients are Hölder continuous, and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.     

On the Reverse Hölder inequality

We obtain exact positive and negative exponents of integrability of a function satisfying the reverse weighted Hölder inequality on parallelepipeds.     

Global integrability of p-superharmonic functions on metric spaces

We consider a metric space equipped with a doubling measure and a length metric. We prove that p-superharmonic functions are integrable with a small exponent on Hölder domains of the space.     

Hölder continuity for a class of X-elliptic equations with singular lower order term

The regularity for a class of X-elliptic equations with lower order term
Lu＋vu=-∑i,j=1 mXj^＊（aij（x）Xiu）＋vu=μ
is studied, where X = {X1,..., Xm} is a family of locally Lipschitz continuous vector fields, v is in certain Morrey type space and μ a nonnegative Radon measure. The HSlder continuity of the solution is proved when μ satisfies suitable growth condition, and a converse result on the estimate of μ is obtained when u is in certain HSlder class.     

Everywhere regularity of certain nonlinear diffusion systems

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left[ \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where ${\Phi(z)}$ is a strictly convex function. I show that when ${\Phi}$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.     

Orlicz spaces,spline systems,and brownian motion

There are three results proved in this paper. The first one characterizes the Hölder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two-dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Hölder functions with exponent 1/2 but in the norm of the Orlicz spaceL _M ^* withM(t)=expt(t ²)?1.