Use this tag for concepts related to Hölder continuity (a generalisation of Lipschitz continuity) and the related Hölder spaces.

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Hölder Space Definition

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows: If $u:U \rightarrow \mathbb{R}$ is bounded and ...
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Regarding Hölder continuity

Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - ...
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Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose ...
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Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the ...
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Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
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On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
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Hölder Condition Implying Uniform Convergence

Define $g(z)=\frac{1}{2\pi i}\int_{-k}^k \frac{h(\zeta)}{\zeta-z}d\zeta$, where $h$ is continuous and defined on $[-k,k]$. Let $|h(x)-h(y)|\leq |x-y|^\alpha$ for all $x,y\in[-k,k]$ and for some ...
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Teacher Challenge - multiple parts

This was his challenge: "I would like you to consider the function $x^{r+\alpha}$, $r$ is an integer, $\alpha$ is a real number between $0$ and $1$. Differentiate it until you get a singularity ...
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Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
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Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
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Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous

Consider the following function defined by a singular integral \begin{equation} F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) ...
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When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
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Convergence of sequence in $C^{k, \alpha}$ composed with $C^\infty$ function

Is it true that if a sequence $u_n \to u$ in $C^{k, \alpha}$ norm, and if you have a function $f \in C^\infty$, then $f(u_n) \to f(u)$ in $C^{k, \alpha}$? I think so, since this is true for ordinary ...
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Proving a basic result about Holder continuous functions

Let $V$ be a open convex set. We will say that a function $m$ has the order of smoothness $p$ on $V$ with $p=l+\gamma$, where $l \geq0$ is an integer and $0<\gamma\leq1$ and will write $m\in ...
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Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$ Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
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Multiplicative constant in inclusions of Hölder spaces

Let $\Sigma(n + \beta, L)$ for $n \in \mathbb{N}_0$, $0 < \beta \le 1$, $L > 0$ be the set of functions $f : \Omega \to \mathbb{R}$ (or $\mathbb{C}$, whatever) whose derivatives up to order $n$ ...
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Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
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Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
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Parabolic holder norms

Let $Q=\Omega\times[0,T]$ be a cylinder with $\Omega$ bounded open set in $\mathbb{R}^N$. N.V.Krylov in "lectures on elliptic and parabolic equations in holder spaces" defines the parabolic holder ...
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Estimates in Hölder spaces

Let $u,v\in C^{2,\alpha}\left(\overline{\Omega}\right)$. Proof that there exists a constant $C>0$ so that \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq ...
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Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in ...
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Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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Regularity of a a transmission problem

How to prove? Let $a_{ij} \in C^{0, \alpha}(B_1 \cap \mathbb{R}^{n}_{+}), b_{ij} \in C^{0,\alpha}(B_1 \cap \mathbb{R}^{n}_{-})$ elliptic matrices and $$ A_{ij}(x) = a_{ij}(x)\chi_{\{ x_n \ge 0 \}} + ...
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The function $f(x) = x^\alpha $ belongs to the Lipschitz class of order $\alpha$

Suppose that we have a Lipschitz function $$ |f(x_1) - f(x_2) |\le M {|x_1 - x_2|}^{\alpha} $$ where M is a constant. Note Lip $\alpha$ denote the set of all functions satisfying a Lipschitz condition ...
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A function with one partial derivative Hölder continuos is Hölder continuos?

I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $(t,x)\mapsto \partial_x^2 u(t,x)$ is $C^{0,\alpha}$; 2. $(t,x)\mapsto \partial_x ...
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Hölder continuity for parabolic equations

What is a good and modern reference for hölder regularity for non-degenerate parabolic equations? To be a bit more precise, I have a degenerate parabolic equation, exhibiting two degeneracies, and can ...
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Show the following extension is lipschitz

$X=S\cup\{x_0\}$, $f:S\rightarrow \mathbb R$ s.t. $|f(s)-f(t)|\leq kd(s,t)$ for $s,t\in X, k>0$. Suppose $x,y \in X$ s.t. $x\in S$ and $y\notin X$then $x=t, t\in S$ and $y=x_0$. I'm trying to ...