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

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Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
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29 views

Is $C^{1,\alpha}\subseteq C^{0,1}$?

It is true that if $f\in C^{1,\alpha}(I)$ than $f\in C^{0,1}(I)$? I mean: if $f$ is bounded and differenciable with bounded and holder continuous derivative, then $f$ is bounded and lipschitz ...
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29 views

Product of Holder functions is Holder

I am having trouble proving that given $u_1 \in C^{k,\alpha}$ and $u_2 \in C^{k,\alpha}$ the product also lies in $C^{k,\alpha}$. I have tried first doing the $k=0$ case but I can't fully understand ...
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Lip $\alpha$ trigonometric series

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} ...
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36 views

Holder norms inequality

I saw this in a proof I am reading, and have been unable to justify the statement. Assume $\Omega$ to be a open bounded set in $R^n$ with smooth boundary, and $u\in C^{2,\alpha}(\bar\Omega)$. Then,for ...
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34 views

Global Holder regularity from DiGiorgi-nash-moser

For a strictly elliptic differential operator $L=\sum_{i,j} D_i (a_{ij}D_ju) $ with strictly elliptic condition on open bounded smooth domain $\Omega$,and bounded coefficients, DiGiorgi-Nash-Moser ...
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Hölder continuity of $\frac1x$

I have a question. Is the function $f(x)=1/x$ Hölder continuous if $x\in (\varepsilon,+\infty),\ \varepsilon>0$?
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C^1 is not dense in Holder space

Let $\overline{\Omega}$ be a bounded, closed and convex set of $\mathbb{R}^n$. Prove that $C^1(\overline{\Omega})$, the space of continuously differentiable functions on $\overline{\Omega}$, is not ...
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rational function of non vanishing Hölder continuous functions

Conside Hölder continuous functions $f_1,…,f_m:\mathbb{R}^n\rightarrow \mathbb{R}$ (with Hölder coefficient $\alpha$). The claum is now that any rational function of $f_1,…,f_m$ with non vanishing ...
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38 views

Is the Hölder space with exponent $\beta$ dense in the space with exponent $\alpha$ for $\alpha<\beta$?

For $0<\alpha\leq 1$ let $\Lambda_{\alpha}([0,1])$ be the space of functions on $[0,1]$ such that $||{f_{\Lambda_{\alpha}}}||<\infty$, where ...
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Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
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75 views

Prove an interpolation inequality

Assume $0 < \beta < \gamma \le 1$. Prove the interpolation inequality $$\|u\|_{C^{0,\gamma}(U)} \le \|u\|_{C^{0,\beta}(U)}^{\frac{1-\gamma}{1-\beta}} ...
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127 views

Proving that a Hölder space is a Banach space

I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, ...
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38 views

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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70 views

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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59 views

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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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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48 views

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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1answer
46 views

Show that the set of uniformly Lipschitz functions vanishing at $0$ is compact in $C[0,1]$

The question is: For $K$ and $\alpha$ fixed, show that $\{f\in \operatorname{Lip}_k \alpha : f(0) = 0\}$ is a compact subset of $C[0,1]$. I was going to attempt this by using by Arzela-Ascoli theorem ...
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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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62 views

Continuity of a composition map between Holder spaces

Let $\varphi\in C^{\infty}_0(\mathbb{R})$, $0<\alpha<1$, $\Omega\subset\mathbb{R}^d$ be a bounded domain. Is it true that the map $\Phi:C^{0,\alpha}(\bar{\Omega})\to ...
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262 views

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for ...
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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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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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57 views

An oscillation property that implies Hölder continuity of function

Let for a function $u: \mathbb R \rightarrow \mathbb R$ and $x_0 \in \mathbb R$, $r>0$: $$ w(u,x_0,r)=\sup_{B(x_0,r)} u-\inf_{B(x_0,r)}u, $$ where $B(x_0,r)=(x_0-r,x_0+r)$. In Wikipedia (see here) ...
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59 views

Question on Non-Lipschitz Functions

I have a question about non-Lipschitz functions. Let $f_1,f_2,...f_n$ be some collection of Holder continuous non-Lipschitz scalar functions defined on a compact subset of $\mathbb{R}^n$. My questions ...
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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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Decay of Fourier Coefficients implies Holder Continuity?

This is an exercise problem. I got stuck here and would like to get a hint. The problem is Suppose $f$ is continuous and $2\pi$-periodic, and $|\hat{f}(n)|\leq |n|^{-3/2}$ for all non-zero ...
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A function with only a partial derivative not Hölder-continuous

I'm looking for a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $x\mapsto u(t,x)$ is $C^{2,\alpha}$; 2. $t\mapsto u(t,x)$ is $C^{1,\alpha}$; 3. $t\mapsto ...
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Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
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For which exponents $\gamma$, the function $|x|^{1/2}$ is $\gamma$-Holder continuous?

I have to prove the following. Let $$u(x):=|x|^{1/2}$$ if $$|x|\le 1$$ For which exponents $\gamma\in (0,1]$, $u\in C^{0,\gamma}([-1,1]).$ The answer should be $\gamma\in (0,\frac{1}{2}]$, but I ...
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Sobolev embedding counterexample

I trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$ for $p>n$ and $\alpha > 1 -\frac{n}{p}$. No clue yet, thanks for your help.
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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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104 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
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About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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39 views

Hölder continuity and uniform boundedness

Is uniform boundedness is related to Hölder continuity of a function? I mean is it necessary to prove first uniform boundeness to prove the Hölder continuity of a function? Also tell me the ...
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27 views

Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
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193 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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56 views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Let $1≤p≤\infty$ and $ f \in L^{p}(a,b)$ such as there is a function $g \in L^{p}(a,b)$ that for all $\phi \in C^{1}(a,b)$ (and continuos in $[a,b]$) with $\phi(a)=\phi(b)=0$ we have: $\int_{a}^{b} ...
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Proving that $f(x) = \vert x \vert^{\alpha}$ is Holder continuous, inequality help

The definition of $\alpha$-Holder continuity for a function $f(x)$ at the point $x_0$ is tha there exist constant $L$ such that for all $x \in D$ \begin{equation} \vert f(x) - f(x_0) \vert \leq L ...
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Inclusion of Holder Spaces

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be defined by, for example, $f(x) = \frac{1}{2}x^2$. Then $f$ belongs to the Holder space $C^{1,1}(\mathbb{R})$. Since $C^{1,1}(\mathbb{R})\subset ...
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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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Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
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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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78 views

Hölder-Zygmund Spaces on compact sets and for integer smoothness parameters

I know from Triebl, Theory of Function Spaces II, that for $\alpha \notin \mathbb{N}$ Hölder-Zygmund Spaces on $\mathbb{R}$ are equal to the classical Hölder Spaces. However, I have two questions ...
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Hölder estimate of solution to linear parabolic PDE

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 ...
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A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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Gilbarg Trudinger: Hölder continuity in chapter 8

I'm trying to track the behaviour of the coefficients in Theorems 8.22 and Theorem 8.24. Particularly, I'm considering the behaviour w.r.t. to the distance from $\Omega'$ to $\partial \Omega$ I'll ...
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65 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
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84 views

Hölder Condition for Fourier Series

So I'm trying to prove that the function (as represented by a Fourier series) $ f(x) = \sum_{k=0}^\infty 2^{-k\alpha}e^{i2^kx}$ satisfies the Hölder Condition: $|f(x+h)-f(x)| \le C|h|^\alpha$, with $0 ...