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

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Difference between local Holder exponent and point-wise Holder exponent

What is the precise difference between local Holder exponent (resp. continuity) and point-wise Holder exponent (res. continuity)? I use the following definition for point-wise Holder continuity: ...
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24 views

How to integrate some function-inequality?

I'm concerned with the subject of integrating function inequalities, namely given a function $r\in C^{1,\alpha}([0,s_{max}];\mathbb{R})$ and a constant $A$ satisfying the ineqality $\begin{align} ...
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$0<\beta < \alpha \leq 1$, unit ball of Hölder space $C^{0,\alpha}[0,1]$ compact in $C^{0,\beta}[0,1]$?

So this is a very basic question on Hölder spaces. Let $0 < \beta < \alpha \leq 1$. Prove that the unit ball of $C^{0,\alpha}[0,1]$ is compact in $C^{0,\beta}[0,1]$. For reference: $\| ...
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Kolmogorov's continuity criterion Ornstein-Uhlenbeck process

Let $(X_t, t \in \mathbb{R})$ be an Ornstein-Uhlenbeck process, i.e. $X_t$ is defined by $$X_t = \sigma \int_{-\infty}^t e^{-\theta(t-s)} dW_s$$ for $t \in \mathbb{R}$ for parameters $\theta, \sigma ...
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the equivalence of two definition of Holder space?

I have seen two different definition of $C^{k,\gamma}(\Omega)$: semi-norm $$[u]_{C^{0,\gamma}}:=\sup_{x,y\in \Omega,x\neq y}\frac{|u(x)-u(y)|}{|x-y|^\gamma}$$ one defines ...
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1answer
31 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
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31 views

Why does a Poisson process hurt the prerequesites of the Kolmogorov-Chentsov theorem

I have a question for you. Obviously, by looking at the sample paths of a Poisson process with parameter $\lambda >0$, this process does not have a hölder-continuous version. But why? I have the ...
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32 views

Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
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1answer
51 views

Holder continuity of power function

I need to compute the coefficient for the Holder continuity of $x^p$ with $x > 0$, that is $$ H(p) := \sup_{x\neq y}\frac{|x^p - y^p|}{|x - y|^p}. $$ I am actually going to apply this in ...
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46 views

Pathwise measurability of Ito integral under supremum norm

I'm doing my first research project on Stochastic Analysis and in order to prove something which is crucial, I need to prove the following claim: LEMMA: Denote by ...
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109 views

What's the term for “the supremum of constants $\alpha$ such that a function is $\alpha$-Hölder continuous”?

The $\alpha$-Hölder norm of a function $f(x)\colon I \to X$ where $I=[0,T]$ and $X$ is some Banach space with norm $\|\cdot\|$ is: $$\|f(t)\|_{\alpha}\colon=\sup_{s \neq t \in ...
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44 views

Notation for subspace of Hölder Space

As mentioned, this is largely a question on notation. I'm reading Fractional Integrals and Derivatives: Theory and Applications by Samko, Kilbas, and Marichev. I'm just starting and I'm curious about ...
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1answer
62 views

Using Hölder condition to find upper bound on Fourier coefficients

First I want to stress that I don't want an answer, perhaps a hint. Let $f(x)$ have period $2\pi$ and let $|f(x) -f(y)| \leq c|x-y|^{\alpha}$, for some constants $c$ and $\alpha$ for all $x$ and $y$. ...
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Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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32 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb ...
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67 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result ...
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75 views

Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
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Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
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44 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
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Calculate a Limit and find the Sup

Supporse that $$ f(x)=\frac{1}{|x|^\alpha+1},\quad\alpha\in(0,1],x\in I:=[-1,1], $$ and $$ g(x,y)=\frac{|f(x)-f(y)|}{|x-y|^\beta},\quad x,y\in I. $$ I can prove if $$ \sup_{x,y\in ...
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174 views

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let ...
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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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45 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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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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59 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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60 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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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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436 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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719 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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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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202 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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71 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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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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48 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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88 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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955 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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49 views

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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1answer
140 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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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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120 views

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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38 views

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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180 views

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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88 views

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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1answer
102 views

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.