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

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If $f^\alpha$ is Lipschitz, what can be said about $f$?

If $f$ is a positive function and $f^{\alpha}$ with $0<\alpha\leq1$ is Lipschitz then what we can say about $f$? Ps: I need to know what is the exponent holder continuity of $f$.
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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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1answer
23 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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1answer
22 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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30 views

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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16 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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33 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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34 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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60 views

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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39 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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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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24 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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2answers
67 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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49 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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35 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.
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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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88 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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1answer
39 views

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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25 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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23 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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98 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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32 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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68 views

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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35 views

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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21 views

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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1answer
23 views

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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52 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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59 views

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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152 views

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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46 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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48 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 ...
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1answer
92 views

Holder continuous derivative on bounded $D$ implies Lipschitz?

Suppose $D$ is a bounded, open, connected subset of $\mathbb{R}^n$. Suppose that $u \in C^{1,\alpha}$ for $\alpha \in (0,1]$, i.e. $u$ has a Holder continuous derivative with exponent $\alpha$. Is it ...
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53 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
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35 views

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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50 views

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 ...
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Interpolation inequality on Holder space

Let $0< \beta < \gamma <1$. Show that the interpolation inequality holds. $$||U||_{C^{0,\gamma}(U)} \le ||U||^{\frac{1-\gamma}{1-\beta}}_{C^{0,\beta}(U)} ...
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147 views

Sobolev, Holder, Lp spaces continuous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
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116 views

Question about Lipschitz function

Suppose $A = (a_{ij})$, $1 \leq i \leq m$, $1 \leq j \leq n$ is an $m \times n$ matrix. then $A: \mathbb{R}^n \to \mathbb{R}^m$ given by $$ A(x) = A(x_1,\dots,x_n) = \left( \sum_{j=1}^n ...
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$\sqrt{x}$ isn't Lipschitz function

A function f such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function show that $f(x)=\sqrt{x}\hspace{3mm} ...
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192 views

Prove that Cantor function is Hölder continuous

Let $C_k$ be the set obtained in the $k-$th stage of building the Cantor set, where $$C_1=[0,\frac{1}{3}]\cup[\frac{1}{3},\frac{2}{3}]$$ ...
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Lipschitz Continuity versus continuity [closed]

Show that if a function $ f : \mathbb{R}^n \rightarrow \mathbb{R} $ is $\gamma $-Lipschitz (with any constant $\gamma > 0$) then it is continuous. Show that there exists $\gamma$-Lipschitz ...
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Lipschitz-continuous $f(x)=x^2\cdot \sin\left(\frac{1}{x}\right)$

How to prove that $f$ is globally Lipschitz-continuous $$ f:\mathbb{R}\longrightarrow \mathbb{R}$$ $$ f(x) = \left\{ \begin{array}{c l} x^2\cdot \sin\left(\frac{1}{x}\right) & ,\quad ...
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42 views

Showing $u \in H^s$ and $\varphi \in C^1$ implies $u\varphi \in H^s$ (product rule)

Let $\Omega$ be bounded and open set in $\mathbb{R}^n$. As a start, I pose this question: For $u \in H^s(\Omega)=W^{s,2}(\Omega)$, define the Holder seminorm type quantity $$F(u) = ...
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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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1answer
49 views

Holder continuity and Hilber space

Let $\Omega\subset \Re^n$ be an open set and let $u \in H^1_{loc}(\Omega)$ be a weak solution of $\Delta u=f $ in $\Omega$, with $f \in C^{0,\alpha}(\Omega)$. Prove that $u \in C^{2,\alpha}(K)$ for ...
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2answers
161 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
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85 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
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277 views

Question on Sobolev Space

In learning the Sobolev space, I have a question why the Sobolev space $W^{k,p}$ could be embedded in the Holder space $C^{k,\alpha}$. Can we find a function in Holder space but not in the Sobolev ...