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

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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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1answer
25 views

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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0answers
57 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 ...
2
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1answer
26 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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1answer
21 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
20 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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1answer
51 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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43 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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0answers
45 views

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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0answers
31 views

Holder continuous spaces

To which holder space $C^{k,\alpha}([-1,1])$ does the function $|x|^{5/2}$ belong? can someone help because I don't really know how to show this any clue will be appreciated
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1answer
51 views

Sobolev, Holder, Lp spaces continous 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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13 views

What is a Holder differentiable function?

What is the definition of Holder differentiability? Is there an example function that is Holder differentiable with the exponent alpha?
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1answer
103 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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5answers
206 views

$\sqrt{x}$ 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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1answer
84 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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2answers
105 views

Lipschitz Continuity versus continuity

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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1answer
75 views

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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1answer
36 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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0answers
29 views

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
30 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
87 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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2answers
73 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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1answer
142 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 ...
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1answer
185 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
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1answer
137 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
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1answer
56 views

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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1answer
243 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
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1answer
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functions in Holder space

Let $U \subset \mathbb R^n$ be open, bounded and connected, and let $ \alpha, \beta \in (0,1]$ with $\alpha < \beta$. If $f \in C^{0,\alpha} (U)$ and $g \in C^{0,\beta}(U)$ then what is ...
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1answer
63 views

If $|f|$ is Hölder continuous, what about $f$?

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $|f|$ is Hölder continuous with exponent $0<\alpha\leq1$. Does it follow that $f$ is also Hölder continuous with ...
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1answer
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Holder Space series term

Holder space $C^{k,\gamma}(\bar{U})$ has the norm $||u||_{C^{k,\gamma}(\bar{U})} := \sum_{|\gamma | \leq k}||D^{\alpha}u||_{C(\bar{U})} + \sum_{|\gamma|=k}[D^{\alpha}u]_{C^{0,\gamma}(\bar{U})}$, where ...
2
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1answer
128 views

How to show pre-compactness in Holder space?

Let $K \in \mathbb{R}^d$ be a compact set and consider the space of Hölder continuous functions $C^{0,\gamma}(K)$ with norm $||f||_{C^{0,\gamma}}:=||f||_{\infty}+\sup_{x,y \in K,x \neq ...
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1answer
30 views

Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
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1answer
77 views

Show: $C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)$.

Show that $$ C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)~~~~~~~\forall0<\lambda\leq 1. $$ Hello, I have some problems to show these ...
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1answer
60 views

Show that $f(x):=\sqrt{\lvert x\rvert}$ belongs to $C^{0,\frac{1}{2}}(\mathbb{R})$

Show that $f(x):=\sqrt{\lvert x\rvert}$ belongs to $C^{0,\frac{1}{2}}(\mathbb{R})$. Hello, when I got it right, I have to show four things: (1) $f\in C(\mathbb{R})$ (2) $f\in ...
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0answers
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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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1answer
133 views

Proving that a holder continuous function always has a smaller exponent.

According to wikipedia if we have $f:X \rightarrow Y$ which is $\alpha$-Holderian then for all $\beta < \alpha$ the function is also $\beta$-Holderian. How do we prove this starting from the fact ...
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45 views

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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1answer
286 views

Showing a function is Hölder Continuous

I'm trying to show that the function $f(x) = x \ln(x)$ is Hölder continuous on $(0,1) $ for $0 < \alpha < 1$. I must be missing something, because I don't really understand how the definition ...
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1answer
153 views

Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into $C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can ...
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1answer
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How to show Zygmund space is Hölder space?

The motivation of this question is to show that Zygmund space is Hölder space, in certain cases. For simplicity, take $s\in (0,1)$, I want to show $$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
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1answer
108 views

Is there a reference for compact imbedding of Hölder space?

Suppose $0<\alpha <\beta$. Then, the Hölder space $C^\beta$ is compactly imbedded to $C^\alpha$. See the Wikipedia article Hölder condition. However, I could not find precise reference from ...
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1answer
249 views

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in ...
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1answer
72 views

lipschitz function and its properties

$f\colon\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\le |x-y|^{\beta}$. Which of the following are correct statements? $\beta=1$ $f$ is differentiable; $\beta>0$ $f$ is uniformly continuous; ...
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1answer
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Approximating Lipschitz funtion by $C^1$ function.

Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
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1answer
213 views

Hölder Continuity between metric spaces

Hölder continuity has never appeared in my formal education and the wikipedia article seems insufficiently general. I want to make sure that this definition of Hölder continuity is correct and ...
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1answer
107 views

Reverse Hölder Continuity and Hausdorff dimension

Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
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Proving something is $1$-Lipschitz

(1) Let $(X,d)$ be a metric space, and let A be a non-empty subset. Show that the function $$D_A :X \to [0,\infty ]$$ defined by $$D_A (x) =\inf \{d(x,y) : y \in A\}$$ is $1$-Lipschitz (when ...
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Show polynomial is a Lipschitz function

If $A\subseteq \mathbb{R}$ is a bounded set and $p$ is a polynomial, then show that $p:A\to \mathbb{R}$ is a Lipschitz function.
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Polynomials not dense in holder spaces

How to prove that the polynomials are not dense in Holder space with exponent, say, $\frac{1}{2}$?
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232 views

Is a function with bounded Holder continuous derivatives itself “more” than just Lipschitz continuous.?

Was wondering about this as I brushed my teeth this morning. I have a differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that has bounded and $\gamma$-Holder continuous derivatives. Can I ...