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Questions tagged [holder-spaces]

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

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18 views

Doubt about an estimate in Hölder spaces.

I am studying a paper about fluid dynamics and there is an estimate that involves norms in Hölder spaces that I do not know why it is true. Let be $R^2=n^2+z^2$, $\theta=\arctan\left(\dfrac{z}{n}\...
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20 views

Holder condition using second difference

I want to find a simple proof for, for $0<\alpha<1$, if $f\in C_c^0(\mathbb R^n)$ satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|x|^\alpha$, then $f\in C^\alpha(\mathbb R^n)$. I saw in Zygmund's ...
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1answer
15 views

Estimate for non-differentiable implicit function theorem

Let $F(x,s)$ be a continuous function $F:\mathbb R^m\times\mathbb R^n\to\mathbb R^m$ such that $\nabla_xF$ is a Holder $C^\alpha$-function, say $\frac12<\alpha<1$. Suppose $F(0,0)=0$ and ...
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1answer
27 views

Is $K_t :\mathbb{T}\to\mathbb{C}, s\mapsto\sum_{n∈\mathbb{Z}}e^{-n^2 t}e^{ins}$ a summability kernel for $t\to 0^+$?

In trying to solve with Fourier series the heat equation on the 1-torus, I stumbled into this family of functions: $$\forall t>0, K_t :\mathbb{T}\to\mathbb{C}, s\mapsto\sum_{n∈\mathbb{Z}}e^{-n^2 t}...
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42 views

Convergence of Fourier series in $\alpha$-Hölder norm.

For $\alpha\in(0,1]$, define the $\alpha$-Hölder space on the 1-torus $\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$ as the space: $$C^{0,\alpha}(\mathbb{T}):=\{f\in C(\mathbb{T})\ |\ \sup_{s,t\in\mathbb{T}\\...
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1answer
40 views

Proving an inequality between $(1, \frac{\alpha}{2})$-Hölder norms of two functions

I have to prove that, given $f\in C^{1, \frac{\alpha}{2}}([a, b])$, such that $\|f\|_{\infty}<L$ for some $L$, $f\geq0$ and $\alpha\in(0, 1)$, there exists a constant $K$ such that $$ \|e^f-1\|_{1,...
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1answer
30 views

Are uniformly equivalent metrics with the same bounded sets Holder equivalent?

This is a follow-up to my question here.  Let $d_1$ and $d_2$ be two metrics on the same set $X$.  Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $...
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1answer
39 views

A Taylor theorem for Hölder continuous function?

Let $ C^{m,s}_{b} $ be the space of bounded function $ u: \mathbb{R} \rightarrow \mathbb{R} $ which satisfies \begin{alignat*}{2} \bigg| u^{(m)}(x) - u^{(m)}(y) \bigg| \leq C | x - y |^{s}. \end{...
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Integrating elements of Holder-Besov spaces using LIttlewood-Payley decomposition

Let $ \mathcal{C}^{s} $ be the Holder-Besov space equipped with the norm $$ \| u \|_{\mathcal{C}^{s}} = \sup_{j \geq -1}\Big| 2^{js} \| \mathit{\Delta_{j}u} \|_{L^{\infty}} \Big|.$$ Suppose that ...
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62 views

Dimension of Holder space

$\textbf{Definition}$ Let $ \Omega $ be open in $\mathbb{R}^n$ and $\alpha \in (0,1]$. \begin{align*} \textrm{For } \alpha\in(0,1],& \newline\\ &[u]_{\alpha,\Omega}:=\sup_{x\neq y}\frac{\...
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0answers
28 views

Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put ...
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32 views

Is this function which shows infinite concussion near $0$ Hölder continuous?

Define $f:\left(0,+\infty \right)\rightarrow \mathbb{R}$ as $$f(x)=x\cos \frac{1}{x}$$ Is $f$ Hölder continuous? I found it uniformly continuous because it is continuous in the interval $(0,1] $ ...
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11 views

Inequality of |Bias(x)| using kernel estimation

$f \in \mathcal H(\beta,L)$ with $ \beta > 1,$ $L >0$ $\mathcal H(\beta,L):$ $f: \Bbb R \to \Bbb R$ is $l= \lfloor \beta \rfloor $- differentiable, and for $f^{(l)} $: $|f^{(l)}(x) - f^{(l)...
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23 views

An inequality in Hölder norms

I am studying particle-trajectory method for solution to the Euler equation, chapter four of Majda-Bertozzi book. Let $$X:\mathbb{R}^n\longrightarrow\mathbb{R}^n$$ be a smooth, invertible ...
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15 views

Holder continuity Equivalence

Suppose $\Omega \subseteq \mathbb{R}^{n}$ is open and connected, $u:\Omega \to \mathbb{R}^{n}$ satisfies \begin{align*} \exists K > 0:\forall x \in \Omega, \exists C_{x}\in\mathbb{R}: \|u-C_x\|_{L^...
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29 views

Singular function that is Holder for all $\alpha<1$

I am asking for an example of a singular continuous function that is Holder for all $\alpha<1$. We know that such function cannot be Lipschitz, otherwise it is absolutely continuous. We also know ...
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36 views

For which $ l \in\mathbb{N} $ and $ \alpha\in[0,1] $ does u belong to the hoelder space $ C^{l,\alpha}(\overline {B_1(0)}) $?

$$ u:\overline {B_1(0)}\subset\mathbb{R^2}\to\mathbb{R} $$ defined by : $$ u(x_1,x_2)=x_1x_2(1-\sqrt{x_1^2+x_2^2}) $$ Can I at first consider $$ x_1x_2 $$ and look if $$ x_1x_2\in C^{l,\alpha}(\...
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1answer
55 views

A question about Hölder space and Sobolev

How to find the norms defined on the Hölder space The Hölder Space $C^{k,\gamma}(\bar{U})$ consisting of the all $u \in C^k(\bar{U})$ for which the norm $$\|u\|_{C^{k,\gamma}(\bar{U})}:= \sum_{|\...
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86 views

Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?

For fixed $d\geq 1$ and $\beta\in (1,2]$, consider the two following classes of functions: Let $\mathcal{H}^\beta$ denote the collection of all $C^1$ functions $\phi:\mathbb{R}^d\to\mathbb{R}$ for ...
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4answers
120 views

Let $f\colon\Bbb{R}^2\to \Bbb{R}$ such that $|f(x)-f(y)|\leq \Vert x-y\Vert^2.$ Prove that $f$ is a constant

Edit: Several questions of this type have been asked here before but not on the same domain $\Bbb{R}^2.$ Please, how do I deal with a function of this type or could anyone show me a reference or a ...
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2answers
75 views

How to prove that Holder space is normed linear space

Can you some one please tell how to prove Holder Space is Normed Linear Space The Holder Space $C^{k,\gamma}(\bar{U})$ consisting of the all $u \in C^k(\bar{U})$ for which the norm $$\|u\|_{C^{k,\...
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2answers
60 views

Equivalence between Hölder norms

Let us consider the following two norms: $$ \left\lVert f\right\rVert_\alpha = \left\lVert f\right\rVert_\infty + \displaystyle{\sup_{\substack{x,y \in U \\ x \neq y}} \frac{\left| f(x) - f(y))\right|...
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0answers
33 views

If $K⊆ℝ^d$ is compact and $f:[0,T]×K×K→ℝ$ is continuous with $f(t,\;⋅\;,\;⋅\;)∈C^{0+β}$, is $t↦\left\|f(t,\;⋅\;,\;⋅\;)\right\|_{C^{0+β}}$ continuous?

Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $K\subseteq\mathbb R^d$ be compact $f:\overline I\times K\times K\to\mathbb R$ be (jointly) continuous $\beta\in(0,1]$ Moreover, let $$\left\|g\right\|_{C^{0+\...
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30 views

Holder Continuity of 1/f (Pink) noise

For $\omega=\omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $\alpha$ such that $$|\omega(t)-\omega(s)|\leq C|t-s|^\alpha $$ for ...
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1answer
40 views

Showing $\|uv\|_{C^\alpha} \leq C(v) \|u\|_{C^\alpha}$ for Hölder functions

Suppose $\Omega \subseteq \subseteq \mathbb{R}^n$ and $0 < \alpha < 1$. As my notes suggest, there is an estimate of the form $$\|uv\|_{C^\alpha} \leq C(v) \|u\|_{C^\alpha}$$ for $u,v \in C^\...
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2answers
73 views

Equivalence of norms for the space of Hölder continuous functions

I'm currently reading an introduction to rough paths by G. Zanco, and in it he defines the Banach space of $\alpha$-Hölder continuous functions from $[0,T]$ into $E$, with norm $$\|X\|_{C^\alpha} := |...
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15 views

Functions $u \in C^{0,1/2}(\mathbb{R}^n)$ satisfy $\|u\|_{C^0(\mathbb{R}^n)} < \infty$

In solving problem sheets for my upcoming functional analysis $2$ exam I encountered the following statement: Functions $u \in C^{0,1/2}(\mathbb{R}^n)$ satisfy $\|u\|_{C^0(\mathbb{R}^n)} < \...
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1answer
76 views

Show that a function of two arguments, which is uniformly Hölder continuous on compact subsets, is jointly continuous

Let $M$ be a metric space $\Lambda\subseteq M$ be open $E$ be a $\mathbb R$-Banach space $g:\Lambda\times\Lambda\to E$ $\alpha\in(0,1]$ Assume $$C_K:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:K}{x\:\ne\:...
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1answer
46 views

Class of functions that satisfy $\frac{f(x+th)-f(x)}{t^{\alpha}} \to d \text{ as $t \to 0$}$

Suppose $f:X \to Y$ is a function between Hilbert spaces that satisfies, given $x, h \in X$, $$\frac{f(x+th)-f(x)}{t^{\alpha}} \to d \text{ as $t \to 0$}$$ for some $d \in Y$. Is there a name for such ...
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2answers
95 views

If $f$ is Hölder continuous, are we able to conclude that $(t,x)\mapsto\int_0^tf(x+sh)\:{\rm d}s$ is Hölder continuous?

Let $X$ be a normed $\mathbb R$-vector space $d$ denote the metric induced by $\left\|\;\cdot\;\right\|_X$ $h\in X$ with $\left\|x\right\|_X=1$ $\Lambda\subseteq X$ be open $E$ be a $\mathbb R$-...
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0answers
22 views

How can we show that this function is Hölder continuous?

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open Moreover, let $$\left\|h\right\|_{\tilde C^{0+\gamma}(K)}:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:K}{x\:\ne\:x',\:y\:\ne\:y'}}\frac{\left|h(x,...
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0answers
27 views

If $f:[0,∞)×\mathbb R^d\to\mathbb R$ is continuous in the first variable and $C^{k+γ}$ in the second variable, is $t\mapsto f(t,\;⋅\;)$ continuous?

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $k\in\mathbb N_0$ $\gamma\in(0,1]$ Moreover, let $$\tilde C^{k+\gamma}(\Lambda):=\left\{g:\Lambda\times\Lambda\to\mathbb R\mid\partial^\...
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1answer
93 views

Continuity of a Hölder continuous function depending on a temporal parameter

Let $(M,d)$ be a metric space $E$ be a normed $\mathbb R$-vector space Now, let $$\left\|f\right\|_{\tilde C^{0+\alpha}(A,\:E)}:=\sup_{\stackrel{x,\:y,\:x',\:y'\:\in\:A}{x\:\ne\:x',\:y\:\ne\:y'}}\...
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49 views

Show that $ C^1(\overline{\Omega}) \hookrightarrow C^{0,\alpha}(\overline{\Omega})$ is compact.

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz-domain and $0 < \alpha < 1 $ arbitrary. Show that the embedding $ C^1(\overline{\Omega}) \hookrightarrow C^{0,\alpha}(\overline{\Omega})...
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1answer
74 views

Proof of inclusion of Holder continuous functions

Let $1\le\alpha\le\beta\le1$ and $-\infty<a<b<\infty$ then $C^{0,\beta}([a,b])\subsetneq C^{0,\alpha}([a,b])$ Proof: First define $\|f\|_{C^{0,\alpha}}:=\|f\|_{C^0([a,b])}+[f]_{C^{0,\...
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0answers
38 views

Show that the integral of a two-parameter function is Hölder continuous

Let $(\Omega,\mathcal A,\mu)$ be a measure space $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space $f:\Omega\times\Lambda\to E$ with $$f(\;\cdot\;,x)\in\mathcal L^1(\mu;...
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0answers
51 views

Why are in the definition of the Hölder space $C^{k+\gamma}$ only the partial derivatives of order $k$ assumed to be $\gamma$-Hölder continuous?

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $E$ be a $\mathbb R$-Banach space $k\in\mathbb N_0$ $\gamma\in(0,1]$ The Hölder space $C^{k+\gamma}(\Lambda,E)$ is defined to be $$\left\{f\...
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1answer
42 views

Best regularity for elliptic PDE with Neumann data

Consider $-\Delta u + \lambda u = f$ on $\Omega$ with BC $\partial_\nu u = 0$ on $\partial\Omega$ where $\Omega$ is a bounded smooth domain and $\lambda > 0$. If $f \in L^\infty(\Omega)$, what's ...
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45 views

Under which circumstances is the space of Hölder continuous functions a Banach space?

Let $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space Let $$\left\|f\right\|_{B(\Lambda,\:E)}:=\sup_{x\in\Lambda}\left\|f(x)\right\|_E\;\;\;\text{for }f:\Lambda\to E.$$...
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1answer
20 views

Fréchet space of Hölder continuous differentiable functions

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $k\in\mathbb N_0$ Now, let $$\left\|f\right\|_{C^k(K)}:=\sup_{x\in K}\frac{|f(x)|}{1+|x|}+\sum_{1\le|\alpha|\le k}\sup_{x\in K}|{\rm D}^\...
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0answers
21 views

Is there a good reference for Hölder spaces of Fréchet differentiable functions between Banach spaces?

The notion of Hölder continuity can be defined for any function between metric spaces. However, the topological properties of the space generated by continuously differentiable functions whose ...
2
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1answer
102 views

If a function is log-lipschitz, then it is $\alpha$-Hölder for all 0<$\alpha$<1 and isn't Lipschitz continuous

Consider $C^{0,\alpha}$ the class of $\alpha$-Hölder continuous functions. Let $\Omega \subset \mathbb{R}$ be a bounded subset and let $u: \Omega \rightarrow \mathbb{R}$ a continuous function. Prove ...
2
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1answer
71 views

Hölder continuity of Weierstrass Function

Let us fix some notations, we will call Weierstrass function a function such that: $$f(x)=\sum_{i=0}^{n}a^{n}\cos(b^{n}x) $$ with $0<a<1, b>1 \text{and} \ ab>1$. Wondering whether a ...
5
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1answer
130 views

Showing a function is in Holder space for some $a \in (0,1] $

Hi Im stuck on this exercise : for which $a \in (0,1]$ is $f(x)=x^{2}\sin(\frac{1}{x^{3}})$ in $C^{a}((0,1])$ This is my attempt so far : $|f(x)| \leq x^{2} $ $|f'(x)| = 2x\sin(1/x^{3})-\frac{3}{...
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1answer
64 views

Finding an $\alpha$-Hölder continuous function $f_{\alpha}$ at $0$, but not $\beta$-Hölder.

Assuming $\alpha\in(0,1]$, which function $f_\alpha:\mathbb{R}\rightarrow\mathbb{R}$ is $\alpha$-Hölder continuous at $0$, but not $\beta$-Hölder continuous for $\beta>\alpha$? I've been thinking ...
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2answers
114 views

Is $\ln(x)+1$ contractive?

I am having trouble understanding the definition of a contractive function. The definition is: Let $(\Bbb R,d)$ be a Metric Space with the standard metric. Let $A\subseteq X$ (with the same metric ...
0
votes
1answer
210 views

Hölder continuous functions are uniformly continuous

I want to show that Hölder continuous functions are uniformly continuous using $\epsilon-\delta$. Is it sufficient to find a $\delta>0$ that does not depend on $"x"$ and depends only on $"\epsilon"...
0
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0answers
17 views

Estimating a difference with a Riemann-Stieltjes integral.

I'm having trouble with this problem Let $X\in \mathcal{C}^{\alpha}[0,T]$ and $Y\in\mathcal{C}^{\beta}[0,T]$ with $\alpha+\beta>1$. (Here $\mathcal{C}^{\gamma}[0,T]$ is the space of Hölder ...
3
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0answers
43 views

Showing that an operator between Hölder spaces is a contraction

I'm having trouble with the following problem: Consider $\beta\in(\frac{1}{2},1]$, $\xi\in\mathbb{R}$, $f:\mathbb{R}\longrightarrow \mathbb{R}$ bounded with first and second derivatives bounded, $X\...
2
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2answers
72 views

Showing that if a function between metric spaces has Hölder exponent $\alpha > 1$, this function must be constant

Let us recall the definition of a Hölder continuous function in the most general setting: Let $(M,d)$ and $(M',d')$ be two metric spaces. A function $f : M \to M'$ is said to be Hölder continuous ...