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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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For which $\gamma \in (0,1]$ does u belong to $C^{1,\gamma}(\overline K_{\beta}) $?

Let \begin{cases}\text{-$\Delta$u=f} \ \ in \ \ K_{\beta} \ \\\text{u=0 } \ \ in \ \ \partial K_{\beta} \\ \end{cases} $K_{\beta} \subset \mathbb{R^2}$ and $\beta \in(0,2\pi)$ define $$K_{\...
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12 views

Holder continuity of function of two variables.

Define $f:[0,1]^2 \rightarrow \mathbb{R}$ by $$f(x,y) = (x^{7/10} + y^{3/5})^{1/4}.$$ I want to find all $\alpha \in (0,1]$ such that $f$ is $\alpha$-Holder. That is, there exists a constant $c >...
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1answer
112 views

Interpolation inequality for Holder continuous functions.

Let $\Omega$ be a bounded open connected set in $\mathbb{R}^n$ with $C^1$ boundary and let $0<\alpha<1$. Then there exists a real number $\sigma_0>0$ and a dimensional constant $C>0$ such ...
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2answers
17 views

Does positive part preserve Holder continuity?

Let $u^+$ denote the positive part of the function $u$ on a bounded domain $\Omega.$ If $u \in C^{0,\alpha}(\bar \Omega)$, is also $u^+ \in C^{0,\alpha}(\bar \Omega)$?
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40 views

Showing $f(x)$ $=$ $\sqrt{x}\sin(\frac{1}{x})$ satisfies a Holder Condition of $\alpha < 1$

I'm learning about functions that satisfy Holder's Condition of order $\alpha$. Specifically, A function $f$ is said to satisfy a Hölder condition of order $\alpha > 0$ if there exist $M$ such ...
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1answer
22 views

Closure of Continuously DifferentiableFunctions in Holder Space

Here is a curious (already submitted) homework problem I had in analysis some time ago: Let $\Omega$ be a convex domain in $\mathbb{R}^n$ with $C^1$ boundary. Let $C^{0,\alpha}(\overline{\Omega})$ ...
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18 views

Construct a Holder continuous compactly supported function

I want to construct a function $f$ satisfying the following properties: 1) Holder continuous with exponent $\alpha \in (0,1)$ so that there exists $C > 0$ such that $$ |f(x+t) - f(x)| \le C|t|^{\...
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19 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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0answers
24 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
23 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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61 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
41 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
35 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
66 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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14 views

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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68 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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31 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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33 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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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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0answers
24 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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0answers
16 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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0answers
33 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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0answers
38 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
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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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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
125 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
86 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
63 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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0answers
32 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
46 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
89 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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0answers
16 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
77 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
51 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
99 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
23 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
94 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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0answers
54 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
85 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
40 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;...
3
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0answers
55 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\...
1
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1answer
50 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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0answers
46 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
21 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
23 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 ...
3
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1answer
134 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
83 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 ...