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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Hölder continuity on Sobolev space $H^1(a,b)$

Let $H^{1}(a, b)=W^{1,2}(a,b)$ be the Sobolev space, that is, the functions $f\in L^2$ such that $f' \in L^2$, where $f'$ denotes the weak derivative of $f$. This is equivalent to $$f(x)=c+\int_a^x f'...
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Some questions in the compactness of Sobolev space and Holder space

Let $B_1$ be the unit ball in $\mathbb{R}^2$. Define $A=\{u \in W^{1,2}(B_1) : \|u\|_{ W^{1,2} } \leq 1\}$, $B= \{u \in L^{2}(B_1) : \|u\|_{ L^2 } \leq 1\}$, $C=\{u \in C^{0,\alpha}(B_1) : \|u\|_{ ...
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Reference for Sobolev-Hölder embedding for unbounded domains

I would like to know whether the following is true and references: If $\Omega\subset\mathbb R^n$ is open (not necessarily bounded) and $k = n/p + r+\alpha$, and $\alpha \in (0,1), r\in \mathbb N, k\...
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Sequence of Lipschitz continuous functions approaching Holder continuous function

Let $\{f_n\}$ be a sequence of uniformly bounded Lipschitz functions with Lipschitz constant $L_n$, i.e. for each $n\in\mathbb{N}$ $$ |f_n(x) - f_n(y)| \leq L_n |x-y| $$ It is easy to see if the ...
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monotone functions agreeing with Holder functions on a large set

Let $\alpha \in (0,1)$, $f:[0,1]\rightarrow \mathbb{R}$ be a continuous monotone function and $\varepsilon>0$. Does there exist a function $\phi_{\varepsilon} \in \mathcal{C}^{\alpha}$ such that $\...
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Intersection of all Holder spaces

Given $0 < \alpha < 1$, consider the Holder space $C^{0,\alpha}(\overline{U})$. Is there any nice subspace of all of these Holder spaces? In other words, whats is $$ \bigcap_{0 < \alpha < ...
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A Lipschitz function of order $1/2$

The following exercise is from an old prelim exam that I was never able to solve when I was a graduate student, and I still think about it every now and then. Suppose $f:\mathbb{R}\rightarrow\mathbb{R}...
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Inclusion of Holder Spaces.

From the wikipedia page on Holder spaces it says that if $0 < \alpha < \beta \leq 1$, then there is an inclusion map $\iota : C^{0 , \beta, }(\Omega) \rightarrow C^{0 , \alpha}(\Omega)$, where $\...
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Regularity of the solution of a 1D parabolic PDE

Let $\alpha \in (0, 1)$. Consider $f = f(t, x)$ a bounded measurable function on $[0, 1] \times \mathbb{R}$, such that $f$ is $\alpha$-Hölder with respect to $x$, uniformly in $t$. Let $G(r) := e^{-r^...
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Complex Hölder Space

Usually the Hölder Space is defined for functions of $\mathbb{R}^n$ to $\mathbb{R}$ which is easily extended to any Banach space $B$. But I came across the definition with the domain being $\mathbb{C}$...
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Proving or disprove $\sigma$-Hölder of $x\mapsto \|x\|^{\sigma-1}x$, $\sigma\in(0,1)$.

Is the following function $\Bbb R^d\ni x\mapsto \|x\|^{\sigma-1}x$, $\sigma$-Hölder continuous for $\sigma\in (0,1)$? That, is I would like to know if there exists a constant $C>0$ such that $$\big\...
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Show u^2|u|^(p-3) is Holder continuous (derivative of power nonlinearity)

I am trying to show that the function $f:\mathbb{C}\rightarrow \mathbb{C}$ given by $f(u)=u^2|u|^{p-3}$ is (uniformly) Holder continuous for $p\in(1,2)$ (so $p-3\in(-2,-1)$, this ensures that $f$ is ...
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Embedding Sobolev space $W^{k,\infty}(U)$ into Hölder space $C^{k-1,\gamma}(U)$

Let $U\subseteq\mathbb{R}^n$ be a Lipschitz domain. I know that we can embed $W^{1,\infty}(U)$ into $C^{0,1}(U)$ with the help of Morrey's inequality. I also know that it is (up to measure zero) ...
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Confusion about a problem from Stein and Shakarchi's Complex analysis

Hölder Condition Implying Uniform Convergence I'm also working on Problem 5(b) from Chapter 3 in Stein's Complex Analysis(Page 110). In the discription of the problem, it assert that when some ...
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A Lipschitz bound on the regularization of Holder continuous compactly supported functions

Let $\alpha \in (0,1]$, $\Omega \subset \mathbb{R}^n$ be an open bounded set and $u \in C_c^{0,\alpha}(\overline{\Omega})$. Let $\{\eta_\varepsilon\}$ be a family of standard (radially symmetric) ...
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Holder regularity for solutions to the Poisson equation

I am dealing with a nonlinear PDE of the form $$-\Delta u=f(u),\quad in\,\,\,\mathbb{R}^n$$ (where $f(u)$ is a nonliear function) I would like to ask you which regularity results do exist in the case ...
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Holder continuity of Ornstein-Uhlenbeck process

Let $X_t$ be a continuous-time Gaussian process on time interval $[0,1]$ with $\mathbb{E}[X_t]=0$ and $$ \operatorname{cov}(X_t,X_s)=\frac{1}{2} \exp(-|t-s|) $$ Hence, $X_t$ is an O-U process. Would ...
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Analysis on levels of a function and characterization of Hölder regularity.

I would like to understand the comment made in the remark 3.1 of the book "Degenerate Parabolic Equations"- Emmanuele DiBenedetto. I understand the requeriment to choose a small $\delta$ ...
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Showing that, if a pointwise-convergent sequence converges in norm in $\mathcal{C}^{2+\alpha}$, it does in $\mathcal{C}^{2+\beta}$ ($\beta < \alpha$)

$ \newcommand{\CC}{\mathcal{C}} \newcommand{\a}{\alpha} \newcommand{\b}{\beta} \newcommand{\n}[1]{\left\| #1 \right\|} \newcommand{\nc}[2]{\left\| #1 \right\|_{\CC^{#2}(\oo)}} \newcommand{\seq}[1]{\...
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Definitions of terms tied to continuous functions and Holder spaces and their norms in PDE theory.

$ \newcommand{\c}[1]{\mathcal{C}^{#1}(\Omega)} $So I've been struggling with this for a while, and I keep getting what feels like a lot of conflicting answers regarding these when I search these up, ...
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Definition of the function space $C^{\alpha,\beta}(\Omega\times [0,T])$

I saw the space $C^{\alpha,\beta}(\Omega\times [0,T])$, with $\alpha, \beta>0$. I do not know what this means. For the context, I infer that this should be a Hölder Space in the time and space but, ...
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What is the definition of Hölder space $C^{\alpha}(\mathbb{R})$.

Are we saying that $\sup _{x, y \in \mathbb{R} \atop x \neq y}\left\{\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}\right\}$ $<\infty$ $(0<\alpha<1)$? So a smooth function on $\mathbb{R}$ is not ...
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What's the definition of $C^\alpha$ norm of a tensor?

Recently I came across $C^{k,\alpha}$ convergence of metrics as well. I am confused how to define this norm and can't find a book on it. Is the harmonic coordinate a necessity? Can someone put a good ...
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Is there an example of Hölder continuous functions such that their Young integral is not unique or does not exist?

Does anybody have a reference to or an example of functions $f \in C^\alpha([0,T],\mathbb{R})$, $g \in C^\beta([0,T],\mathbb{R})$ with $\alpha,\beta \in (0,1)$, $\alpha+\beta \leq 1$, such that the ...
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The sum of Hölder continuous functions is Hölder continuous

I want to show that the sum of two Hölder continuous functions is Hölder continuous. Definition: $f:[0, 1] \to \mathbb R$ is Hölder continuous if there exist $\alpha,M>0$, s.t. $\forall x,y\in[0,1]$...
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Composition of $C^{k,\alpha}$ Holder Functions

Let $k,\ell \geq 0$ be integers and $\alpha,\beta \in [0,1]$. Let $\Omega\subseteq \mathbb R^n$ be a closed set and set $f \in C^{k,\alpha}(\mathbb R;\mathbb R)$ and $g \in C^{\ell,\beta}(\Omega;\...
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Invert quantificator in Kolmogorov continuity theorem

The Kolmogorov continuity theorem can be stated as If $(X_t)_{t \geq 0}$ is a stochastic process valued in a complete metric space for which there exists constants $q,\epsilon,C > 0$ such that $$ \...
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Why Do We Care About Hölder Continuity?

I have often encountered Hölder continuity in books on analysis, but the books I've read tend to pass over Hölder functions quickly, without developing applications. While the definition seems natural ...
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Hölder continuous functions from $L^1(\mathbb{R})$ [closed]

Let $f\in L^1(\mathbb{R})$ be a Hölder continuous function of order $\alpha$. Can we always find constants $\varepsilon>0$ and $C>0$ such that $$|f(x)-f(y)|\leq C\frac{|x-y|^\alpha}{(1+|x|+|y|)^{...
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Definition of semi-norms on $C^{k,r}(\mathbb{R}^n,\mathbb{R}^m)$

When the set $C^{k,r}(\mathbb{R}^n,\mathbb{R})$ is equipped with the usual semi-norm topology (https://en.wikipedia.org/wiki/Hölder_condition) it is known as the Hölder space. However, how do we ...
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Riemann Lebesgue equivalent of Fourier Transform for Hölder Continuity

I am currently working with a paper using the following norm: \begin{equation} \|F\|_{B^{s/2}}=\int_{-\infty}^{\infty}|\hat{F}(\tau)|(1+|\tau|)^{s/2}\,\mathrm{d}\tau\\ \end{equation} for all $C_c^\...
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Which Hölder-Spaces on the d-dimensional Torus Ensure Absolute Convergence of Fourier Series?

Suppose $f : \mathbb{T}^d \to \mathbb{R}$. I would like to know under what circumstances the Fourier series of $f$ is absolutely convergent, i.e., $\sum_{k \in \mathbb{Z}^d} |\hat f(k)| < \infty.$ ...
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$\overline{C^{\infty}}^{||\cdot||_{\alpha}} \neq C^{\alpha}$; lower bound of $ \frac{|(s-\delta/2)^{\alpha} - s^{\alpha}|}{(\delta/2)^{\alpha}}$

I would like to prove myself that: $$\overline{C^{\infty}}^{||\cdot||_{\alpha}} \neq C^{\alpha}$$ where the left hand side is the closure with respect to the semi-norm of Holder continuous functions (...
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1 vote
1 answer
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A question on Hölder condition.

Suppose $f:U\subset\mathbb{R}^n\to \mathbb{R}^{m}$ satisfies the Hölder condition, i.e. there are constant $\alpha>0$ and $L>0$ such that $$ \|f(x) - f(y)\|\leq L\cdot \|x - y\|^{\alpha},\quad \...
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Understanding an argument involving Holder spaces

I'm trying to better understand the following argument: The function $u$ is in $L^q(M)$ for some $q > 2^\star=\frac{2n}{n-2}$. By standard elliptic theory, the $u$ is in $C^{2,\theta}(M)$, $0 < \...
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How big is the set of Hölder continuous functions?

Let $C^\delta$ be all $\delta$-Hölder continuous functions on $[0,1]$. Is $\cup_{\delta\in (0,1)} C^\delta$ a dense subset of $C^0$, the set of continuous functions on $[0,1]$ in the uniform metric? ...
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Proof that Hölder Space is Banach

Fix some open bounded convex $U\subseteq\mathbb{C}$ and some $0<\alpha<1$. Let $F: U \to B$, where $B$ is some Banach space, define, $$\lVert F \rVert_{\infty}=\sup_{z\in U} \lVert F(z)\rVert \...
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Show that the composition of these functions is Hölder continuous

Let $I$ be a finite open interval, $d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$ be bounded and open and $$|f|_{\alpha,\:\beta}:=\sup_{\substack{(s,\:x),\:(t,\:y)\:\in\:\overline I\times\overline\...
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How can we show $C_c^\infty(\Omega)\subseteq C^{0,\:\alpha}(\overline\Omega)$ for all $\alpha\in(0,1]$?

Let $E_i$ be a normed $\mathbb R$-vector space and $\Omega\subseteq E_1$ be open. We can easily show that if $f:\Omega\to E_2$ be Fréchet differentiable $\Omega$ is convex and ${\rm D}f$ is bounded$^...
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Extension of function in Hölder class outside of compact set

Let $K_1 \subsetneq K_2\subseteq \mathbb{R}^n$ be two connected and compact subsets. Suppose we have $\beta > 0$ and a function $f: K_1 \rightarrow \mathbb{R}$ satisfying (using the Multi-Index-...
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Frechet differentiability with a Hölder continuous map

A bit of context: I try to verify that the following map is Frechet differentiable everywhere: Let $\Omega\subset\mathbb{R}^n$ be convex, bounded and open. Furthermore let $0<\alpha<\beta\leq 1$ ...
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2 votes
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Show that a set of $\alpha$-Holder functions is compact

Let $(K,d_K)$ be a compact metric space and consider $Z=\mathbb{R}^d$, with the usual metric. Let $L,\alpha>0$ and $(t,z_0) \in K\times \mathbb{R}^d$. Show that the set $$\mathcal{H}_{L,\alpha,t_0,...
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Is a Hölder continuous function differentiable almost everywhere?

As per the title. I know that a Lipschitz continuous function is differentiable almost everywhere (see the Rademacher Theorem). I was wondering if something similar was true for Hölder continuous ...
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3 votes
1 answer
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Fixed points for operators in Hölder spaces: Why is merely being a self-map of a ball not enough?

Let $D \subseteq \mathbb{R}^n$ be a bounded domain with smooth boundary and $C^{\alpha}(\overline{D})$, $\alpha \in (0,1)$, the space of Hölder continuous functions with the usual norm $\|\cdot\|_{C^{\...
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estimate of holder norm of a product

In a book 'Elliptic Partial Differential Equations of Second Order' by Gilbarg and Trudinger I stumbled upon the following inequality for two functions $f, g$ and their Holder norms in bounded domains ...
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4 votes
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Given a coin that produces the probability $\lambda$, what is a constructive way to produce $f(\lambda)$, where $f$ is continuous non-Hölder?

Given a coin of "bias" $\lambda$, sample the probability $f(\lambda)$. This is the Bernoulli factory problem, and it can be solved only for certain functions $f$. (For example, flipping the ...
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Is the primitive of a density function Hölder continuous?

Let $I=[-1,1]$ and let $f\in L^1(I)$. Is it true that the function $F:I\to\mathbb{R}^+$ defined as $$F(s)=\int_{-1}^s |f(x)|dx$$ belongs to $C^{0,\alpha}(I)$ for a certain $\alpha>0$?
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About the norm in $C^{\ell, \alpha}(\overline{\Omega})$.

In my functional analysis course, we defined for $\ell \in \mathbb N$ and $\alpha \in (0, 1]$, \begin{align*} C^{\ell, \alpha}(\Omega) &= \left\{u \in C^\ell(\Omega)~\bigg|~ \max_{|\beta| = \ell}\...
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4 votes
2 answers
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Is $x\log x$ Hölder continuous on $[0,1]$?

Consider the function $f:[0,1]\rightarrow \mathbb{R}$ given by $$f(x)= x\log(x).$$ This function is not Lipschitz continuous at zero, but apparently Hölder continuous for any $\alpha<1$, i.e. there ...
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Does the Hölder coefficient tend to zero?

A continuous fucntion $f: \Omega\rightarrow \mathbb{C}$, with $\Omega$ subset of $\mathbb{R}^n$ or $\mathbb{C}^n$, is a $\alpha$-Hölder continuous with $0< \alpha\leq 1$, if \begin{equation} [f]_{\...
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