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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If $f$ is Holder of order $\alpha$, then $f'$ is holder of order $\alpha-1$
Let $D\subset\mathbb{R}$. We say that $f:D\to \mathbb{R}$ is Holder of order $\alpha>1$ if for the largest integer $l<\alpha$ (i.e. $\alpha-l\in (0,1)$:
$f$ is $l$ times differentiable, and ...
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Regularity of functions approximated by polynomials
I want to ask how to prove the following result:
Suppose $u\in C(\overline{B_{1}})$, if there exists a universal constant M>0, such that for any $x_{0}\in \overline{B_{1}} $, we have a quadratic ...
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Reverse Holder functions
For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that
$$|f(x) - f(y)| \geq c|x-y|^\alpha. $$
The ...
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Holder continuity in an interval implies boundedness
Let $\alpha\in (0, 1)$ and $-\infty<a<b<+\infty$. Assume that $u\in C^{0, \alpha}((a, b), \mathbb R)$. Here
$$|u|_{C^{0, \alpha}((a, b))} = \|u\|_{L^\infty((a, b))} +\sup_{x, y\in (a, b), x\...
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What is the relationship between Hölder spaces and differentiability?
Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources informally describe these spaces as functions having at least "$k + \alpha$" derivatives. How ...
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A property of $C^{0, \alpha}$ functions
I am going through $C^{0, \alpha}(\Omega)$ regularity, $\alpha\in (0, 1)$, where $\Omega$ is a regular domain (not bounded, in general).
I can not find much material to study from, but I found this ...
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definition of BUC($\alpha,\frac{\alpha}{2})$ [closed]
i found in the book of P;Souplet and P;Quiitner the definition and norm of space $BUC^{(\alpha,\frac{\alpha}{2})}$
.here is the definition:
Let $Q=Q_T=\Omega \times(0, T)$ where $\Omega$ is an ...
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For a Holder-continuous function, is the quotient $\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y\rvert^{\alpha}}$ jointly continuous for $x \neq y$?
This seems a bit trivial but there is no explicit proof for the statement nor can I prove it directly myself.
Let $f: X \to \mathbb{R}$ be $\alpha$-Holder continuous for some $\alpha \in (0,1)$, where ...
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A Holder-continuous function differentiable a.e. is absolutely continuous?
Let $f : [0,1] \to \mathbb{R}$ be a Holder-continuous function of an exponent $\alpha \in (0,1)$ and differentiable a.e. at the same time.
Assume further that the derivative $f'$ is integrable on $[0,...
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Embedding of differentiable functions into Holder Functions
I have the following problem.
Consider $\Omega \subset \mathbb{R}^{n}$, fix $\alpha \in (0,1) $. Consider the space $C^1(\overline \Omega)$ of differentiable functions equipped with the norm $${\| u \|...
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Does differential operator $D^{\alpha} : W^{m+|\alpha|, p} \rightarrow W^{m, p}$ have a closed image in $W^{m, p}$
I actually have have $\Omega \subseteq \mathbb{R}^{d_1+d_2}$ and set multiindex set
$$\mathcal{A}:= \{\alpha = (\alpha_1, \alpha_2)\in \mathbb{N}^{d_1+d_2} : 2|\alpha_1| + |\alpha_2|\leq2 \}.$$
And ...
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Showing that spaces $C_b^{(1,2)}$ and $W^{(1,2),p}$ are Banach spaces.
So in essence there are a few propositions in the beginning.
For arbitrary $T$ the space $B(T;\mathbb{C})$ is a space of all (real or) complex functions $f: T\rightarrow \mathbb{C}$ with sup norm, ...
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Does $C^{1,\alpha}$ imply Holder's continuity with exponent $1+\alpha$ in some cases? [duplicate]
My analysis professor said that if $V \colon \mathbb R^n \to \mathbb R$ is $\mathcal C^{1,\alpha}$ for $\alpha >0$, i.e. it is differentiable with Holder's continuous gradient of order $\alpha$ ...
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Extending Holder functions
If $(X, d)$ is a metric space and $A \subseteq X$, McShane's Extenstion Theorem states that every $L$-Lipschitz function $f \colon A \to \mathbb{R}$ can be extended to $L$-Lipschitz function $f' \...
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What is the space $BUC^s$?
I am reading Elliptic operators with infinite-dimensional state spaces, Herber Amann. I am confuse with the norm of the space $BUC^s$ with $s\in \mathbb R_{+}$
For me i guess that $||u||_{BUC^s}=\sup_{...
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The property of Hölder continuous density function
I am reading the article but I don't understand some lines in page 10.
First, assumptions on the density $h$ are needed. Of course, since $h$ is the density of a symmetric probability distribution on ...
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Hölder domains are locally graphs
I'm currently reading chapter 6 of Gilbarg + Trundinger book on second order elliptic pde. They define $C^{k,\alpha}$ domain in the following manner:
A domain $\Omega\subset\mathbb{R}^{n}$ and its ...
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Example of little $\alpha$ Hölder function that is not $\beta$ Hölder, for any $\beta>\alpha$?
What functions are in the set $$ c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$
A single example will do, but the more the merrier.
This question was natural to me after writing this ...
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Littlewood-Paley Characterisation of Hölder Regularity
I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
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Little Hölder spaces
In this question, we have a definition of the above spaces:
What is little Holder space?
How can we prove that the closure of the “smooth” functions, i.e. the little Hölder space, is not the whole ...
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Relation between Minimum Holder Coefficient and Holder Semi-norm
I have the following two definitions pertaining to Holder continuity:
For both, let $\alpha \in (0,1]$ and consider the path $X:[0,T] \rightarrow \mathbb{R}^d$
$X$ is $\alpha$-Holder continuous if ...
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Hölder continuity of Brownian Sheet/Wiener Field
It is known from P. Lévy, Théorie de l’addition des variables aléatoires. Monographies des Probabilités ; calcul des probabilités et ses applications 1, Paris (1937), that the sample paths of the ...
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Hölder regularity in the quantitative manner
Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
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Meaning of $D^kf$ and of $D^{\beta}f$ (with $k$ nonnegative integer and $\beta$ multi-index)
What is the meaning of $D^kf$ in this context? I'm a bit confused because I know that $Df(a)$ normally means the total derivative at $a$. Moreover, at some point the text treats $Df(x)$ and $\nabla f(...
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$C^2$ regularity for $-\Delta u$ Hölder?
If $u \in H^1_0(\Omega)$ is such that $- \Delta u$ is $C^{0,\alpha}$, then do we have $u \in C^{2,\alpha}$ ?
The regularity theory I know tells me that
$$
u \in \bigcap_{1 < p < \infty} W^{2,p}(\...
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Distributional Derivative is a Holder function
For $\alpha \in \mathbb{R}_+ \setminus \mathbb{N}$, I say that $f \in \mathcal{D}'(\mathbb{R}^d)$ is $\alpha$-Holder (and write $f \in C^\alpha(\mathbb{R}^d)$) if for every $x \in \mathbb{R}^d$ there ...
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If the derivative of a function is $\alpha$-Hoelder continous, is the original function $\alpha$-Hoelder continuous?
Let $I \subseteq \mathbb R$ be some interval and let $\alpha \in (0,1]$.
A function $f : I \rightarrow \mathbb R$ satisfies the Hoelder condition if there exists a constant $C > 0$ such that
$$\...
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Bounded increasing differentiable and right-continuous function is Hölder continuous for some exponent $\gamma$?
Consider for arbitrary $\gamma=\alpha+\beta, \alpha=0,1, \ldots, 0<\beta \leq 1$, the Hölder class $\Lambda_\gamma$ of functions $g(y), y \geq 0$ having finite norm
$$
|| g ||_{\gamma}=\sup _{y \...
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Proof verification : inequality for solution of $\Delta u = f $ et $u|_{\delta \Omega} = \psi$ with special norm on $\mathcal{C}^{k,\alpha}(\Omega)$
Hello everyone (sorry for the messy title!) ! I am currently working with Gilbarg & Trudinger, Elliptic Partial Differential Equations of Second Order (so I use their notation and definition). I ...
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Difference Between Holder Continuous and Locally Holder Continuous functions in Holder Spaces
I am reading "Holder and Locally Holder Continuous Functions and Open Sets of Class $C^k,C^{k,\lambda}$" by Renato Fiorenza.
I have seen that a function is Locally Holder Continuous on a set ...
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Is the limit of the scaling for Lipschitz functions unique?
Let $ B(0,1) $ be a ball with center $ 0 $ and radius $ 1 $. If $ u $ is a $ C^{0,1}(\overline{B(0,1)}) $ function such that $ u(0)=0 $ and $ [u]_{C^{0,1}(B(0,1))}\leq M $ with $ M>0 $. Let $ u^r(x)...
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A compact embedding for sobolev space H^{1} (to holder space ?)
In an article I am reading, the author proves that a function sequence $g_{n}$ is bounded in $H^{1}([0,1];\mathbb{R}^{k})$, then he uses Rellich-Kondrachov Theorem to obtain the relative compactness ...
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Sum of Hölder functions with disjoint support
Fix some $d\in\mathbb{N}$, $m\in \mathbb{N}_0$ and $0<a\leq 1$. Let $f_1,..., f_n$ be functions on $[0,1]^d$ with Hölder smoothness $m+a$. That is, they are $m$-times continuously differentiable ...
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Does uniform convergence imply Hölder convergence?
Let $\alpha\in(0,1)$ and define the $\alpha$-Hölder norm of a function $f : [0,1]\to\mathbb{R}$ by
$$
\lVert f\rVert_\alpha:=|f(0)| + \sup_{0\le s<t\le 1}\frac{|f(t)-f(s)|}{|t-s|^\alpha}.
$$
Let $...
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Is $f(x) = \frac{1}{\sqrt{x}}$ Hölder continuous on its domain?
The question is the one in the title. I was wondering if $f(x) = \frac{1}{\sqrt{x}}$ is Hölder continuous because I know that for $x^{\beta}$ with $\beta \in (0,1]$ we have Hölder continuity. But when ...
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Is $e^{\gamma t}$ Hölder continuous, with $\gamma<0$.
Is $e^{\gamma t}$ Hölder continuous?, with $\gamma<0$.
This question appear in something that i am working, i am not sure if the answer is yes or not.
My only attempt is for definition $|e^{\gamma ...
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About a Lipschitz property $\|f^{\mu}(x)-f^{\mu}(y)\| \leq L^{\mu}\|x-y\|^{\mu}$
For a function $f(x)=[f_1(x),f_2(x),\cdots,f_m(x)]^T:\mathbb{R}^n\to\mathbb{R}^m$ where $f_i(x):\mathbb{R}^n\to \mathbb{R}$ is a real-value function, $f(x)$ is Lipschitz, i.e., there exists a positive ...
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Scaling properties of Hölder seminorms
Let $T>1$ and $W$ be a Banach space. Assume $X:= [0,T]\to W$ is a path. For $[s,t]\subset[0,T]$, let $\| X\|_{\alpha,[s,t]}:=$$\sup_{s\le u<v\leq t} \frac{|X_v-X_u|}{|v-u|^\alpha}$ be the well-...
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Embedding of $W^{1,\infty}$ in $C^{0,1}$
Theorem 5 of pag 283 of the text "Partial Differential Equations - Second Edition (Lawrence C. Evans)" states
Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ ...
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Compact Embedding Theorem. Metric Space. Santambrogio. (reference request).
In the book `Optimal Transport for Applied Mathematicians' by Santambrogio, on page 289 it is written that
In this setup, the space $X=(X,d)$ is a metric space, $\{\tilde{x}^\tau\}_{\tau>0}$ is a ...
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Show that $x\mapsto |x|^s_2x$ on $\mathbb{R}^n$ is globally $C^{1,s}$ for $0<s<1$
Recently I had to make use of the fact that the map $x\mapsto |x|^s_2$ is globally $s$-Hölder continuous $\left(|\cdot|_2\text{ is the Euclidean distance}\right)$, which can be reduced to the $1$-...
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A inclusion of sobolev space $W^{2,p}$ in a Holder space
Let $\Omega \subset \mathbb{R}^{N}$ be a bounded smooth domain and $L$ a uniformly elliptic operator given by
$$
Lu = -div(A(x) \nabla u) + \langle b(x), \nabla u\rangle + c(x) u,
$$
where $b = (b_{1},...
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$f \in H^{s}(\mathbb{R}^{n})$, $s > n/2$, then $f$ is uniformly Holder continuous of $\alpha \in (0,1)$ order??
How do I show that if $f \in H^{s}(\mathbb{R}^{n})$, $s > n/2$, then $f$ is uniformly Holder continuous of $\alpha \in (0,1)$ order??
I know I need to show that
$$|f(x) - f(y)| \leq C\|f\|_{H^{s}}|...
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linear operator Hölder continuous
Let $p\in(1,\infty)$ and $\beta:=1-\frac{1}{p}$ and define $Tf:[0,1]\rightarrow \mathbb R$, $x\mapsto \int_{[0,x]}fd\lambda$. Show:
$A_p:=\{\alpha \in (0,1): Tf\text{ Hölder continuous for the ...
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Green operator of Dirichlet problem is compact
Let $\alpha \in (0,1]$, $\Omega \subset \mathbb R^n$ open bounded and the problem
$$ \left\lbrace \begin{array}{r c l c l}
- \Delta u&=&0&\operatorname{on}& \Omega \\
u&= & h&...
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L1 and L2 norm not equivalent on sequence of functions where H^1 norm is uniformly bounded
I am looking for a sequence of $f_n\in L^2(0,T)$ where
$\|f_n\|_{H^1(0,T)}\leq M \quad \forall n \in \mathbb{N}$
$(\forall C>0)(\exists n\in \mathbb{N})(\|f_n\|_{L^2(0,T)} > C\|f_n\|_{L^1(0,T)})...
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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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