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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23 views

Hölder regularity for $f^2$

Let $f$ be Hölder-continuous, which means in that case $$\sup_{|x-y|\leq \tau}|f(x)-f(y)|\leq C\tau^\alpha,$$ for a $\alpha\in(1/2,1]$ and $\tau < 1$ as well as $x,y\in[0,1]$. I'm trying to show, ...
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20 views

Estimate product of Hölder functions

Let $U \subseteq \mathbb{R}^n$ be the unit ball, let $k \in \mathbb{Z}_{\ge 0}$, and let $0 < \alpha < 1$. Let $f, g \in C^{k, \alpha}(U)$ be Hölder functions, i.e. $f$ and $g$ are of class $C^k$...
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19 views

Hölder continuity of quocient of Hölder functions

Let $u,v:[a,b]\longrightarrow \mathbb{R}$ two $\alpha$-Hölder continuous functions for some $\alpha \in (0,1]$ . If $v$ is bounded away from zero, it is very easy to prove that the funtion $w(x) \...
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1answer
30 views

A function that is $1/p^{\prime}$-Hölder continuous on $[a,b]$

Let $g \in L^p([a,b])$ for some $p \in (1, \infty]$. Define $$f(x) = f(a) + \int_a^xg(s)\ ds$$ for any $x \in [a,b]$. The claim is that $f$ is $\frac{1}{p^{\prime}}$-Hölder continuous on $[a,b]$, ...
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Holder function and weak derivatives

Let $B_1$ the $n$-dimensional unit ball and $u\in C^{0,\alpha}(B_1)$ be an $\alpha$-Holder continuous function, for some $\alpha \in (0,1)$, such that $u(0)=0$. Is it true that $$ \int_{B_{1/2}} |x \...
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1answer
88 views

Lemma 8.23 (Gilbarg-Trudinger)

My doubt is in a step of the following lemma Lemma Let w be a non-decreasing function on an interval (0, Ro] satisfying, for all $R \le R_0$, the inequality \begin{equation} w(\tau R) \le \gamma w(R) ...
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Why do we obtain $C^{1+\alpha,\frac{1+\alpha}{2}}(U\times(0,T))$ regularity?

Let $U$ be a $C^3$ compact manifold in $\mathbb R^3$ and assume $f \in L^2(0,T;H^1(U)) \cap H^1(0,T;H^1(U)^*) \cap W^{2,1}_p(U\times (0,T))$ for all $p \in[1,\infty)$, where $H^1(U)^*$ denotes the ...
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1answer
31 views

Controlling $C^1$ norm by Hölder norm and $C^0$ norm

I've been trying to do this problem for a little while and have not made much progress. I want to show that for any $\epsilon>0$, there exists some $C_{\epsilon}>0$ such that $$ \Vert u'\Vert_{L^...
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A type of Hölder function

I was doing the following exercise: if $ f: \mathbb{R}^n \setminus \{ 0 \} $ defined by $ f (x) = | x |^\alpha x_i $ for a fixed i, then $ f \in C^{0, \beta} (B_1 (0)) $, for $ 0<\alpha\leq 1$. I ...
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Holder estimate on ball from estimate in full space

I’m working through a proof of the Schauder estimates for Elliptic PDE. Basically, right now I’ve gotten to the point where I have an estimate of the form: $$ |D^2 v|_{\alpha} \leq C |\sum_{ij} a_{ij} ...
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37 views

Weak partial derivative versus strong derivative

Let $f\in W^{1,p}(U)\,,U \subset\mathbb{R}^n$ open. If $f$ satisfies: $$| D^\beta f(x) - D^\beta f(y)|\leq C |x-y|^\gamma $$ a. e. for some $0\leq \gamma \leq 1,\, x,y \in U$ and $|\beta|\leq 1$. Then ...
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Function $f$ such that $|f(x)-f(y)|\leq \sqrt {|x-y|}, \forall x,y\in\Bbb R.$

Let $f$ be a real function such that such that $|f(x)-f(y)|\leq \sqrt {|x-y|}, \forall x,y\in\Bbb R.$ Does this condition imply that $f$ will be differentiable ? If Lipschitz order is greater than $1$...
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Uniform convergence of $C^{0, \alpha}$ functions implies convergence in $C^{0, \alpha}$ topology

I am looking for a reference to a proof for the following result: let $m \in \mathbb{N}, \alpha \in (0,1)$ and $\Omega \subset \mathbb{R}^m$ be a bounded open set. Let $f_n: \Omega \to \mathbb{R}$ be ...
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If $\Theta$ is continuously embedded into $C^1(E,E)$ and $t\mapsto f(t,\;\cdot\;)$ is in $C^0([0,\tau],\Theta)$, then ${\rm D}_2f$ is continuous

Let $\tau>0$ and $E$ be a $\mathbb R$-Banach space. If $f:[0,\tau]\times E\to E$, write $\left.f\right|_1$ for the function $$[0,\tau]\to E^E\;,\;\;\;t\mapsto f(t,\;\cdot\;)\tag1$$ and $\left.f\...
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$u\in C^{\beta}[0,a]$ then $\int_{0}^{x} u(t) dt \in C^{\beta+1}[0,a]$

If $u\in C^{\beta}[0,a]$ for $0<\beta<1$ then prove that $\int_{0}^{x} u(t) dt \in C^{\beta+1}[0,a]$. where $C^{\beta}[0,a]$ is space of Holder Continuous functions in [0,a]. $\textbf{I TRIED}$ $...
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Notation of Holder space

The question is from book Abel Integral Equation, Sergio and Rudolf. From section 4.2.1 they denoted Holder space as $C^{\alpha}[0,a]$. I know holder space as $C^{k,\alpha}[0,a]$. Are both same? or is ...
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Whitney extension theorem for Hölder spaces

The usual Whitney extension theorem says that Whitney data with remainders like $R_\alpha=o(x-y)^{k-|\alpha|}$ extends to a $C^k$ function. If we also have $R_\alpha=o(x-y)^{k+\lambda-|\alpha|}$ for ...
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fractional integral of a distribution - reference request

Suppose I have $f:[0,T]\to\mathbb{R}$ which is $\beta$-Holder continuous, $\beta\in (0,1)$, $\alpha\in (0,1)$ and I want to consider the fractional integral $$ g(t):=\Gamma(\alpha) \int_0^t (t-s)^{\...
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Generalization of “Lipschitz implies bounded derivatives” for Holder functions

Is there a generalization of the following statement for Holder continuous functions: For $f:\mathbb{R}^d \rightarrow \mathbb{R}$ Lipschitz continuous, $\|\nabla f\|$ is uniformly bounded. What I am ...
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Inclusion map $i : C^{0,\beta}[0, 1] \rightarrow C^{0,\alpha}[0, 1] $ is linear, and therefore compact

Q) Given $0 < \alpha < \beta \leq 1$. Show that the inclusion map $i : C^{0,\beta}[0, 1] \rightarrow C^{0,\alpha}[0, 1] $ is compact. Ans) Let {$u_n$}$_1^\infty$ ⊂ $C^{0,β}[0,1]$ such that $\|...
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Is $(C^{\alpha +\epsilon}(0,T, \mathbb{R}^d), d_{\alpha})$ a complete metric space?

Let $0< \alpha < \frac{1}{2}$ and $C^{\alpha +\epsilon}(0,T, \mathbb{R}^d)$ be the space of Hölder-continuous functions $f:[0,T] \to \mathbb{R}^d$ of exponent $\alpha+ \epsilon$ for $0<\...
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35 views

Holder Extension Operator

As we all know, every holder continuous function is Bounded and uniform continuity in domain $\omega$. Now we give a more large domain $X$, then $\omega \subset \subset X$. Can we find a extension ...
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Differential operator given by the Fréchet derivative of functions over a Banach space

Let $E$ be a $\mathbb R$-Banach space, $\Omega\subseteq E$ be open and$^1$ $$\left\|f\right\|_{C^1(\Omega)}:=\left\|f\right\|_\infty+\left\|{\rm D}f\right\|_\infty\;\;\;\text{for Fréchet ...
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If $f$ is Fréchet differentiable, bound the Lipschitz seminorm by the supremum of the norms of $f$ and ${\rm D}f$

Let $E$ be a $\mathbb R$-Banach space, $$d(x,y):=\min(1,\left\|x-y\right\|_E)\;\;\;\text{for }x,y\in E,$$ $\Omega\subseteq E$ be open, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:\Omega\...
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Show for a $\gamma-$holder map $\mathcal{H}^{s/\gamma}(f(A))\le MH^s(A)$

Suppose $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is a $\gamma-$Holder continuous map $(0<\gamma\le1)$, that is:} $$|f(x)-f(y)|\le C |x-y|^\gamma, \ \ \ \ \text{for some }C>0, \text{ and all }x,...
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Proof verification for $\alpha,\beta$ Holder spaces

Let $X\subseteq \mathbb{R}^n$ be a compact set. Also, we have $f:X\rightarrow \mathbb{R}$ such that $f$ is $\alpha$ Holder. which means that there exists $0<\alpha \leq 1\;$ and $K>0\;$ such ...
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1answer
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Showing the map from the middle $\lambda$ cantor set to the $\nu$ cantor set is $\gamma$-hölder continuous

Let $C_\lambda$ and $C_\nu$ be the middle $\lambda$ and $\nu$ cantor sets, respectively. I want to show the map $\Pi_{\lambda,\nu}:C_\lambda\rightarrow{C_\nu}$ is $\gamma$-hölder continuous, with $\...
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Trapezoidal quadrature convergence for Holder continuous functions

Suppose $f \in C^{k, \alpha}(a,b)$, with $f^{(j)}(a) = f^{(j)}(b)$ for $j = 0,1,\ldots, k$. We have at least that the error of the trapezoidal rule is $$ |I[f] - T_{h}[f]| \le h^{k}\int_{a}^{b} \tilde{...
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Non-homogeneous transport equation with Hölder continuous coefficients

I'm working with an non-homogeneous transport equation with Hölder Continuous coefficient. I'm seeking for some reference in this subject. Precisely, I'm concerned with the following problem: \begin{...
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21 views

$C^k$ functions for any $k\in\mathbb R$

My question is simple: how is the space of $C^k$ functions defined for any $k\in\mathbb R$
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Convergence of an infinite series $\sum_{n\geq 2}v_nw_n n^3$

Original Question: $\displaystyle v=\sum_{n\neq 0}v_ne^{in\theta}, w=\sum_{n\neq 0}w_ne^{in\theta}$ are $C^{3/2+\epsilon}$ functions on $S^1$ ($v_n$ and $w_n$ are Fourier coefficients of $v$ and $w$), ...
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2answers
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Continuous Function and lipschitz continuity

Let $C(a,b)^\alpha$ the set of Hölder function continuous. If $\alpha<\beta$ ,under this condition I must prove that $C(a,b)^\alpha \subset C(a,b)^\beta$ or $C(a,b)^\beta \subset C(a,b)^\alpha$. ...
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6 views

Combining Holder continuous functions on Whitney cylinders

Let $u$ be a bounded function and let a closed set $E$ be given. The compliment of $E$ can be covered with a Whitney type covering $B_i$ such that the following are satisfied: 1) $E^c \subset \...
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Holder regularity of Daubechies wavelets

In this pdf, table 2, the H\"older regularity of Daubechies wavelets is listed. The citation is given as Two-Scale difference equations I and Two-Scale difference equations II. However, in the ...
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Holder semi norm estimate for a solution of Poisson's equation in the half space

Let $u\in C^{2,\alpha}_0(\overline{\mathbb{R}^n_+})$, $0<\alpha<1$, such that $$-\Delta u=f \quad\text{in }\quad \overline{\mathbb{R}^n_+}$$ $$ u=0 \quad\text{in }\quad \partial\overline{\...
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When is $x^a\sin (x^{-b}) $ $\alpha$-Hölder Continuous on $[0,1]$?

The question is very direct. What should $\alpha$ be for $f(x)=\left\{ \begin{array} Xx^a\sin (x^{-b})&x \in (0,1]\\ 0&x= 0 \end{array} \right.$ to be $\alpha$-Hölder continuous? It ...
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1answer
35 views

If a path is nowhere Holder continuous for any rational $\gamma> 1/2$ then it is nowhere Holder continuous for any $\gamma > 1/2$

Let $(B_t)_{t \ge 0}$ be a $d$-dimensional Brownian motion. Then almost every $\omega$ in the underlying probability space $\Omega$ has this property: For each $\gamma > 1/2,$ the path $t \mapsto ...
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1answer
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Convergence of Lipschitz functions in $L^\infty$ implies the convergence in $H^{1}$

Suppose $f_n \to f$ in $L^\infty(U)$ where $f_n, f \in C^{1,1}(\bar{U})$, (Holder space with $k=1, \alpha =1$, i.e, Lipschitz functions) where $U$ is an open and bounded in $\mathbb{R}^n$. Q: Can we ...
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Show that the set of $\alpha$-H$\ddot{\text{o}}$lder continuous functions is a Banach space

A function $f$ is $\alpha$-H$\ddot{\text{o}}$lder continuous if $|f(x) - f(y)| \leq C |x - y|^\alpha $ for some $C \geq 0$ and $\forall$ $x, y \in \mathbb{R}$ such that $|x - y| < 1$. Define the ...
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Holder equivalent metrics

Suppose we have a compact metrizable space $X$ and two metrics $d,h$ such that there is $a\in (0,1)$ and $c>0$ with $$h(x,y)\leq d(x,y)\leq c h(x,y)^a.$$ Can we deduce that $d,h$ are Holder ...
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30 views

Difference quotient for Hölder continuous functions

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in C^\alpha_{\mathrm{loc}}(\Omega)$. For $h>0$, $1\leq k\leq n$, let $$D_k^hu(x)=\frac{u(x+he_k)-u(x)}{h}$$ where $e_k$ is the $k$-th ...
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Boundedness of Riesz Transform on (subsets of) Hölder spaces?

Definition and setup The Riesz transform for say $C^\infty_c(\mathbb R^d)$ functions $f$ is defined by a principal value integral, $$ Rf(x) := c_d \operatorname{pv}\!\!\!\int_{\mathbb R^d} \frac{y}{|y|...
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If $f \in C_{u}(U \times [0,T])$ then is it true that $f(\cdot,t) \in C_{u}([0,T])$?

Let $f:=f(x,t)$ belong to the parabolic Hölder space $C^{1,\beta}(U\times [0,T])$ where $U$ is a compact subset of $\mathbb R^2$ and $0<\beta <1$. This yields that $f$ is a Hölder continuous ...
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1answer
24 views

Are bounded sequences in $C^{1,\alpha}(B_1)$ precompact in $C^{1,\beta}(B_1)$ for $\alpha>\beta$?

Are bounded sequences in $C^{1,\alpha}(B_1)$ precompact in $C^{1,\beta}(B_1)$ for $0<\beta<\alpha<1$? I.e., if $(u_n)_n\subset C^{1,\alpha}(B_1)$ with $||u_n||_{C^{1,\alpha}(B_1)}\leq C$ ...
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1answer
91 views

Bounded and Continuous but not Holder [duplicate]

This is a follow-up question to this post. Does there exist an example of a bounded and continuous but not necessarily uniformly continuous function defined on all of $\mathbb{R}$, which is not H\"{o}...
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27 views

Limit of function in $H^1(\mathbb{R}_+;\mathbb{C})$

Let $H^1(\mathbb{R}_+;\mathbb{C})$ the space of function in $L^2(\mathbb{R}_+;\mathbb{C})$ with weak derivative in the same space. Show that \begin{equation} lim_{x\rightarrow \infty} f(x) = 0 \end{...
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1answer
148 views

Sharpness of Kolmogorov-Chentsov

The Kolmogorov-Chentsov continuity theorem is a general way to estimate the Hölder continuity of a process $X$ (up to taking a different version $\tilde{X}$ of $X$ ). However, I would like to know if ...
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1answer
50 views

Show that if $|\hat{f}(n)|\le\frac{C}{|n|^{1+\alpha}}(n\ne0),\alpha\in(0,1)$ and $f\in C(T)$

Show that if $|\hat{f}(n)|\le\frac{C}{|n|^{1+\alpha}}(n\ne0),\alpha\in(0,1)$ and $f\in C(T)$ then $f$ is holder continuous of order $\alpha$ on $T$.
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1answer
48 views

Compact ball in a closed subspace of X

Consider the $B$-space $X = C([0, 1])$ equipped with the norm $\Vert · \Vert_{\infty}$. And let $F$ be a closed subspace of $X$ such that every function $u\in F$ is Holder continuous, namely, for each ...
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78 views

Reference on Sobolev-Slobodeckij spaces

I'm currently studying for a course about Mean Field Theory and other scaling limits topics in probability. An instrument that has been introduced is the Sobolev-Slobodeckij space $W^{\alpha,p}(0,T;\...

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