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

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38 views

Why are $C^0(\mathbb{R})$ and $C^{0,0} (\mathbb{R})$ the same spaces?

$C^0(\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})}$. $C^{0,0} (\mathbb{R})$ has the norm $\Vert f \Vert_{C^0(\mathbb{R})} + \sup_{x,y \in \mathbb{R}, x \neq y} |f(x) - f(y)|$. I don't ...
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24 views

A proof of Holder continuity

I'm studying the following proof by I'm not able to understand the main step. Let \begin{equation} \sigma(x)=\sqrt{\beta x(N-x)+\alpha x} \end{equation} defined for $x \in [0, N+\frac{\alpha}{\beta}]$....
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13 views

Hölder continuity on space variable

Is it correct to consider Hölder continuity on space variable for solution of stochastic differential equation? Let $$dx(t)=f(t,x(t))dt+g(t,x(t)dB(t)).$$ If $t\mapsto x(t)$ is Hölder continuous i.e.,...
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18 views

Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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52 views

Reverse Holder continuity

Consider a function $f(x)$ with a point-wise Holder exponent $\beta \leq 1$. Definition of point-wise Holder exponent: $$ \beta_x: = \sup \left\lbrace \beta: \limsup_{h \rightarrow 0^+} \left|\frac{...
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2answers
62 views

Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions, $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that $$ \|u\|_{H^s}^2=\sum_{k\in\mathbb Z}(1+k^2)^{...
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1answer
55 views

Let $X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,\ f(0)=0\right\}$. Show $(X,\lVert\cdot\rVert_X)$ is complete.

The following is a problem on an old Analysis preliminary exam at my institution; I'm prepping for the prelim. The problem is: Let $$X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,...
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22 views

Showing any bounded sequence in Holder space $C^{1/2}$ has a convergent subsequence in Holder space $C^{1/3}.$ [duplicate]

Prove that any bounded sequence in $C^{1/2}([0,1])$ admits a convergent subsequence in $C^{1/3}([0,1]),$ where we say that $f \in C^{\alpha}([0,1])$ if $f$ is Holder continuous of order $\alpha.$ The ...
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37 views

Continuity of Holder functions

If a function taking values in $\mathbb{R}^n$ is $\alpha$-Holder continuous along lines parallel to the axes (uniformly on a compact set), is it continuous?
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70 views

The $Hol$ operator is a continuous function?

Let $\Omega$ be a compact space, and consider $C(\Omega)$ the space of the continuous functions over $\Omega$, consider also, $C^\gamma(\Omega)$ the space of all $\gamma$-holder continuous functions,...
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1answer
28 views

Is this function Hölder continuous?

Define $f:[0,1]\rightarrow \mathbb{R}$ as $$f(x)=x^\alpha \int_x^1 y^{-\alpha-1}(y-x)^{-\alpha}dy, \quad x\in [0,1],$$ where $\alpha\in (0,1/2)$ is some fixed parameter. Is $f$ Hölder continuous of ...
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1answer
18 views

Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all $r&...
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1answer
27 views

initial value problems and non- uniqueness of solution (doesn't satisfy Lipschitz condition)

Let 0<ε<1 for fixed ε and the following initial value problem : { y'(t)=abs(y(t))^(1-ε) & y(0)=0 for 0<=t<=b show that the problem does ...
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23 views

Prove that $\,C^1 \subset H_1 \subset H_\mu\,$ (Hölder Space) [duplicate]

I need to prove that continuously differentiable functions on $\,\left[a,b\right]\,$ are a subset of Hölder space of order $1$ and that the Hölder space of order $1$ is a subset of the Hölder space of ...
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47 views

About the proof of De Giorgi Theorem

here we discuss homogeneous equations with no lower-order terms. $Lu \equiv -D_{j}(a_{ij}(x)D_{j}u)$in $B_{1}(0)\subset \mathbb{R}^n$ where $a_{ij} \in L^{\infty}(B_{1})$ satisfies $\lambda |\xi|^...
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1answer
111 views

Upper bounding a double sum — tightening Holder's inequality under extra assumptions?

I have tuples of numbers $(x_{i,a})_{1\leq a \leq M, 1\leq i \leq n}, (x_{i})_{1\leq i \leq n}\subseteq [0,1]$ satisfying the following, for some fixed parameter $\beta \in (0,1)$: $$ \begin{align} \...
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0answers
29 views

Prove inequality to reject differentiability

Suppose there is a function sequence defined on $[0,1]$ and $f_1(t)=t$. For each $f_n(t)$, there is a set of points $T_n=\{0,2^{-n},2\times2^{-n},\cdots,1\}$, such that between each of these points, $...
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22 views

Hölder continuous function in one of the multiple variables

A function $f: I^2\rightarrow R$ is called Hölder continuous if $|f(\mathbf{x})-f(\mathbf{y})|\le||\mathbf{x}-\mathbf{y}||^\alpha$. However, what is the meaning of "f is $1/2$-Hölder continuous in ...
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1answer
27 views

Lipschitizianity of the square root of a positive $C^2$ function

I was trying to solve this exercise. Let $f\in C^2(\mathbb{R})$ a strictly positive function such that $f''$ is bounded. Then prove that $\sqrt{f}$ is Lipschitz. A first idea was to prove that it's ...
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75 views

Difference between local Holder exponent and point-wise Holder exponent

What is the precise difference between local Holder exponent (resp. continuity) and point-wise Holder exponent (res. continuity)? I use the following definition for point-wise Holder continuity: ...
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1answer
49 views

How to integrate some function-inequality?

I'm concerned with the subject of integrating function inequalities, namely given a function $r\in C^{1,\alpha}([0,s_{max}];\mathbb{R})$ and a constant $A$ satisfying the ineqality $\begin{align} \...
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1answer
85 views

$0<\beta < \alpha \leq 1$, unit ball of Hölder space $C^{0,\alpha}[0,1]$ compact in $C^{0,\beta}[0,1]$?

So this is a very basic question on Hölder spaces. Let $0 < \beta < \alpha \leq 1$. Prove that the unit ball of $C^{0,\alpha}[0,1]$ is compact in $C^{0,\beta}[0,1]$. For reference: $\| ...
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78 views

Kolmogorov's continuity criterion Ornstein-Uhlenbeck process

Let $(X_t, t \in \mathbb{R})$ be an Ornstein-Uhlenbeck process, i.e. $X_t$ is defined by $$X_t = \sigma \int_{-\infty}^t e^{-\theta(t-s)} dW_s$$ for $t \in \mathbb{R}$ for parameters $\theta, \sigma &...
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1answer
42 views

the equivalence of two definition of Holder space?

I have seen two different definition of $C^{k,\gamma}(\Omega)$: semi-norm $$[u]_{C^{0,\gamma}}:=\sup_{x,y\in \Omega,x\neq y}\frac{|u(x)-u(y)|}{|x-y|^\gamma}$$ one defines $$\|u\|_{C^{k,\gamma}(\...
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1answer
71 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
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1answer
38 views

Why does a Poisson process hurt the prerequesites of the Kolmogorov-Chentsov theorem

I have a question for you. Obviously, by looking at the sample paths of a Poisson process with parameter $\lambda >0$, this process does not have a hölder-continuous version. But why? I have the ...
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50 views

Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
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1answer
94 views

Holder continuity of power function

I need to compute the coefficient for the Holder continuity of $x^p$ with $x > 0$, that is $$ H(p) := \sup_{x\neq y}\frac{|x^p - y^p|}{|x - y|^p}. $$ I am actually going to apply this in ...
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62 views

Pathwise measurability of Ito integral under supremum norm

I'm doing my first research project on Stochastic Analysis and in order to prove something which is crucial, I need to prove the following claim: LEMMA: Denote by $(C_{0}[0,\,T],\,||\centerdot||_{\...
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What's the term for “the supremum of constants $\alpha$ such that a function is $\alpha$-Hölder continuous”?

The $\alpha$-Hölder norm of a function $f(x)\colon I \to X$ where $I=[0,T]$ and $X$ is some Banach space with norm $\|\cdot\|$ is: $$\|f(t)\|_{\alpha}\colon=\sup_{s \neq t \in I}\frac{\|f(t)-f(s)\|}{|...
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1answer
51 views

Notation for subspace of Hölder Space

As mentioned, this is largely a question on notation. I'm reading Fractional Integrals and Derivatives: Theory and Applications by Samko, Kilbas, and Marichev. I'm just starting and I'm curious about ...
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1answer
83 views

Using Hölder condition to find upper bound on Fourier coefficients

First I want to stress that I don't want an answer, perhaps a hint. Let $f(x)$ have period $2\pi$ and let $|f(x) -f(y)| \leq c|x-y|^{\alpha}$, for some constants $c$ and $\alpha$ for all $x$ and $y$. ...
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Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and $\langle\;\cdot\;,\;\cdot\;\...
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1answer
39 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb N}\...
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1answer
92 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result $$\widehat{f}(n)=O\left(\frac{1}{|n|^{\alpha}}\...
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1answer
114 views

Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
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Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
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47 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
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Calculate a Limit and find the Sup

Supporse that $$ f(x)=\frac{1}{|x|^\alpha+1},\quad\alpha\in(0,1],x\in I:=[-1,1], $$ and $$ g(x,y)=\frac{|f(x)-f(y)|}{|x-y|^\beta},\quad x,y\in I. $$ I can prove if $$ \sup_{x,y\in I}g(x,y)<+\...
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229 views

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let $\varepsilon>...
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1answer
159 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq |x-y|^{1-1/...
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1answer
56 views

Is $C^{1,\alpha}\subseteq C^{0,1}$?

It is true that if $f\in C^{1,\alpha}(I)$ than $f\in C^{0,1}(I)$? I mean: if $f$ is bounded and differenciable with bounded and holder continuous derivative, then $f$ is bounded and lipschitz ...
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51 views

Lip $\alpha$ trigonometric series

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} |...
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73 views

Holder norms inequality

I saw this in a proof I am reading, and have been unable to justify the statement. Assume $\Omega$ to be a open bounded set in $R^n$ with smooth boundary, and $u\in C^{2,\alpha}(\bar\Omega)$. Then,for ...
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76 views

Global Holder regularity from DiGiorgi-nash-moser

For a strictly elliptic differential operator $L=\sum_{i,j} D_i (a_{ij}D_ju) $ with strictly elliptic condition on open bounded smooth domain $\Omega$,and bounded coefficients, DiGiorgi-Nash-Moser ...
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58 views

Hölder continuity of $\frac1x$

I have a question. Is the function $f(x)=1/x$ Hölder continuous if $x\in (\varepsilon,+\infty),\ \varepsilon>0$?
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112 views

C^1 is not dense in Holder space

Let $\overline{\Omega}$ be a bounded, closed and convex set of $\mathbb{R}^n$. Prove that $C^1(\overline{\Omega})$, the space of continuously differentiable functions on $\overline{\Omega}$, is not ...
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2answers
27 views

rational function of non vanishing Hölder continuous functions

Conside Hölder continuous functions $f_1,…,f_m:\mathbb{R}^n\rightarrow \mathbb{R}$ (with Hölder coefficient $\alpha$). The claum is now that any rational function of $f_1,…,f_m$ with non vanishing ...
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1answer
74 views

Is the Hölder space with exponent $\beta$ dense in the space with exponent $\alpha$ for $\alpha<\beta$?

For $0<\alpha\leq 1$ let $\Lambda_{\alpha}([0,1])$ be the space of functions on $[0,1]$ such that $||{f_{\Lambda_{\alpha}}}||<\infty$, where $$\|{f}\|_{\Lambda_{\alpha}}=|f(0)|+\text{sup}_{x,y\...
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36 views

Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...