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

learn more… | top users | synonyms (1)

2
votes
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} \...
2
votes
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 ...
1
vote
1answer
68 views

Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the ...
1
vote
1answer
101 views

Hölder Space Definition

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows: If $u:U \rightarrow \mathbb{R}$ is bounded and ...
1
vote
1answer
142 views

Regarding Hölder continuity

Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - x_0|^{\...
0
votes
1answer
77 views

Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose $...
-1
votes
1answer
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{...
4
votes
0answers
196 views

Hölder Condition Implying Uniform Convergence

Define $g(z)=\frac{1}{2\pi i}\int_{-k}^k \frac{h(\zeta)}{\zeta-z}d\zeta$, where $h$ is continuous and defined on $[-k,k]$. Let $|h(x)-h(y)|\leq |x-y|^\alpha$ for all $x,y\in[-k,k]$ and for some $\...
3
votes
0answers
22 views

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)<+\...
3
votes
0answers
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} |...
3
votes
0answers
179 views

Hölder estimate of solution to linear parabolic PDE

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 \end{...
3
votes
0answers
123 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
3
votes
0answers
223 views

Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous

Consider the following function defined by a singular integral \begin{equation} F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) \...
3
votes
0answers
300 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
3
votes
0answers
287 views

Convergence of sequence in $C^{k, \alpha}$ composed with $C^\infty$ function

Is it true that if a sequence $u_n \to u$ in $C^{k, \alpha}$ norm, and if you have a function $f \in C^\infty$, then $f(u_n) \to f(u)$ in $C^{k, \alpha}$? I think so, since this is true for ordinary $...
2
votes
0answers
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 $m\in \mathbb{Z}^+$, ...
2
votes
0answers
77 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 &...
2
votes
0answers
72 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
73 views

Proving a basic result about Holder continuous functions

Let $V$ be a open convex set. We will say that a function $m$ has the order of smoothness $p$ on $V$ with $p=l+\gamma$, where $l \geq0$ is an integer and $0<\gamma\leq1$ and will write $m\in H_{p}(...
2
votes
0answers
68 views

Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$ Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
2
votes
0answers
26 views

Multiplicative constant in inclusions of Hölder spaces

Let $\Sigma(n + \beta, L)$ for $n \in \mathbb{N}_0$, $0 < \beta \le 1$, $L > 0$ be the set of functions $f : \Omega \to \mathbb{R}$ (or $\mathbb{C}$, whatever) whose derivatives up to order $n$ ...
2
votes
0answers
55 views

Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have \...
2
votes
0answers
117 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, \...
1
vote
0answers
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|^...
1
vote
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, $...
1
vote
0answers
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||_{\...
1
vote
0answers
38 views

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\;\...
1
vote
0answers
13 views

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 ...
1
vote
0answers
284 views

The function $f(x) = x^\alpha $ belongs to the Lipschitz class of order $\alpha$

Suppose that we have a Lipschitz function $$ |f(x_1) - f(x_2) |\le M {|x_1 - x_2|}^{\alpha} $$ where M is a constant. Note Lip $\alpha$ denote the set of all functions satisfying a Lipschitz condition ...
1
vote
0answers
36 views

Estimates in Hölder spaces

Let $u,v\in C^{2,\alpha}\left(\overline{\Omega}\right)$. Proof that there exists a constant $C>0$ so that \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq C\left(\|u\|_2+\|...
1
vote
0answers
30 views

Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in ...
1
vote
0answers
68 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
1
vote
0answers
103 views

Regularity of a a transmission problem

How to prove? Let $a_{ij} \in C^{0, \alpha}(B_1 \cap \mathbb{R}^{n}_{+}), b_{ij} \in C^{0,\alpha}(B_1 \cap \mathbb{R}^{n}_{-})$ elliptic matrices and $$ A_{ij}(x) = a_{ij}(x)\chi_{\{ x_n \ge 0 \}} + ...
0
votes
0answers
23 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 $x \in [0, N+\frac{\alpha}{\beta}]$....
0
votes
0answers
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.,...
0
votes
0answers
36 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?
0
votes
0answers
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 ...
0
votes
0answers
72 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: ...
0
votes
0answers
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 ...
0
votes
0answers
51 views

Hölder continuity for parabolic equations

What is a good and modern reference for hölder regularity for non-degenerate parabolic equations? To be a bit more precise, I have a degenerate parabolic equation, exhibiting two degeneracies, and can ...