Articles of holder spaces

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 numerical scheme, so I am interested in finding $H(p)$ itself, rather than upper bounds, or at least an upper […]

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 other way around : if we look at elements of Sobolev spaces, then we get a Hölder regularity depending on the dimension $n$ and […]

How to show pre-compactness in Holder space?

Let $K \in \mathbb{R}^d$ be a compact set and consider the space of Hölder continuous functions $C^{0,\gamma}(K)$ with norm $||f||_{C^{0,\gamma}}:=||f||_{\infty}+\sup_{x,y \in K,x \neq y}\frac{|f(x)-f(y)|}{|x-y|^{\gamma}}$. Assume we have a bounded sequence $\{f_n\} \subset C^{0,\gamma}(K)$, i.e. $\exists C>0$ s.t. $\sup_{n}||f_n||_{C^{0,\gamma}} \leq C$. Under what conditions can we say the sequence $\{f_n\}$ is pre-compact in $C^{0,\gamma}(K)$? In other […]

Fundamental Theorem of Calculus for distributions.

Consider a function $F \in C^{\alpha}( \mathbb{R})$ for $0 < \alpha < 1.$ Then we can take it’s distributional derivative. We can say $f = F’ \in C^{\alpha -1}( \mathbb{R} )$. My issue is going back. Say I have a $\alpha -1 -$Hölder function $f$, then how can I “integrate” it to get a primitive […]

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for $0<\alpha<1$, then $u$ is uniformly continuous for sure. However, I can not prove $u\in C^{0,\alpha}(I)$ then $u\in AC(I)$, nor $u\in […]

Let the function $f: \to \mathbb R$ be Lipschitz. Show that $f$ maps a set of measure zero onto a set of measure zero

Let the function $f:[a,b] \to \mathbb R$ be Lipschitz, that is, there is a constant $c \geq 0$ such that for all $u,v \in [a,b]$, $|f(u)-f(v)| \leq c|u-v|$. Show that $f$ maps a set of measure zero onto a set of measure zero. Show that $f$ maps an $F_\sigma$ set onto an $F_\sigma$ set. Conclude […]

Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{align} Lu&=f,&in \quad U,\\ u&=g,&on \quad \partial U.\tag{1} \end{align} where $Lu:=a_{ij}(x)u_{x_ix_j}+b_i(x)u_{x_i}+c(x)u.$ is a strictly elliptic operator. I have known that the $C^{2,\alpha}$-regularity from Gilbarg&Trudinger’s book and the $H^2$-regularity from Evans’book. Now I wonder that can the $C^2$-regularity is also available?Namely,can we […]

Locally continuously differentiable implies locally Lipschitz

I am interesting in the following result: Let $X$ be a normed space and $f : X \to \mathbb{R}$. If $f$ is continuously differentiable in a neighborhood $V$ of a point $x_0 \in X$, then $f$ is locally Lipschitz at $x_0$. Using mean value theorem, for all $x,y$ in an open ball $B \subset V$, […]

Hölder continuity definition through distributions.

I am trying to prove that for a given Hölder parameter $\alpha \in (0, 1)$ and a distribution $f \in \mathcal{D}'(\mathbb{R}^d)$ the following are equivalent: $f \in C^{\alpha}$ For any $x$ there exists a polynomial $P_x$ such that $| \langle f – P_x, \phi_x^{\lambda} \rangle | \le C \lambda^{\alpha.}$ Where the latter estimate holds uniformly […]

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in \mathbb{Z}$. Also there exists a constant $C$ such that $$ |f(x+h)-f(x)| \leq C|h|^{\alpha}$$ for all $x$ and $h$. […]