Articles of holder spaces

Holder continuous derivative on bounded $D$ implies Lipschitz?

Suppose $D$ is a bounded, open, connected subset of $\mathbb{R}^n$. Suppose that $u \in C^{1,\alpha}$ for $\alpha \in (0,1]$, i.e. $u$ has a Holder continuous derivative with exponent $\alpha$. Is it true that $u$ is Lipschitz? My thoughts are that $u \in C^{1,\alpha}$ implies $\nabla u$ is uniformly continuous, and hence bounded since $D$ is […]

Proving something is $1$-Lipschitz

(1) Let $(X,d)$ be a metric space, and let A be a non-empty subset. Show that the function $$D_A :X \to [0,\infty ]$$ defined by $$D_A (x) =\inf \{d(x,y) : y \in A\}$$ is $1$-Lipschitz (when $[0,\infty)$ is given the standard metric. (2) Now suppose that A is compact. Prove that, for each $x \in […]

Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into $C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can we reach those embeddings and for which exponents $s, \alpha$ are these embeddings compact? $H^s \subset C^{(\alpha)} $ continuous for $s > \alpha + [?]$ $H^s \subset C^{(\alpha)} $ compact for $s […]

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 […]