# Questions tagged [holder-spaces]

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

447 questions
Filter by
Sorted by
Tagged with
24 views

### If $f$ is Holder of order $\alpha$, then $f'$ is holder of order $\alpha-1$

Let $D\subset\mathbb{R}$. We say that $f:D\to \mathbb{R}$ is Holder of order $\alpha>1$ if for the largest integer $l<\alpha$ (i.e. $\alpha-l\in (0,1)$: $f$ is $l$ times differentiable, and ...
35 views

### Regularity of functions approximated by polynomials

I want to ask how to prove the following result: Suppose $u\in C(\overline{B_{1}})$, if there exists a universal constant M>0, such that for any $x_{0}\in \overline{B_{1}}$, we have a quadratic ...
1 vote
40 views

### Reverse Holder functions

For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that $$|f(x) - f(y)| \geq c|x-y|^\alpha.$$ The ...
45 views

1 vote
29 views

### Does differential operator $D^{\alpha} : W^{m+|\alpha|, p} \rightarrow W^{m, p}$ have a closed image in $W^{m, p}$

I actually have have $\Omega \subseteq \mathbb{R}^{d_1+d_2}$ and set multiindex set $$\mathcal{A}:= \{\alpha = (\alpha_1, \alpha_2)\in \mathbb{N}^{d_1+d_2} : 2|\alpha_1| + |\alpha_2|\leq2 \}.$$ And ...
87 views

### Showing that spaces $C_b^{(1,2)}$ and $W^{(1,2),p}$ are Banach spaces.

So in essence there are a few propositions in the beginning. For arbitrary $T$ the space $B(T;\mathbb{C})$ is a space of all (real or) complex functions $f: T\rightarrow \mathbb{C}$ with sup norm, ...
1 vote
35 views

### Does $C^{1,\alpha}$ imply Holder's continuity with exponent $1+\alpha$ in some cases? [duplicate]

My analysis professor said that if $V \colon \mathbb R^n \to \mathbb R$ is $\mathcal C^{1,\alpha}$ for $\alpha >0$, i.e. it is differentiable with Holder's continuous gradient of order $\alpha$ ...
1 vote
68 views

55 views

### The property of Hölder continuous density function

I am reading the article but I don't understand some lines in page 10. First, assumptions on the density $h$ are needed. Of course, since $h$ is the density of a symmetric probability distribution on ...
1 vote
38 views

### Hölder domains are locally graphs

I'm currently reading chapter 6 of Gilbarg + Trundinger book on second order elliptic pde. They define $C^{k,\alpha}$ domain in the following manner: A domain $\Omega\subset\mathbb{R}^{n}$ and its ...
1 vote
55 views

### Example of little $\alpha$ Hölder function that is not $\beta$ Hölder, for any $\beta>\alpha$?

What functions are in the set $$c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$ A single example will do, but the more the merrier. This question was natural to me after writing this ...
31 views

### Littlewood-Paley Characterisation of Hölder Regularity

I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
1 vote
114 views

### Little Hölder spaces

In this question, we have a definition of the above spaces: What is little Holder space? How can we prove that the closure of the “smooth” functions, i.e. the little Hölder space, is not the whole ...
129 views

### Relation between Minimum Holder Coefficient and Holder Semi-norm

I have the following two definitions pertaining to Holder continuity: For both, let $\alpha \in (0,1]$ and consider the path $X:[0,T] \rightarrow \mathbb{R}^d$ $X$ is $\alpha$-Holder continuous if ...
1 vote
86 views

### Hölder continuity of Brownian Sheet/Wiener Field

It is known from P. Lévy, Théorie de l’addition des variables aléatoires. Monographies des Probabilités ; calcul des probabilités et ses applications 1, Paris (1937), that the sample paths of the ...
37 views

### Hölder regularity in the quantitative manner

Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
59 views

156 views

### Embedding of $W^{1,\infty}$ in $C^{0,1}$

Theorem 5 of pag 283 of the text "Partial Differential Equations - Second Edition (Lawrence C. Evans)" states Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ ...
1 vote
56 views

### Compact Embedding Theorem. Metric Space. Santambrogio. (reference request).

In the book `Optimal Transport for Applied Mathematicians' by Santambrogio, on page 289 it is written that In this setup, the space $X=(X,d)$ is a metric space, $\{\tilde{x}^\tau\}_{\tau>0}$ is a ...
68 views

### Show that $x\mapsto |x|^s_2x$ on $\mathbb{R}^n$ is globally $C^{1,s}$ for $0<s<1$

Recently I had to make use of the fact that the map $x\mapsto |x|^s_2$ is globally $s$-Hölder continuous $\left(|\cdot|_2\text{ is the Euclidean distance}\right)$, which can be reduced to the $1$-...
1 vote