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Questions tagged [holder-spaces]

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

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Extension of a Hölder function on the product space

Let $k\geq0, \alpha>0$ and let $\Omega\subset \mathbb{R}^d$ be a $C^{k,\alpha}$ convex domain. Suppose $f:[0,1]\times \overline{\Omega} \to \mathbb{R}$ satisfies the following: For each $t\in[0,1]$...
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Example for $u\in C^\infty(\mathbb{R}^3\setminus \{0\})\cap C^{1,\alpha}\cap L^2$

I am looking for an example for $u(x,y,z)\not=\text{ const}$, such that $$u\in C^\infty(\mathbb{R}^3\setminus \{0\})\cap C^{1,\alpha}\cap L^2.\tag{1}$$ Does (1) really mean: $$u\in C^\infty(\mathbb{R}^...
mike's user avatar
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A Hölder norm of square root of a $C^2$ function

Background I am reading a proof of the Calabi-Yau theorem from these notes. In page 15 he claims the following statement without proof (calling it elementary): Let $M$ be a compact manifold. There ...
Or Kedar's user avatar
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Little Hölder spaces - Reference request

The little Hölder space $c^\gamma(\mathbb{R}),$ for $0<\gamma<1,$ is usually defined in one of the following ways: $c^\gamma(\mathbb{R})$ is the clousure of $C^\infty(\mathbb{R})$ in the usual ...
Marc Magaña's user avatar
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Uniform extension of Holder functions

Let $\{\Omega_t\}_{t\in [0,1]} \subseteq \Omega'$ where $\Omega_t$ is a $C^{k,\alpha}$ uniformly convex domain and $\Omega'$ is open and bounded. Suppose $u_t \in C^{k,\alpha}(\Omega_t)$, then one may ...
Stephen_lamb's user avatar
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About the Hölder continuity of the power of a function

Let $\Omega$ be a bounded domain and assume that $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}$ is a function in the space $C^{2,\alpha}(\overline{\Omega})$, $0 < \alpha <1$, such that $f(x) &...
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Are $C^1$ functions Holder Continuous on general subsets of $\mathbb{R}^2$

So suppose that $D$ is an open and connected subset of $\mathbb{R}^n$ for some n. Given a $f\in C^k(D,\mathbb{R})$ then we know that that all of the derivatives up to order $k$ exist and they are ...
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Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$

Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following: Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
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How to show that a convolution of function with the heat kernel can only improve its Holder regularity

Let $D$ be a closed connected set in $\mathbb{R}^d$ and let $f\colon D\to\mathbb{R}$ be an $\alpha$-Holder continuous function. I want to show that if I convolve this function with a heat kernel $p_t$,...
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A uniform bounded in a bounded $\Omega\subset \mathbb{R}$

I’m in trouble with some inequality. I want to prove that $$\|\Delta u-\frac{c_1}{\epsilon^3}\int_{\mathbb{R}}J(\frac{x-y}{\epsilon})(u(y)-u(x))dy\|_{L^{\infty}(\Omega)}\leq C\epsilon^\alpha,$$ where $...
Luiza Camile's user avatar
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Question about the proof of Theorem 5 in the subsection 5.6.2 of Evans' PDE book (page 283 in the 2nd version)

This is the Theorem 5 in the subsection 5.6.2 of Evans' PDE book (second edition): Picture 1: page 283 of Evans' book Picture 2: page 284 of Evans' book It's a theorem related to Morrey's inequality (...
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Estimates on Holder Norms and $C$^k norms

I'm dealing with some estimates. There are some very useful estimates on Holder norms in a paper by Hormander. But I can't relate them immediately to the usual $C^k$ norms. The estimates appear in ...
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Adaptive estimation of Hölder-continuous signal

Consider the Gaussian white noise model given by, $$ dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$$ with $W$ a $d-$parameters Wiener field, $f:[0,1]^{d}\...
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How does the mean value theorem with Hölder seminorms depend on dimension?

I am reading through Krylov's Lectures on Elliptic and Parabolic Equations in Hölder Spaces. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to ...
IdenticallyEulerian's user avatar
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How do we show that on a parabolic Hölder space, the polynomial and Hölder seminorms are equivalent?

I am currently working through Krylov's Lectures on Elliptic and Parabolic Equations in Holder Spaces. One of the key points of chapter 8 is that the two seminorms $$[u]_{1+\delta/2,2+\delta;U} := \...
IdenticallyEulerian's user avatar
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Estimate on $\alpha$-Hölder norm of path signature

Let $N \geq \lfloor 1/\alpha \rfloor > 0$ and consider a weakly geometric $\alpha$-Hölder rough path $\textbf{x}$ that preserves the origin, i.e. an element $\textbf{x} \in C^{\alpha\text{-Höl}}_o([...
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Best constant for Sobolev inequality from Sobolev space to Hölder space

Let $u(x) \in W^{1,p}(\mathbb{R}^n)$ such that the Sobolev embedding from $W^{1,p}(U)$ to $C^{0,\alpha}(U)$ holds for some $n$, $p$, $\alpha$ and bounded domain $U$. We have the Sobolev inequality $$ \...
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Continuous bounded variation path that fails to be Lipschitz

Let $C^{1\text{-var}}([0,T];\mathbb{R}^d)$ denote the space of continuous bounded variation paths taking values in $\mathbb{R}^d$. Similarly, let $C^{1\text{-Höl}}([0,T];\mathbb{R}^d)$ denote the ...
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What is a closed subspace of infinite codimension within the Banach space of Hölder continuous functions?

What property does a function that belongs to a closed subspace of infinite codimension within the Banach space of Hölder continuous functions have?
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Comment on the uniqueness of the solution of ODE

Consider the first-order equation given by $y' = f(x, y)$ in $I := [x_0, b]$ subject to the initial condition $y(x_0) = y_0$, assuming $f$ is continuous everywhere it is defined and there exists $M &...
R_Squared's user avatar
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Sobolev inequality and C^{1,1} estimate

I read in an article the following statement: I have question about how we get the inequality (2.19) :it is written that we uses $C^{1,1}$ estimate but here $p\geq 2 $ and not necesserly bigger ...
RIM's user avatar
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How to bound the Hölder norm?

Let $\rho:\mathbb R^n\to\mathbb R$ be a function supported on a ball of radius $r$ with $$\begin{array}{rcl} \int_{\mathbb R^n}|\rho(y)|dy & \lesssim & r^m \\ \int_{\mathbb R^n}|\nabla \rho(y)|...
enihcamemit's user avatar
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Proving the Compact Embedding of $C^{0, \alpha}(K)$ into $C^{0, \beta}(K)$

My question: Let $K \subset \mathbb{R}^d$ be compact and $0<\beta<\alpha \leq 1$. Let $\gamma=\theta \alpha+(1-\theta) \beta$ for $\theta \in[0,1]$. Show $$ [f]_{C^{0, \gamma}(K)} \leq[f]_{C^{0,...
Mathematiker's user avatar
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Interpolation Inequalities in Lebesgue Spaces and Compactness in Hölder Spaces: Proving Norm Relations and Compact Embeddings [duplicate]

I am required to solve this: Let $\Omega \subset \mathbb{R}^d$ be open and let $p, q \in[1, \infty]$ with $p \leq q$. Let $f \in L^p(\Omega) \cap L^q(\Omega)$. Show that for $r \in(p, q)$ it holds $$ \...
Mathematiker's user avatar
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Canonical lift of a smooth path as a rough path

Let $X_t$ be say, differentiable, importantly it is guaranteed to be at least more than $\frac{1}{2}$ Holder continuous. Then I can define the iterated integrals in the Riemann-Stieljes/Young sense as ...
Theo Diamantakis's user avatar
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Is there a constant $c>0$ such that $\| (1-\Delta)^{-\alpha} P^\kappa_t f\|_{C^{n,\beta}_b}\le c t^{-(\frac{n}{2} - \alpha)^+} \| f \|_{C^{\beta}_b}$?

For $n \in \mathbb N$ and $\alpha \in (0, 1)$, let $C^{n, \alpha}_b (\mathbb R^d)$ be the space of $n$-times continuously differentiable real-valued functions $f$ on $\mathbb R^d$ that admit the ...
Analyst's user avatar
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Is there a constant $c>0$ such that $\|(1-\Delta)^{\frac{\alpha}{2}} f\|_\infty\le c\|f\|_{C^{0,\alpha}}$ for all $f \in C^{0,\alpha} (\mathbb{R}^d)$?

For $n \in \mathbb{N}$ and $\alpha \in(0,1)$, let $C^{n, \alpha} (\mathbb{R}^d)$ be the Hölder space of real-valued functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\{\nabla^i f\}_{0 \leq ...
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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 ...
mathematica's user avatar
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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 ...
Fan's user avatar
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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 ...
Bihu Duo's user avatar
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Holder continuity in an interval implies boundedness

Let $\alpha\in (0, 1)$ and $-\infty<a<b<+\infty$. Assume that $u\in C^{0, \alpha}((a, b), \mathbb R)$. Here $$|u|_{C^{0, \alpha}((a, b))} = \|u\|_{L^\infty((a, b))} +\sup_{x, y\in (a, b), x\...
C. Bishop's user avatar
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What is the relationship between Hölder spaces and differentiability?

Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources informally describe these spaces as functions having at least "$k + \alpha$" derivatives. How ...
CBBAM's user avatar
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A property of $C^{0, \alpha}$ functions

I am going through $C^{0, \alpha}(\Omega)$ regularity, $\alpha\in (0, 1)$, where $\Omega$ is a regular domain (not bounded, in general). I can not find much material to study from, but I found this ...
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definition of BUC($\alpha,\frac{\alpha}{2})$ [closed]

i found in the book of P;Souplet and P;Quiitner the definition and norm of space $BUC^{(\alpha,\frac{\alpha}{2})}$ .here is the definition: Let $Q=Q_T=\Omega \times(0, T)$ where $\Omega$ is an ...
RIM's user avatar
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For a Holder-continuous function, is the quotient $\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y\rvert^{\alpha}}$ jointly continuous for $x \neq y$?

This seems a bit trivial but there is no explicit proof for the statement nor can I prove it directly myself. Let $f: X \to \mathbb{R}$ be $\alpha$-Holder continuous for some $\alpha \in (0,1)$, where ...
Keith's user avatar
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A Holder-continuous function differentiable a.e. is absolutely continuous?

Let $f : [0,1] \to \mathbb{R}$ be a Holder-continuous function of an exponent $\alpha \in (0,1)$ and differentiable a.e. at the same time. Assume further that the derivative $f'$ is integrable on $[0,...
Keith's user avatar
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2 votes
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Embedding of differentiable functions into Holder Functions

I have the following problem. Consider $\Omega \subset \mathbb{R}^{n}$, fix $\alpha \in (0,1) $. Consider the space $C^1(\overline \Omega)$ of differentiable functions equipped with the norm $${\| u \|...
Adriano Banchieri's user avatar
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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 ...
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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, ...
Volburin's user avatar
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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 $\alpha$ ...
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Extending Holder functions

If $(X, d)$ is a metric space and $A \subseteq X$, McShane's Extenstion Theorem states that every $L$-Lipschitz function $f \colon A \to \mathbb{R}$ can be extended to $L$-Lipschitz function $f' \...
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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 ...
Pipnap's user avatar
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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 ...
wannabecapablanca's user avatar
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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 ...
Calvin Khor's user avatar
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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 ...
Tham's user avatar
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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 ...
Martin Geller's user avatar
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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 ...
Jamal's user avatar
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1 answer
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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 ...
BabaUtah's user avatar
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43 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 ...
Cézar Bezerra's user avatar
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1 answer
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Meaning of $D^kf$ and of $D^{\beta}f$ (with $k$ nonnegative integer and $\beta$ multi-index)

What is the meaning of $D^kf$ in this context? I'm a bit confused because I know that $Df(a)$ normally means the total derivative at $a$. Moreover, at some point the text treats $Df(x)$ and $\nabla f(...
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