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21 votes
1 answer
3k views

Density of polynomials in $C^k(\overline\Omega)$

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
user111's user avatar
  • 4,034
15 votes
1 answer
1k views

Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
Piotr Hajlasz's user avatar
8 votes
2 answers
1k views

Extension theory with bump function

Let $B_t(0)$ denote the $n$ dimensional ball of radius $t$ centered at the origin. Does there exist a $\phi\in C(\mathbb{R}^n)$ function with the properties: $ \phi (x) = \begin{cases} 1&x\in B_r(...
alext87's user avatar
  • 3,217
7 votes
2 answers
3k views

Arzelà-Ascoli theorem and Hölder spaces

Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$. Does there exist ...
asv's user avatar
  • 21.8k
7 votes
1 answer
771 views

Famous but unavailable paper of Jan Boman

The following paper is well known, but hard to find: J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982. In this paper ...
Piotr Hajlasz's user avatar
6 votes
2 answers
1k views

Exercise 8.13 - Brezis

Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set $$ D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0 $$ Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
user253963's user avatar
6 votes
1 answer
253 views

Littlewood-Paley theory and dense property of Sobolev spaces [duplicate]

I'm learning the Littlewood-Paley theory by myself and I encounter the following claim: Pick a smooth function $\chi$ such that: $$\chi(\xi) = \begin{cases} 1 &|\xi| \leq \frac{1}{2}\\ 0 &|\...
randombeaver's user avatar
5 votes
1 answer
657 views

Showing integrability of a locally integrable function on a bounded domain under some additional assumptions

Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies Assumption A: For all $g\in C_0^1(\...
Ben Ciotti's user avatar
4 votes
1 answer
225 views

Approximate constant function

Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$ Does there exist a constant $c>0$ such that any such function ...
Kung Yao's user avatar
  • 192
4 votes
2 answers
364 views

Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following. Use pseudo-differential operators on $M$...
kuuga's user avatar
  • 71
4 votes
1 answer
1k views

Sobolev-Slobodeckij spaces for p=infinity

For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then, $ \...
alext87's user avatar
  • 3,217
4 votes
0 answers
802 views

Reproducing kernel Hilbert space of Matérn kernels

I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv) On the top of ...
seeker_after_truth's user avatar
3 votes
2 answers
279 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
Lewandowski's user avatar
3 votes
2 answers
657 views

A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e

I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is not differentiable a.e. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,...
Dorian's user avatar
  • 2,641
3 votes
1 answer
132 views

Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that $$ \lvert u\rvert_s = \...
smyrlis's user avatar
  • 2,933
3 votes
1 answer
317 views

Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...
Qwertuy's user avatar
  • 251
3 votes
1 answer
111 views

Sobolev inequalities and Wiener algebra

It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$) such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and $$ \...
Bazin's user avatar
  • 16.2k
3 votes
0 answers
542 views

Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative. It is easy to show that $f *g$ is defined pointwise when $s_1+s_2\...
Zorgoth's user avatar
  • 256
2 votes
1 answer
724 views

What is the motivation of the $L^p$ differentiability?

I was reading some papers and come up with the next definition : A function is differentiable in the $L^p$ sense at $x$ if there exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^...
Jingeon An-Lacroix's user avatar
2 votes
2 answers
450 views

The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$

Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
Alexandru Pirvuceanu's user avatar
2 votes
1 answer
93 views

Reference needed: estimate of the second order derivatives

In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions) $$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
Michael Perelmuter's user avatar
2 votes
1 answer
154 views

Hölderness of the inverse to a $W^{1,p}$-homeomorphism (with additional conditions) of a Lipschitz domain

Let $\Omega_1 \subset \mathbb R^2$ be a bounded simply-connected Lipschitz domain, and $f: \bar \Omega_1 \rightarrow \bar \Omega_2$ be a homeomorphism, which is a diffeomorphism on $\Omega_1$ such ...
Taras's user avatar
  • 208
2 votes
0 answers
225 views

Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and $$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$ for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
Desura's user avatar
  • 233
2 votes
0 answers
67 views

Traceless sobolev forms on compact manifolds with boundary

Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
H1ghfiv3's user avatar
  • 1,255
2 votes
0 answers
64 views

The continuity of $L^2$ gradient on moving domain

I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem... Let $I:=(...
JumpJump's user avatar
  • 679
2 votes
0 answers
82 views

Properties of a Sobolev bound

I am interested in computing $$ A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2} $$ where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound. ...
rrr's user avatar
  • 193
1 vote
1 answer
179 views

On the compact embedding of Sobolev space

In dimension three, we know that the Sobolev space $W^{\frac{13}{11},11}(D)$ is compactly embedded into $W^{1,11}(D)$, where $D$ is a bounded domain in $R^3$ with smooth boundary. My question is: Does ...
ABCD's user avatar
  • 31
1 vote
2 answers
322 views

Variational formulation of abstract Cauchy problem, possible?

Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial ...
avg_ali's user avatar
  • 59
1 vote
1 answer
169 views

Higher integrability for Sobolev functions - part 2

This is a follow-up to the question asked in Higher integrability for Sobolev functions Updated question: Given the very helpful counterexamples and the ideas, I have the following question: Suppose ...
Adi's user avatar
  • 455
1 vote
1 answer
251 views

Is maximum principle valid in the case of non-smooth boundaries?

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad. Denote $U=U_2 \...
user101829's user avatar
1 vote
2 answers
183 views

Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$. Question 1: Can we ...
user84068's user avatar
  • 169
1 vote
1 answer
399 views

How to prove Poincaré inequality by using extension theorem?

Poincaré inequality is stated as follows: Let $ \Omega $ be bounded, connected, open subset pf $ \mathbb{R}^n $, with a $ C^1 $ boundary $ \partial \Omega $. Assume that $ 1\leq p<\infty $ and $ u\...
Luis Yanka Annalisc's user avatar
1 vote
0 answers
21 views

Regularity of solutions of a 2nd order singular integro-differential operator

I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
BlueCharlie's user avatar
1 vote
0 answers
130 views

Fractional Sobolev embedding theorem

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
user173196's user avatar
0 votes
1 answer
104 views

If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?

Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality \begin{equation*} \int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
JZS's user avatar
  • 481
0 votes
1 answer
257 views

Explicit expression for the fractional Laplacian of $1/(1+|x|^2)^s$

For $s\in (0,1)$, is there are an explicit expression for $$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$ Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\...
Student's user avatar
  • 537
0 votes
1 answer
53 views

Exponentially weighted norms are not equivalent

Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show ...
Lars Siemer's user avatar
0 votes
1 answer
294 views

Bound for the product of Sobolev functions in $W^{s,1}$

I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$. $$ \lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert. $$ It seems reasonable to want to use Hölder's ...
johann's user avatar
  • 1
0 votes
1 answer
110 views

Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$

Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
Hiro's user avatar
  • 131
0 votes
0 answers
112 views

Vector field connecting two points

I'm now working on somehow an inverse problem of an ODE: Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t). Now there is a ...
Sqr's user avatar
  • 1
0 votes
0 answers
134 views

How can i show that the last equality in the text is true?

Suppose that $v$ is critical point of $$ f(u)=\frac{1}{2} \int_D|\nabla u|^2-\frac{1}{p+1} \int_D|u|^{p+1}, \quad u \in H_{0, \text{rad}}^1(D),\quad D(r, d)=\left\{z \in R^N: r^2<|z|^2<(r+d)^2\...
Pádua's user avatar
  • 69
0 votes
0 answers
211 views

Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
Muniain's user avatar
  • 101