All Questions
21 questions
8
votes
3
answers
1k
views
Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains
I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
- 81
7
votes
2
answers
536
views
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...
- 2,670
6
votes
0
answers
113
views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
- 395
5
votes
0
answers
341
views
Real interpolation of weighted Sobolev spaces with different weights
Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
4
votes
1
answer
423
views
Real interpolation for vector-valued Sobolev spaces
I am having trouble finding any results concerning real interpolation of vector-valued Sobolev spaces. Namely, I would like to know if a continuous embedding of the type,
$$
L^p(0,T;X_1)\cap W^{1,p}(0,...
- 213
4
votes
0
answers
164
views
A modern reference for the "Intermediate Derivatives Theorem"
In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...
- 1,247
3
votes
0
answers
209
views
Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
2
votes
1
answer
404
views
Trace of a function
Let $T,L> 0$ two real numbers and we consider the Sobolev space $X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$. My question is:
Given $f \in X$, the trace $ t \mapsto f(t,L)$ belongs to what ...
- 29
2
votes
1
answer
800
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
- 843
2
votes
0
answers
151
views
Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
- 989
2
votes
0
answers
124
views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 21
2
votes
0
answers
121
views
Interpolation of Sobolev/Besov spaces in the limiting case q = ∞
I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$)
$$
X=(L_{p_0}(0;1),W^1_{p_1}(0,1))_{\theta,q}\quad\text{with}\quad q=\infty\ \ \text{and}\ \ p_0\ne p_1 .
$$
It ...
- 61
2
votes
0
answers
223
views
Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
- 81
2
votes
0
answers
114
views
What is $(L^2(M), H^1_0(M))_{\frac 12}$ on a smooth manifold with boundary?
Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space
$$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$
(see Taylor's book on PDE for example). Suppose $M$ has a ...
- 251
2
votes
0
answers
52
views
About norm on $H^{\frac 12}(M \times \{0,1\})$
Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.
Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space ...
- 171
1
vote
0
answers
65
views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
1
vote
0
answers
174
views
Interpolation of Sobolev spaces with constraints
Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
- 81
1
vote
0
answers
123
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
1
vote
0
answers
82
views
Extreme case of K-interpolation
Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
1
vote
1
answer
277
views
Intersection of the kernel with the interpolation space
$\DeclareMathOperator\Ker{Ker}$Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow ...
- 11