All Questions
13 questions
3
votes
0
answers
53
views
Bounds on Besov norms for mollification of a bounded Lipschitz function
Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
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
0
votes
0
answers
149
views
Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
0
votes
0
answers
103
views
Extension for fractional Sobolev spaces, s>0
In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...
- 93
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
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
2
votes
0
answers
121
views
Interpolation of Sobolev/Besov spaces in the limiting case q = ∞
I'm interested in the interpolation space ($1\le p_0,p_1\le\infty$, $0<\theta<1$)
$$
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 .
$$
It ...
- 61
2
votes
0
answers
225
views
Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$
Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and
$$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$
for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
- 233
0
votes
0
answers
247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
4
votes
0
answers
99
views
Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?
Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
- 5,405
2
votes
0
answers
65
views
Generalized Besov spaces with different integrability and smoothness in space and time?
Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$.
Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{...
- 2,306
7
votes
2
answers
536
views
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...
- 2,670
12
votes
0
answers
476
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
- 8,086
4
votes
1
answer
3k
views
Besov and Triebel-Lizorkin spaces
Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$,
$$
1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad \varphi_{\nu}...
- 16.2k