Questions tagged [bessel-potential]
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15
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For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
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An identity about Bessel potential operators
I'm reading this paper where I encounter below equality, i.e.,
$$
\begin{aligned} & \left|\int_{\mathbb{R}^d}\left\{\ell_s^2\left(a_s^{\gamma^2}-a_s^{\gamma^1}\right)_{i j} \partial_i \partial_j ...
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Reference request: Hölder regularity of $(1-\Delta)^{\frac{\alpha}{2}}$ for $\alpha >0$
Let $j \in \mathbb N$ and $\alpha \in (0, 1)$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\mathbb R}^d)$ the usual Hölder space. For convenience, we denote $H^{\alpha} := H^{j + \alpha}$ for the ...
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How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
...
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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 ...
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Riesz potential and homogeneous Sobolev spaces
Consider the Bessel potential and Riesz potential spaces $$L^{s,p}=\{f:f=(I-\Delta)^{-s/2}g,\, g\in L^p\},$$ $$\dot{L}^{\alpha,p}=\{f:f=(-\Delta)^{-s/2}g,\, g\in L^p\},$$
where $(I-\Delta)^{s/2}$ and $...
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Exponential decay of a convolution
Let $z=(x,y) \in \mathbb{R}^N \times (0,+\infty)$, and let
$$
P_m(z)=y^{2s} |z|^{-\frac{N+2s}{2}} K_{\frac{N+2s}{2}}(m|z|),
$$
where $N \geq 3$ is an integer, $0<s<1$ and $K_{\frac{N+2s}{2}}$ ...
- 459
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Comparison of Bessel Capacities
The Bessel kernels $G_{\alpha},\, \alpha>0$ are defined by their Fourier transform
$
\hat G_{\alpha}(\xi):= \frac 1 { (1+4\pi ^{2}\vert \xi \vert ^{2})^{\alpha/2}}.
$
Bessel $(\alpha, p)$-capacity ...
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Is the distribution $f\mapsto \int_{S} \frac{\partial^i }{\partial \nu^i}f\,\mathrm{dvol}$ in a Bessel potential space?
In order to finish a paper on 'metric space magnitude' I need to prove that a certain distribution on $\mathbb{R}^{2p+1}$ is in Mark Meckes' weighting space (see
Magnitude, Diversity, Capacities, and ...
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Some detail in Fefferman's thesis
Recently, I am reading "Inequalities for strongly singular convolution operator" written by Fefferman. I have some question on the detail of proof of Theorem 2'.
Let $\theta \in (0,1)$.
Let $f \in ...
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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 ...
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Description of Bessel potential spaces
Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by
$$
\mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n).
$$...
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Sum of two random variables following K0 (modified 2nd kind Bessel) distributions
If X and Y follow independently a density distribution represented by the function $\tfrac{1}{\pi} K_0\left(\tfrac{|x|}{a^2}\right)$ (a modified Bessel function of the second kind), then the sum $Z = ...
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Bessel identities
Please help me prove the following identity
$$
a(J_1(a)Y_0(a)-J_0(a)Y_1(a))=\frac{2}{\pi}
$$
for any $a$.
$J$ and $Y$ are bessel functions of the first and second kind respectively.
Thank you.
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Bessel Potential Space inequality
The Bessel Potential Space is defined for $s\in\mathbb{R}$ as,
$H^s(\mathbb{R}^d) = \{f\in L_2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L_2(\mathbb{R}^n)\}.
$
This defines a Hilbert space ...
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