All Questions
4 questions
0
votes
1
answer
105
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,...
- 481
3
votes
2
answers
413
views
A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
- 351
3
votes
1
answer
262
views
Low/high-frequency estimates in $\mathrm{L}^\infty$ for Lipschitz nonlinearities
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. ...
- 63
-3
votes
1
answer
230
views
$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...
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