All Questions

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,...

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 ...

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 &|\...

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 ...