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The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm $$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...

Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...

I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while; Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable ...