The Bourgain space $X^{s,b}$ is the closure of the set of Schwartz functions $\mathcal S_{t,x}$ under the norm
$$\|u\|_{X^{s,b}}:=\|\langle\xi\rangle^s\langle\tau-h(\xi)\rangle^b\hat u(\tau,\xi)\|_{L_{\tau,\xi}^2}$$
for $\{s,b\}\subset\mathbb R$ with $\langle\cdot\rangle:=1+|\cdot|$ and a continuous function $h$. Define its time-restricted variant as
$$\|u\|_{X_I^{s,b}}:=\inf\{\|v\|_{X^{s,b}}:u=v\;\text{on}\;I\}$$
for a time interval $I$. It is clear from the definition that $\|u\|_{X_I^{s,b}}\le\|u\|_{X^{s,b}}$, but I am confused about what, loosely speaking, the $v$ that gives the infimum, say $v_{\inf}$, should look like. Since $\|𝟙_I(t)f(x)\|_{X^{s,b}}=\infty$ for $\|f\|_{L^2}>0$ and $b\ge1/2$, $v_{\inf}$ cannot be $𝟙_I(t)u$, at least for $b\ge1/2$. Say $\eta_i\in\mathcal S(\mathbb R)$ for $i\in\mathbb N$ such that $\eta_i=1$ on $I$. We can pick $\eta_i$ such that $\|\eta_i(t)u\|_{X^{s,b}}\to\|u\|_{X_I^{s,b}}$ as $i\to\infty$. Here are my questions. For convenience let $I=[0,1]$.
- Does $\eta_i$ point-wise converge to the characteristic function $𝟙_I$ as $i\to\infty$?
- Can we pick $\eta_i$ such that they are supported on $[-\epsilon,1+\epsilon]$ for arbitrarily small $\epsilon>0$?
- If $\eta_i(t)\ge\eta_j(t)$ for all $t\in\mathbb R$, can we say $\|\eta_i(t)u\|_{X^{s,b}}\ge\|\eta_j(t)u\|_{X^{s,b}}$?
- Could it possibly be $\|u\|_{X_I^{s,b}}=\|𝟙_Iu\|_{X^{s,b}}$ for all $u\in X^{s,b}$ for some $b<1/2$?
Best Answer
Consider for example, $X^{0,1}_{[0,1]}$ and for simplicity assume that $h=0$. Suppose that for $u \in X^{0,1}_{[0,1]}$, we want to find some $\tilde{u} \in X^{0,1}$ such that $\lVert \tilde{u}\rVert_{X^{0,1}}=\lVert u\rVert_{X^{0,1}_{[0,1]}}$. Since $\lVert w\rVert^{2}_{X^{0,1}}=\lVert w\rVert^{2}_{L^{2}}+\lVert \partial_{t}w\rVert^{2}_{L^{2}}$ this problem is equivalent to minimization problems \begin{equation*} \inf_{w(1,x)=u(1,x)}\int_{[1,\infty)\times\mathbb{R}}|\partial_{t}w|^{2}+|w|^{2}dtdx \end{equation*} and \begin{equation*} \inf_{w(0,x)=u(0,x)}\int_{(-\infty,0]\times\mathbb{R}}|\partial_{t}w|^{2}+|w|^{2}dtdx \end{equation*} which can be solved using standard variational techniques. Solving this, we can see that for \begin{equation*} \tilde{u}(t,x)=\begin{cases} e^{t}u(0,x) &t\leq 0\\ \\ u(t,x) &0<t<1\\ \\ e^{-t+1}u(1,x) &t\geq 1 \end{cases}, \end{equation*} we have $\lVert \tilde{u}\rVert_{X^{0,1}}=\lVert u\rVert_{X^{0,1}_{[0,1]}}$.