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Where can I find embedding results for the Bourgain spaces $X^{s,b}$ (for a definition see the bottom of page 2 here).

In particular, I'd like to know if, for $s$ sufficiently large, it is contained in the space of regular functions $C^k(\mathbb{R}^2)$.

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    $\begingroup$ Not unless $b$ is large as well. The decay at $\infty$ in the $\tau$ direction for fixed $\xi$ is controlled by $b$ only, ergo... $\endgroup$
    – fedja
    Commented Dec 1, 2017 at 1:57
  • $\begingroup$ @fedja Does it suffice to take $b \in (1/2,1)$ to get at least $C^1$ regularity with respect to the time variable? $\endgroup$
    – user60665
    Commented Dec 1, 2017 at 2:03
  • $\begingroup$ For a weight $w$, you have $\mathcal F(L^2(w))\subset C^1$ if and only if $\int u(y)^2w(y)\,dy<+\infty$ implies $\int u(y)|y|\,dy<+\infty$. To figure out whether the implication holds or not is a task for business calculus students. We usually prefer to challenge them with much harder exercises like finding patient's temperature from the formula that gives the result above the boiling point of water if the patient manages to survive beyond the first day in the hospital. That one only a genius can solve, but if you can read Bourgain, the task in question should be no brainer for you ;-) $\endgroup$
    – fedja
    Commented Dec 1, 2017 at 15:48
  • $\begingroup$ I meant $1+|y|$, of course :) $\endgroup$
    – fedja
    Commented Dec 1, 2017 at 15:58

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