X^s,b spaces

The ${\displaystyle X^{s,b}}$ spaces (also called ${\displaystyle H^{s,\theta }}$ spaces) are function spaces for functions on spacetime, which are adapted to linear constant-coefficient dispersive or wave operators such as the Schrodinger operator ${\displaystyle i\partial _{t}+\Delta }$ or the d'Lambertian ${\displaystyle \Box }$ in much the same way that Sobolev spaces are adapted to elliptic operators such as the Laplacian ${\displaystyle \Delta }$.

Given a dispersion relation ${\displaystyle \tau =h(\xi )}$, the ${\displaystyle X^{s,b}(\mathbb {R} \times \mathbb {R} ^{d})}$ space is defined for real ${\displaystyle s,b}$ as the closure of the test functions under the norm

${\displaystyle \|u\|_{X^{s,b}}=\|<\xi >^{s}<\tau -h(\xi )>^{b}{\hat {u}}(\tau ,\xi )\|_{L_{\tau ,\xi }^{2}}.}$

For second-order equations such as the wave equation, where the dispersion relation takes on two values (e.g. ${\displaystyle \tau =\pm |\xi |}$), one must modify this slightly, leading to the norm

${\displaystyle \|u\|_{X^{s,b}}=\|<\xi >^{s}<|\tau |-|\xi |>^{b}{\hat {u}}(\tau ,\xi )\|_{L_{\tau ,\xi }^{2}}.}$

Also it turns out to be useful in the wave equation setting to introduce the variant norm

${\displaystyle \|u\|_{{\mathcal {X}}^{s,b}}:=\|\nabla _{t,x}u\|_{X^{s-1,b}}.}$

These spaces are especially useful for establishing a strong local-wellposedness theory in regularities that are only barely above the critical regularity, and they can often cope with semilinear equations which contain derivatives (in contrast to Strichartz spaces, which often needs much more regularity or cannot close an iteration at all). At critical regularities one needs more refined Besov-type blends of the ${\displaystyle X^{s,b}}$ and Strichartz spaces.

For the local-in-time analysis one often needs to localize the ${\displaystyle X^{s,b}}$ space to a spacetime slab such as ${\displaystyle I\times \mathbb {R} ^{d}}$.

History

These spaces and estimates first appear in the context of the Schrodinger estimates in Bo1993b, although the analogous spaces for the wave equation appeared earlier RaRe1982, Be1983 in the context of propagation of singularities. See also Bo1993, KlMa1993.