View source for X^s,b spaces

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The '''<math>X^{s,b}</math> spaces''' (also called '''<math>H^{s,\theta}</math> spaces''')
are function spaces for functions on spacetime, which are adapted to linear
constant-coefficient dispersive or wave operators such as the Schrodinger operator <math>i\partial_t + \Delta</math> or the d'Lambertian <math>\Box</math> in much the same way that Sobolev spaces are adapted to elliptic operators such as the Laplacian <math>\Delta</math>.

Given a [[dispersion relation]] <math>\tau = h(\xi)</math>, the <math>X^{s,b}(\R \times \R^d)</math> space is defined for real <math>s,b</math> as the closure of the test functions under the norm
<center><math>\| u \|_{X^{s,b}} = \| <\xi>^s <\tau-h(\xi)>^b \hat{u}(\tau, \xi) \|_{L^2_{\tau, \xi}}.</math></center>
For second-order equations such as the wave equation, where the dispersion relation takes on two values (e.g. <math>\tau = \pm |\xi|</math>), one must modify this slightly, leading to the norm
<center><math>\| u \|_{X^{s,b}} = \| <\xi>^s <|\tau|-|\xi|>^b \hat{u}(\tau, \xi) \|_{L^2_{\tau, \xi}}.</math></center>
Also it turns out to be useful in the wave equation setting to introduce the variant norm
<center><math>\|u\|_{\mathcal{X}^{s,b}} := \| \nabla_{t,x} u \|_{X^{s-1,b}}.</math></center>

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 estimates|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 <math>X^{s,b}</math> and Strichartz spaces.

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

== 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]].

[[Category:Concept]]