Schrodinger estimates: Difference between revisions

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[[Category:Estimates]]
[[Category:Estimates]]
===Schrodinger Trilinear estimates===
==Schrodinger Trilinear estimates==


* On R we have the following refinement to the <math>L^6/,</math> Strichartz inequality [Gr-p2]:
* On R we have the following refinement to the <math>L^6/,</math> Strichartz inequality [Gr-p2]:

Revision as of 18:29, 28 July 2006

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms or , or in spaces, defined by

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].



Schrodinger Linear estimates

[More references needed here!]

On :

  • If , then
    • (Energy estimate)
    • (Strichartz estimates) references:Sz1997 Sz1977.
      • More generally, f is in whenever , and
        • The endpoint is true for references:KeTa1998 KeTa1998. When it fails even in the BMO case references:Mo1998 Mo1998, although it still is true for radial functions references:Ta2000b Ta2000b, [Stv-p].In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable references:Ta2000b Ta2000b, although there is a limit as to low little regularity one can impose [MacNkrNaOz-p].
        • In the radial case there are additional weighted smoothing estimates available references:Vi2001 Vi2001
        • When one also has
        • When one can refine the assumption on the data in rather technical ways on the Fourier side, see e.g. references:VaVe2001 VaVe2001.
        • When the estimate has a maximizer [Kz-p2].This maximizer is in fact given by Gaussian beams, with a constant of [Fc-p4].Similarly when with the estimate, which is also given by Gaussian beams with a constant of
    • (Kato estimates) references:Sl1987 Sl1987, references:Ve1988 Ve1988
      • When one can improve this to
    • (Maximal function estimates) In all dimensions one has for all
      • When one also has
      • When one also has The can be raised to references:TaVa2000b TaVa2000b, with the corresponding loss in the exponent dictated by scaling. Improvements are certainly possible.
    • Variants of some of these estimates exist for manifolds, see [BuGdTz-p]
  • Fixed time estimates for free solutions:
    • (Energy estimate) If , then is also .
    • (Decay estimate) If , then has an norm of
    • Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

  • embds into references:Bo1993 Bo1993 (see also references:HimMis2001 HimMis2001).
  • embeds into references:Bo1993 Bo1993. One cannot remove the from the exponent, however it is conjectured in references:Bo1993 Bo1993 that one might be able to embed into

On :

  • When embeds into (this is essentially in references:Bo1993 Bo1993)
    • The endpoint is probably false in every dimension.

Strichartz estimates are also available on [#manifold more general manifolds], and in the [#potential presence of a potential].Inhomogeneous estimates are also available off

the line of duality; see [Fc-p2] for a discussion.

Schrodinger Bilinear Estimates

and references:BkOgPo1998 BkOgPo1998

Also, if u has frequency and v has frequency then we have (see e.g. [CoKeStTkTa-p4])

and similarly for .

  • The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R2 estimates. The analogues on are also known references:KnPoVe1996b KnPoVe1996b:

Schrodinger Trilinear estimates

  • On R we have the following refinement to the Strichartz inequality [Gr-p2]:

Schrodinger Multilinear estimates

  • In R2 we have the variant

where each factor is allowed to be conjugated if desired. See references:St1997b St1997b, references:CoDeKnSt-p CoDeKnSt-p.