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==Schrodinger estimates==
==Schrodinger estimates==


Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms <math>L^q_t L^r_x</math> or <math>L^r_x L^q_t</math>, or in [[X^s,b spaces|X^{s,b} spaces]] defined by
::<math>\| u \|_{X^{s,b}} = \| u \|_{s,b} := \| \langle \xi\rangle^s  \langle \tau -|\xi|^2\rangle^b \hat{u} \|_{L^2_{\tau,\xi}}.</math>


Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms <math>L^q_t L^r_x</math> or <math>L^r_x L^q_t<\math>, or in <math>X^{s,b}<\math> 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 <math>X^{s,b}</math> spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. 
 
[[Category:Estimates]]
 
==Schrodinger Linear estimates==


|| u ||s,b = || <x>s  <t-|x|2>^b \hat{u} ||2.
[More references needed here!]


Note that these spaces are not invariant under conjugation.
On <math>R^d</math>:
 
* If <math> f \in X^{0,1/2+}_{}</math>, then
** (Energy estimate) <math>f  \in L^\infty_t L^2_x.</math>
** (Strichartz estimates) <math>f \in L^{2(d+2)/d}_{x,t}</math> [[Sz1977]].
*** More generally, <font face="Symbol">f</font> is in <math>L^q_t L^r_x</math> whenever <math>1/q+n/2r = n/4, r < \infty</math>, and <math>q > 2\,.</math>
**** The endpoint <math>q=2, r = 2d/(d-2)\,</math> is true for <math>d  \ge  3\,</math>[[KeTa1998]]. When <math>d=2\,</math> it fails even in the BMO case [[Mo1998]], although it still is true for radial functions [[Ta2000b]], [[Stv-p]].In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable [[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 [[Vi2001]]
**** When <math>d=1\,</math> one also has <math>f \in L^4_tL^\infty_x.</math>
**** When <math>d=1\,</math> one can refine the <math>L^2\,</math> assumption on the data in rather technical ways on the Fourier side, see e.g. [[VaVe2001]].
**** When <math>d=1\,,</math> the <math>L^6_{t,x}</math> estimate has a maximizer [[Kz-p2]]. This maximizer is in fact given by Gaussian beams, with a constant of <math>12^{-1/12}\,</math> [[Fc-p4]]. Similarly when <math>d=2\,</math> with the <math>L^4\,</math> estimate, which is also given by Gaussian beams with a constant of <math>2^{-1/2}\,.</math>
** (Kato estimates) When <math>d > 1</math>, <math>D^{1/2}\,</math> <math>f \in L^2_{x,loc}L^2_t</math> [[Sl1987]], [[Ve1988]], [[ConSau1988]].
*** When <math>d=1\,</math> we instead have <math>\dot D^{1/2}\,</math> <math>f \in L^\infty_xL^2_t.</math>
** (Maximal function estimates) In all dimensions one has <math>D^{-s} f \in L^2_{x,loc}L^\infty_t</math> for all <math>s > 1/2.\,</math>
*** When <math>d=1\,</math> one also has <math>D^{-1/4}\,</math> <math>f \in L^4_{x}L^\infty_t.</math>
*** When <math>d=2\,</math> one also has <math>D^{-1/2}\,</math> <math>f \in L^4_{x}L^\infty_t.</math> The <math>-1/2\,</math> can be raised to <math>-1/2+1/32+ \epsilon\,</math> [[TaVa2000b]], with the corresponding loss in the <math>L^4\,</math> 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 <math>f \in L^4</math>, then <math>f\,</math> is also <math>\in L^2\,</math>.
** (Decay estimate) If <math>f(0) \in L^1</math>, then <math>f(t)\,</math> has an <math>L^\infty</math> norm of <math>O(t^{-d/2}).\,</math>
** Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.
 
On T:
 
* <math>X^{0,3/8}\,</math> embds into <math>L^4_{x,t}</math> [[Bo1993]] (see also [[HimMis2001]]).
* <math>X^{0+,1/2+}\,</math> embeds into <math>L^6_{x,t}</math> [[Bo1993]]. One cannot remove the <math>+\,</math> from the <math>0+\,</math> exponent, however it is conjectured in [[Bo1993]] that one might be able to embed <math>X^{0,1/2+}\,</math> into <math>L^{6-}_{x,t}.</math>
 
On <math>T^d\,</math>:
 
* When <math>d  \ge  1, X^{d/4 - 1/2+,1/2+}\,</math> embeds into <math>L^4_{x,t}</math> (this is essentially in [[Bo1993]])
** The endpoint <math>d/4 - 1/2\,</math> is probably false in every dimension.
 
Strichartz estimates are also available on [[NLS on manifolds|more general manifolds]], and in the [[NLS with potential|presence of a potential]]. Inhomogeneous estimates are also available off
the line of duality; see [[Fc-p2]] for a discussion.
[[Category:Estimates]]
 
==Schrodinger Bilinear Estimates==
 
* On R<sup>2</sup> we have the bilinear Strichartz estimate [[Bo1999]]:
 
<center><math>\| uv \|_{X^{1/2+, 0}} \leq \| u \|_{X^{1/2+, 1/2+}} \| v \|_{X^{0+, 1/2+}}</math></center>
 
* On R<sup>2</sup> [[St1997]], [[CoDeKnSt-p]], [[Ta2001]] we have the sharp estimates
 
<center><math>\| \underline{u}\underline{v} \|_{X^{0, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}</math></center>
 
<center><math>\| \underline{u}\underline{v} \|_{X^{-1/2-, -1/2+}} \leq \| u \|_{X^{-3/4+, 1/2+}} \| v \|_{X^{-3/4+, 1/2+}}</math></center>
 
<center><math>\| uv \|_{X^{-1/2-, -1/2+}} \leq \| u \|_{X^{-3/4+, 1/2+}} \| v \|_{X^{-3/4+, 1/2+}}</math></center>
 
<center><math>\| u\underline{v} \|_{X^{-1/4+, -1/2+}} \leq \| u \|_{X^{-1/4+, 1/2+}} \| v \|_{X^{-1/4+, 1/2+}}</math></center>
 
* On R [[KnPoVe1996b]] we have
 
<center><math>\| \underline{u}\underline{v} \|_{X^{-3/4-, -1/2+}} \leq \| u \|_{X^{-3/4+, 1/2+}} \| v \|_{X^{-3/4+, 1/2+}}</math></center>
 
<center><math>\| uv \|_{X^{-3/4+, -1/2+}} \leq \| u \|_{X^{-3/4+, 1/2+}} \| v \|_{X^{-3/4+, 1/2+}}</math></center>
 
<center><math>\| u\underline{v} \|_{X^{-1/4+, -1/2+}} \leq \| u \|_{X^{-1/4+, 1/2+}} \| v \|_{X^{-1/4+, 1/2+}}</math></center>
 
and [[BkOgPo1998]]
 
<center><math>\| uv \|_{L^\infty_t H^{1/3}_x} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
 
Also, if u has frequency <math>|\xi| \approx R\,</math> and v has frequency <math>|\eta| << R\,</math> then we have (see e.g. [[CoKeStTkTa2003b]])
 
<center><math>\| uv \|_{X^{1/2, 0}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
 
and similarly for <math>\underline{u}v, u\underline{v}, \underline{uv}\,</math> . <br />
 
* The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R<sup>2</sup> estimates. The analogues on <math>T</math> are also known [[KnPoVe1996b]]:
 
<center><math>\| \underline{u}\underline{v} \|_{X^{-1/2-, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}</math></center>
 
<center><math>\| uv \|_{X^{-3/4+, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}</math></center>
 
<center><math>\| u\underline{v} \|_{X^{0, -1/2+}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
 
In two dimensions, the endpoint linear Strichartz estimate continues to fail in the bilinear setting [[Ta2006c]].
 
[[Category:Estimates]]
 
==Schrodinger Trilinear estimates==
 
* On R we have the following refinement to the <math>L^6</math> Strichartz inequality [[Gr-p2]]:
 
<center><math>\| uvw \|_{X^{0, 0}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{-1/4, 1/2+}} \| w \|_{X^{1/4, 1/2+}}</math></center>
[[Category:Estimates]]
 
==Schrodinger Multilinear estimates==
 
* In R<sup>2</sup> we have the variant
 
<center><math>\| u_{1}...u_{n} \|_{X^{1/2+, 1/2+}} \leq \| u_1 \|_{X^{1+, 1/2+}}...\| u_n \|_{X^{1+, 1/2+}}</math></center>
 
where each factor <math>u_i\,</math> is allowed to be conjugated if desired. See [[St1997b]], [[CoDeKnSt-p]].


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].
[[Category:Estimates]]
[[Category:Schrodinger]]

Latest revision as of 20:29, 17 July 2007

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms or , or in X^{s,b} 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 spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.

Schrodinger Linear estimates

[More references needed here!]

On :

  • If , then
    • (Energy estimate)
    • (Strichartz estimates) Sz1977.
      • More generally, f is in whenever , and
        • The endpoint is true for KeTa1998. When it fails even in the BMO case Mo1998, although it still is true for radial functions Ta2000b, Stv-p.In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable 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 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. 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) When , Sl1987, Ve1988, ConSau1988.
      • When we instead have
    • (Maximal function estimates) In all dimensions one has for all
      • When one also has
      • When one also has The can be raised to 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 Bo1993 (see also HimMis2001).
  • embeds into Bo1993. One cannot remove the from the exponent, however it is conjectured in Bo1993 that one might be able to embed into

On :

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

Strichartz estimates are also available on more general manifolds, and in the presence of a potential. Inhomogeneous estimates are also available off the line of duality; see Fc-p2 for a discussion.

Schrodinger Bilinear Estimates

  • On R2 we have the bilinear Strichartz estimate Bo1999:

and BkOgPo1998

Also, if u has frequency and v has frequency then we have (see e.g. CoKeStTkTa2003b)

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 KnPoVe1996b:

In two dimensions, the endpoint linear Strichartz estimate continues to fail in the bilinear setting Ta2006c.

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 St1997b, CoDeKnSt-p.