Schrodinger estimates: Difference between revisions

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** (Strichartz estimates) <math>f \in L^{2(d+2)/d}_{x,t}</math> [[references:Sz1997 Sz1977]].
** (Strichartz estimates) <math>f \in L^{2(d+2)/d}_{x,t}</math> [[references:Sz1997 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>
*** 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 >= 3\,</math>[[references:KeTa1998 KeTa1998]]. When <math>d=2\,</math> 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].
**** The endpoint <math>q=2, r = 2d/(d-2)\,</math> is true for <math>d >= 3\,</math>[[Bibliography#KeTa1998|KeTa1998]]. When <math>d=2\,</math> it fails even in the BMO case [[Bibliography#Mo1998|Mo1998]], although it still is true for radial functions [[Bibliography#Ta2000b|Ta2000b]], [Stv-p].In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable [[Bibliography#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]]
**** In the radial case there are additional weighted smoothing estimates available [[Bibliography#Vi2001|Vi2001]]
**** When <math>d=1\,</math> one also has <math>f \in L^4_tL^\infty_x.</math>
**** 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. [[references:VaVe2001 VaVe2001]].
**** 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. [[Bibliography#VaVe2001|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>
**** 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) <math>D^{1/2}\,</math> <math>f \in L^2_{x,loc}L^2_t</math> [[references:Sl1987 Sl1987]], [[references:Ve1988 Ve1988]]
** (Kato estimates) <math>D^{1/2}\,</math> <math>f \in L^2_{x,loc}L^2_t</math> [[Bibliography#Sl1987|Sl1987]], [[Bibliography#Ve1988|Ve1988]]
*** When <math>d=1\,</math> one can improve this to <math>D^{1/2}\,</math> <math>f \in L^\infty_xL^2_t.</math>
*** When <math>d=1\,</math> one can improve this to <math>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>
** (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=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> [[references:TaVa2000b TaVa2000b]], with the corresponding loss in the <math>L^4\,</math> exponent dictated by scaling. Improvements are certainly possible.
*** 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> [[Bibliography#TaVa2000b|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]
** Variants of some of these estimates exist for manifolds, see [BuGdTz-p]
* Fixed time estimates for free solutions:
* Fixed time estimates for free solutions:
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On T:
On T:


* <math>X^{0,3/8}\,</math> embds into <math>L^4_{x,t}</math> [[references:Bo1993 Bo1993]] (see also [[references:HimMis2001 HimMis2001]]).
* <math>X^{0,3/8}\,</math> embds into <math>L^4_{x,t}</math> [[Bibliography#Bo1993|Bo1993]] (see also [[Bibliography#HimMis2001|HimMis2001]]).
* <math>X^{0+,1/2+}\,</math> embeds into <math>L^6_{x,t}</math> [[references:Bo1993 Bo1993]]. One cannot remove the <math>+\,</math> from the <math>0+\,</math> exponent, however it is conjectured in [[references:Bo1993 Bo1993]] that one might be able to embed <math>X^{0,1/2+}\,</math> into <math>L^{6-}_{x,t}.</math>
* <math>X^{0+,1/2+}\,</math> embeds into <math>L^6_{x,t}</math> [[Bibliography#Bo1993|Bo1993]]. One cannot remove the <math>+\,</math> from the <math>0+\,</math> exponent, however it is conjectured in [[Bibliography#Bo1993|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>:
On <math>T^d\,</math>:


* When <math>d >= 1, X^{d/4 - 1/2+,1/2+}\,</math> embeds into <math>L^4_{x,t}</math> (this is essentially in [[references:Bo1993 Bo1993]])
* When <math>d >= 1, X^{d/4 - 1/2+,1/2+}\,</math> embeds into <math>L^4_{x,t}</math> (this is essentially in [[Bibliography#Bo1993|Bo1993]])
** The endpoint <math>d/4 - 1/2\,</math> is probably false in every dimension.
** The endpoint <math>d/4 - 1/2\,</math> is probably false in every dimension.


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==Schrodinger Bilinear Estimates==
==Schrodinger Bilinear Estimates==


* On R<sup>2</sup> we have the bilinear Strichartz estimate [[references:Bo1999 Bo1999]]:
* On R<sup>2</sup> we have the bilinear Strichartz estimate [[Bibliography#Bo1999|Bo1999]]:


<center><math>\| uv \|_{X^{1/2+, 0}} \leq \| u \|_{X^{1/2+, 1/2+}} \| v \|_{X^{0+, 1/2+}}</math></center>
<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> [[references:St1997 St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]], [[references:Ta-p2 Ta-p2]] we have the sharp estimates
* On R<sup>2</sup> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]], [[references:Ta-p2 Ta-p2]] 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^{0, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}</math></center>
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<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>
<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 [[references:KnPoVe1996b KnPoVe1996b]] we have
* On R [[Bibliography#KnPoVe1996b|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>\| \underline{u}\underline{v} \|_{X^{-3/4-, -1/2+}} \leq \| u \|_{X^{-3/4+, 1/2+}} \| v \|_{X^{-3/4+, 1/2+}}</math></center>
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<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>
<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 [[references:BkOgPo1998 BkOgPo1998]]
and [[Bibliography#BkOgPo1998|BkOgPo1998]]


<center><math>\| uv \|_{L^\infty_t H^{1/3}_x} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
<center><math>\| uv \|_{L^\infty_t H^{1/3}_x} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
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and similarly for <math>\underline{u}v, u\underline{v}, \underline{uv}\,</math> . <br />
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 [[references:KnPoVe1996b KnPoVe1996b]]:
* 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 [[Bibliography#KnPoVe1996b|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>\| \underline{u}\underline{v} \|_{X^{-1/2-, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}</math></center>
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<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>
<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 [[references:St1997b St1997b]], [[references:CoDeKnSt-p CoDeKnSt-p]].
where each factor <math>u_i\,</math> is allowed to be conjugated if desired. See [[Bibliography#St1997b|St1997b]], [[references:CoDeKnSt-p CoDeKnSt-p]].


<div class="MsoNormal" style="text-align: center"><center>
<div class="MsoNormal" style="text-align: center"><center>

Revision as of 20:29, 28 July 2006

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^q_t L^r_x} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^r_x L^q_t} , or in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^{s,b}} spaces, defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \| u \|_{X^{s,b}} = \| u \|_{s,b} := \| \langle \xi\rangle^s \langle \tau -|\xi|^2\rangle^b \hat{u} \|_{L^2_{\tau,\xi}}.}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^d} :

  • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in X^{0,1/2+}_{}} , then
    • (Energy estimate) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^\infty_t L^2_x.}
    • (Strichartz estimates) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^{2(d+2)/d}_{x,t}} references:Sz1997 Sz1977.
      • More generally, f is in whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/q+n/2r = n/4, r < \infty} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q > 2\,.}
        • The endpoint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q=2, r = 2d/(d-2)\,} is true for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d >= 3\,} KeTa1998. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2\,} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1\,} one also has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^4_tL^\infty_x.}
        • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1\,} one can refine the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
        • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1\,,} the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^6_{t,x}} estimate has a maximizer [Kz-p2].This maximizer is in fact given by Gaussian beams, with a constant of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 12^{-1/12}\,} [Fc-p4].Similarly when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2\,} with the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^4\,} estimate, which is also given by Gaussian beams with a constant of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^{-1/2}\,.}
    • (Kato estimates) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D^{1/2}\,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^2_{x,loc}L^2_t} Sl1987, Ve1988
      • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1\,} one can improve this to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D^{1/2}\,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^\infty_xL^2_t.}
    • (Maximal function estimates) In all dimensions one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D^{-s} f \in L^2_{x,loc}L^\infty_t} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1/2.\,}
      • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1\,} one also has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D^{-1/4}\,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^4_{x}L^\infty_t.}
      • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2\,} one also has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D^{-1/2}\,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^4_{x}L^\infty_t.} The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1/2\,} can be raised to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1/2+1/32+ \epsilon\,} TaVa2000b, with the corresponding loss in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^4\,} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \in L^4} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f\,} is also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \in L^2\,} .
    • (Decay estimate) If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(0) \in L^1} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)\,} has an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^\infty} norm of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O(t^{-d/2}).\,}
    • Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^{0,3/8}\,} embds into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^4_{x,t}} 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 [#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

  • 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. [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 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 St1997b, references:CoDeKnSt-p CoDeKnSt-p.

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