Schrodinger estimates: Difference between revisions

Revision as of 21:50, 9 August 2006

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$ or $L_{x}^{r}L_{t}^{q}$ , or in $X^{s,b}$ spaces, defined by

\|u\|_{X^{s,b}}=\|u\|_{s,b}:=\|\langle \xi \rangle ^{s}\langle \tau -|\xi |^{2}\rangle ^{b}{\hat {u}}\|_{L_{\tau ,\xi }^{2}}.

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 $R^{d}$ :

If $f\in X_{}^{0,1/2+}$ $f\in X_{}^{0,1/2+}$ , then
- (Energy estimate) $f\in L_{t}^{\infty }L_{x}^{2}.$
- (Strichartz estimates) $f\in L_{x,t}^{2(d+2)/d}$ $f\in L_{x,t}^{2(d+2)/d}$ Sz1977.
  - More generally, f is in $L_{t}^{q}L_{x}^{r}$ $L_{t}^{q}L_{x}^{r}$ whenever $1/q+n/2r=n/4,r<\infty$ $1/q+n/2r=n/4,r<\infty$ , and $q>2\,.$ $q>2\,.$
    - The endpoint $q=2,r=2d/(d-2)\,$ is true for $d\geq 3\,$ KeTa1998. When $d=2\,$ 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 $d=1\,$ one also has $f\in L_{t}^{4}L_{x}^{\infty }.$
    - When $d=1\,$ one can refine the $L^{2}\,$ assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
    - When $d=1\,,$ the $L_{t,x}^{6}$ estimate has a maximizer Kz-p2. This maximizer is in fact given by Gaussian beams, with a constant of $12^{-1/12}\,$ Fc-p4. Similarly when $d=2\,$ with the $L^{4}\,$ estimate, which is also given by Gaussian beams with a constant of $2^{-1/2}\,.$
- (Kato estimates) $D^{1/2}\,$ $D^{1/2}\,$ $f\in L_{x,loc}^{2}L_{t}^{2}$ $f\in L_{x,loc}^{2}L_{t}^{2}$ Sl1987, Ve1988
  - When $d=1\,$ one can improve this to $D^{1/2}\,$ $f\in L_{x}^{\infty }L_{t}^{2}.$
- (Maximal function estimates) In all dimensions one has $D^{-s}f\in L_{x,loc}^{2}L_{t}^{\infty }$ $D^{-s}f\in L_{x,loc}^{2}L_{t}^{\infty }$ for all $s>1/2.\,$ $s>1/2.\,$
  - When $d=1\,$ one also has $D^{-1/4}\,$ $f\in L_{x}^{4}L_{t}^{\infty }.$
  - When $d=2\,$ one also has $D^{-1/2}\,$ $f\in L_{x}^{4}L_{t}^{\infty }.$ The $-1/2\,$ can be raised to $-1/2+1/32+\epsilon \,$ TaVa2000b, with the corresponding loss in the $L^{4}\,$ 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 $f\in L^{4}$ , then $f\,$ is also $\in L^{2}\,$ .
- (Decay estimate) If $f(0)\in L^{1}$ , then $f(t)\,$ has an $L^{\infty }$ norm of $O(t^{-d/2}).\,$
- Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

$X^{0,3/8}\,$ embds into $L_{x,t}^{4}$ Bo1993 (see also HimMis2001).
$X^{0+,1/2+}\,$ embeds into $L_{x,t}^{6}$ Bo1993. One cannot remove the $+\,$ from the $0+\,$ exponent, however it is conjectured in Bo1993 that one might be able to embed $X^{0,1/2+}\,$ into $L_{x,t}^{6-}.$

On $T^{d}\,$ :

When $d\geq 1,X^{d/4-1/2+,1/2+}\,$ $d\geq 1,X^{d/4-1/2+,1/2+}\,$ embeds into $L_{x,t}^{4}$ $L_{x,t}^{4}$ (this is essentially in Bo1993)
- The endpoint $d/4-1/2\,$ 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:

\|uv\|_{X^{1/2+,0}}\leq \|u\|_{X^{1/2+,1/2+}}\|v\|_{X^{0+,1/2+}}

On R² St1997, CoDeKnSt-p, Ta-p2 we have the sharp estimates

\|{\underline {u}}{\underline {v}}\|_{X^{0,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}

\|{\underline {u}}{\underline {v}}\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}

\|uv\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}

\|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}

On R KnPoVe1996b we have

\|{\underline {u}}{\underline {v}}\|_{X^{-3/4-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}

\|uv\|_{X^{-3/4+,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}

\|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}

and BkOgPo1998

\|uv\|_{L_{t}^{\infty }H_{x}^{1/3}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{0,1/2+}}

Also, if u has frequency $|\xi |\approx R\,$ and v has frequency $|\eta |<<R\,$ then we have (see e.g. CoKeStTkTa-p4)

\|uv\|_{X^{1/2,0}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{0,1/2+}}

and similarly for ${\underline {u}}v,u{\underline {v}},{\underline {uv}}\,$ .

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 $T$ are also known KnPoVe1996b:

\|{\underline {u}}{\underline {v}}\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}

\|uv\|_{X^{-3/4+,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}

\|u{\underline {v}}\|_{X^{0,-1/2+}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{0,1/2+}}

Schrodinger Trilinear estimates

On R we have the following refinement to the $L^{6}/,$ Strichartz inequality Gr-p2:

\|uvw\|_{X^{0,0}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{-1/4,1/2+}}\|w\|_{X^{1/4,1/2+}}

Schrodinger Multilinear estimates

In R² we have the variant

\|u_{1}...u_{n}\|_{X^{1/2+,1/2+}}\leq \|u_{1}\|_{X^{1+,1/2+}}...\|u_{n}\|_{X^{1+,1/2+}}

where each factor $u_{i}\,$ is allowed to be conjugated if desired. See St1997b, CoDeKnSt-p.

@@ Line 1: / Line 1: @@
 ==Schrodinger estimates==
-r
 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
 ::<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>
@@ Line 7: / Line 6: @@
 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.  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].
+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.  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]]
@@ Line 19: / Line 18: @@
 * 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> [[Bibliography#Sz1977 |Sz1977]].
+** (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>[[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].
+**** 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 [[Bibliography#Vi2001|Vi2001]]
+**** 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. [[Bibliography#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. [[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> [[Bibliography#Sl1987|Sl1987]], [[Bibliography#Ve1988|Ve1988]]
+** (Kato estimates) <math>D^{1/2}\,</math> <math>f \in L^2_{x,loc}L^2_t</math> [[Sl1987]], [[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>
 ** (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> [[Bibliography#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> [[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:
 ** (Energy estimate) If <math>f \in L^4</math>, then <math>f\,</math> is also <math>\in L^2\,</math>.
@@ Line 39: / Line 38: @@
 On T:
-* <math>X^{0,3/8}\,</math> embds into <math>L^4_{x,t}</math> [[Bibliography#Bo1993|Bo1993]] (see also [[Bibliography#HimMis2001|HimMis2001]]).
+* <math>X^{0,3/8}\,</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> [[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>
+* <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 [[Bibliography#Bo1993|Bo1993]])
+* 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 [#manifold more general manifolds], and in the [#potential presence of a potential].Inhomogeneous estimates are also available off
+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.
-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 [[Bibliography#Bo1999|Bo1999]]:
+* 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> [[Bibliography#St1997|St1997]], [[Bibliography#CoDeKnSt-p |CoDeKnSt-p]], [[Bibliography#Ta-p2 |Ta-p2]] we have the sharp estimates
+* On R<sup>2</sup> [[St1997]], [[CoDeKnSt-p]], [[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>
@@ Line 68: / Line 66: @@
 <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 [[Bibliography#KnPoVe1996b|KnPoVe1996b]] we have
+* 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>
@@ Line 76: / Line 74: @@
 <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 [[Bibliography#BkOgPo1998|BkOgPo1998]]
+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>|\epsilon| \approx R\,</math> and v has frequency <math>|\epsilon| << R\,</math> then we have (see e.g. [CoKeStTkTa-p4])
+Also, if u has frequency <math>|\xi| \approx R\,</math> and v has frequency <math>|\eta| << R\,</math> then we have (see e.g. [[CoKeStTkTa-p4]])
 <center><math>\| uv \|_{X^{1/2, 0}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
@@ Line 86: / Line 84: @@
 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 [[Bibliography#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 [[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>
@@ Line 94: / Line 92: @@
 <center><math>\| u\underline{v} \|_{X^{0, -1/2+}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}</math></center>
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
 [[Category: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]]:
 <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>
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
 [[Category:Estimates]]
@@ Line 115: / Line 106: @@
 <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 [[Bibliography#St1997b|St1997b]], [[Bibliography#CoDeKnSt-p |CoDeKnSt-p]].
+where each factor <math>u_i\,</math> is allowed to be conjugated if desired. See [[St1997b]], [[CoDeKnSt-p |CoDeKnSt-p]].
 [[Category:Estimates]]
 [[Category:Schrodinger]]

Schrodinger estimates: Difference between revisions

Revision as of 21:50, 9 August 2006

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