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) $f\in L_{x,t}^{2(d+2)/d}$ $f\in L_{x,t}^{2(d+2)/d}$ references:Sz1997 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 $d>=3\,$ references:KeTa1998 KeTa1998. When $d=2\,$ 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 $d=1\,$ 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. references:VaVe2001 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) $D^{1/2}\,$ $D^{1/2}\,$ $f\in L_{x,loc}^{2}L_{t}^{2}$ $f\in L_{x,loc}^{2}L_{t}^{2}$ references:Sl1987 Sl1987, references:Ve1988 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 $-1/2+1/32+\epsilon \,$ references:TaVa2000b 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 $L_{x,t}^{4}$ references:Bo1993 Bo1993 (see also references:HimMis2001 HimMis2001).
$X^{0+,1/2+}\,$ embeds into $L_{x,t}^{6}$ references:Bo1993 Bo1993. One cannot remove the $+\,$ from the $0+\,$ exponent, however it is conjectured in references:Bo1993 Bo1993 that one might be able to embed $X^{0,1/2+}\,$ into $L_{x,t}^{6-}.$

On $T^{d}\,$ :

When $d>=1,X^{d/4-1/2+,1/2+}\,$ $d>=1,X^{d/4-1/2+,1/2+}\,$ embeds into $L_{x,t}^{4}$ $L_{x,t}^{4}$ (this is essentially in references:Bo1993 Bo1993)
- The endpoint $d/4-1/2\,$ 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 references:Bo1999 Bo1999:

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

On R² references:St1997 St1997, references:CoDeKnSt-p CoDeKnSt-p, references:Ta-p2 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 references:KnPoVe1996b 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 references:BkOgPo1998 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 $|\epsilon |\approx R\,$ and v has frequency $|\epsilon |<<R\,$ 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 references:KnPoVe1996b KnPoVe1996b: