# Difference between revisions of "Schrodinger estimates"

## Schrodinger estimates

r 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 $f\in X_{}^{0,1/2+}$ , then
• (Energy estimate) $f\in L_{t}^{\infty }L_{x}^{2}.$ • (Strichartz estimates) $f\in L_{x,t}^{2(d+2)/d}$ Sz1977.
• More generally, f is in $L_{t}^{q}L_{x}^{r}$ whenever $1/q+n/2r=n/4,r<\infty$ , and $q>2\,.$ • The endpoint $q=2,r=2d/(d-2)\,$ is true for $d>=3\,$ 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) $D^{1/2}\,$ $f\in L_{x,loc}^{2}L_{t}^{2}$ 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 $D^{-s}f\in L_{x,loc}^{2}L_{t}^{\infty }$ for all $s>1/2.\,$ • 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]
• (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 $d>=1,X^{d/4-1/2+,1/2+}\,$ embeds into $L_{x,t}^{4}$ (this is essentially in Bo1993)
• The endpoint $d/4-1/2\,$ is probably false in every dimension.

Strichartz estimates are also available on [#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:
$\|uv\|_{X^{1/2+,0}}\leq \|u\|_{X^{1/2+,1/2+}}\|v\|_{X^{0+,1/2+}}$ $\|{\underline {u}}{\underline {v}}\|_{X^{0,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}$ $\|{\underline {u}}{\underline {v}}\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}$ $\|uv\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}$ $\|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}$ $\|{\underline {u}}{\underline {v}}\|_{X^{-3/4-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}$ $\|uv\|_{X^{-3/4+,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}$ $\|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}$ 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 $|\epsilon |\approx R\,$ and v has frequency $|\epsilon |< 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.