Schrodinger estimates

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Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms $L^q_t L^r_x$ or $L^r_x L^q_t$ , or in X^{s,b} spaces defined by

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

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 $X s, b$ spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.

Schrodinger Linear estimates

[More references needed here!]

On $R d$ :

If , then
- (Energy estimate) $f \in L^\infty_t L^2_x.$
- (Strichartz estimates) Sz1977.
  - More generally, f is in whenever , and
    - The endpoint $q=2, r = 2d/(d-2)\,$ is true for $d \ge 3\,$ 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 $f \in L^4_tL^\infty_x.$
    - 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 $L^6_{t,x}$ 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) When d > 1, Sl1987, Ve1988, ConSau1988.
  - When $d=1\,$ we instead have $\dot D^{1/2}\,$ $f \in L^\infty_xL^2_t.$
- (Maximal function estimates) In all dimensions one has for all
  - When $d=1\,$ one also has $D^{-1/4}\,$ $f \in L^4_{x}L^\infty_t.$
  - When $d=2\,$ one also has $D^{-1/2}\,$ $f \in L^4_{x}L^\infty_t.$ 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 $L^4_{x,t}$ Bo1993 (see also HimMis2001).
$X^{0+,1/2+}\,$ embeds into $L^6_{x,t}$ 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 $L^{6-}_{x,t}.$

On $T^d\,$ :

When embeds into (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, Ta2001 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

$\| uv \|_{L^\infty_t H^{1/3}_x} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}$

Also, if u has frequency $|\xi| \approx R\,$ and v has frequency $|\eta| << R\,$ then we have (see e.g. CoKeStTkTa2003b)

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

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:

$\| \underline{u}\underline{v} \|_{X^{-1/2-, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}$ $\| uv \|_{X^{-3/4+, -1/2+}} \leq \| u \|_{X^{-1/2+, 1/2+}} \| v \|_{X^{-1/2+, 1/2+}}$ $\| u\underline{v} \|_{X^{0, -1/2+}} \leq \| u \|_{X^{0, 1/2+}} \| v \|_{X^{0, 1/2+}}$

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 $L 6$ 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.

Schrodinger estimates

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