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

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
spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.
Schrodinger Linear estimates
[More references needed here!]
On
:
- If
, then
- (Energy estimate)

- (Strichartz estimates)
Sz1977.
- More generally, f is in
whenever
, and
- The endpoint
is true for
KeTa1998. When
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
one also has 
- When
one can refine the
assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
- When
the
estimate has a maximizer Kz-p2. This maximizer is in fact given by Gaussian beams, with a constant of
Fc-p4. Similarly when
with the
estimate, which is also given by Gaussian beams with a constant of 
- (Kato estimates) When
,
Sl1987, Ve1988, ConSau1988.
- When
we instead have

- (Maximal function estimates) In all dimensions one has
for all
- When
one also has

- When
one also has
The
can be raised to
TaVa2000b, with the corresponding loss in the
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
, then
is also
.
- (Decay estimate) If
, then
has an
norm of 
- Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.
On T:
embds into
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 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:
and BkOgPo1998
Also, if u has frequency
and v has frequency
then we have (see e.g. CoKeStTkTa2003b)
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:
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
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, CoDeKnSt-p.