# Schrodinger estimates

## Schrodinger estimates

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

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

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

## Schrodinger Linear estimates

[More references needed here!]

On ${\displaystyle R^{d}}$:

• If ${\displaystyle f\in X_{}^{0,1/2+}}$, then
• (Energy estimate) ${\displaystyle f\in L_{t}^{\infty }L_{x}^{2}.}$
• (Strichartz estimates) ${\displaystyle f\in L_{x,t}^{2(d+2)/d}}$ Sz1977.
• More generally, f is in ${\displaystyle L_{t}^{q}L_{x}^{r}}$ whenever ${\displaystyle 1/q+n/2r=n/4,r<\infty }$, and ${\displaystyle q>2\,.}$
• The endpoint ${\displaystyle q=2,r=2d/(d-2)\,}$ is true for ${\displaystyle d\geq 3\,}$KeTa1998. When ${\displaystyle d=2\,}$ 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 ${\displaystyle d=1\,}$ one also has ${\displaystyle f\in L_{t}^{4}L_{x}^{\infty }.}$
• When ${\displaystyle d=1\,}$ one can refine the ${\displaystyle L^{2}\,}$ assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
• When ${\displaystyle d=1\,,}$ the ${\displaystyle L_{t,x}^{6}}$ estimate has a maximizer Kz-p2. This maximizer is in fact given by Gaussian beams, with a constant of ${\displaystyle 12^{-1/12}\,}$ Fc-p4. Similarly when ${\displaystyle d=2\,}$ with the ${\displaystyle L^{4}\,}$ estimate, which is also given by Gaussian beams with a constant of ${\displaystyle 2^{-1/2}\,.}$
• (Kato estimates) When ${\displaystyle d>1}$, ${\displaystyle D^{1/2}\,}$ ${\displaystyle f\in L_{x,loc}^{2}L_{t}^{2}}$ Sl1987, Ve1988, ConSau1988.
• When ${\displaystyle d=1\,}$ we instead have ${\displaystyle {\dot {D}}^{1/2}\,}$ ${\displaystyle f\in L_{x}^{\infty }L_{t}^{2}.}$
• (Maximal function estimates) In all dimensions one has ${\displaystyle D^{-s}f\in L_{x,loc}^{2}L_{t}^{\infty }}$ for all ${\displaystyle s>1/2.\,}$
• When ${\displaystyle d=1\,}$ one also has ${\displaystyle D^{-1/4}\,}$ ${\displaystyle f\in L_{x}^{4}L_{t}^{\infty }.}$
• When ${\displaystyle d=2\,}$ one also has ${\displaystyle D^{-1/2}\,}$ ${\displaystyle f\in L_{x}^{4}L_{t}^{\infty }.}$ The ${\displaystyle -1/2\,}$ can be raised to ${\displaystyle -1/2+1/32+\epsilon \,}$ TaVa2000b, with the corresponding loss in the ${\displaystyle L^{4}\,}$ exponent dictated by scaling. Improvements are certainly possible.
• Variants of some of these estimates exist for manifolds, see BuGdTz-p
• (Energy estimate) If ${\displaystyle f\in L^{4}}$, then ${\displaystyle f\,}$ is also ${\displaystyle \in L^{2}\,}$.
• (Decay estimate) If ${\displaystyle f(0)\in L^{1}}$, then ${\displaystyle f(t)\,}$ has an ${\displaystyle L^{\infty }}$ norm of ${\displaystyle O(t^{-d/2}).\,}$
• Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

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

On ${\displaystyle T^{d}\,}$:

• When ${\displaystyle d\geq 1,X^{d/4-1/2+,1/2+}\,}$ embeds into ${\displaystyle L_{x,t}^{4}}$ (this is essentially in Bo1993)
• The endpoint ${\displaystyle d/4-1/2\,}$ 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:
${\displaystyle \|uv\|_{X^{1/2+,0}}\leq \|u\|_{X^{1/2+,1/2+}}\|v\|_{X^{0+,1/2+}}}$
${\displaystyle \|{\underline {u}}{\underline {v}}\|_{X^{0,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}}$
${\displaystyle \|{\underline {u}}{\underline {v}}\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}}$
${\displaystyle \|uv\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}}$
${\displaystyle \|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}}$
${\displaystyle \|{\underline {u}}{\underline {v}}\|_{X^{-3/4-,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}}$
${\displaystyle \|uv\|_{X^{-3/4+,-1/2+}}\leq \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}}$
${\displaystyle \|u{\underline {v}}\|_{X^{-1/4+,-1/2+}}\leq \|u\|_{X^{-1/4+,1/2+}}\|v\|_{X^{-1/4+,1/2+}}}$

and BkOgPo1998

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

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

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

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

• 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 ${\displaystyle T}$ are also known KnPoVe1996b:
${\displaystyle \|{\underline {u}}{\underline {v}}\|_{X^{-1/2-,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}}$
${\displaystyle \|uv\|_{X^{-3/4+,-1/2+}}\leq \|u\|_{X^{-1/2+,1/2+}}\|v\|_{X^{-1/2+,1/2+}}}$
${\displaystyle \|u{\underline {v}}\|_{X^{0,-1/2+}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{0,1/2+}}}$

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 ${\displaystyle L^{6}}$ Strichartz inequality Gr-p2:
${\displaystyle \|uvw\|_{X^{0,0}}\leq \|u\|_{X^{0,1/2+}}\|v\|_{X^{-1/4,1/2+}}\|w\|_{X^{1/4,1/2+}}}$

## Schrodinger Multilinear estimates

• In R2 we have the variant
${\displaystyle \|u_{1}...u_{n}\|_{X^{1/2+,1/2+}}\leq \|u_{1}\|_{X^{1+,1/2+}}...\|u_{n}\|_{X^{1+,1/2+}}}$

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