Schrodinger estimates

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.

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 $d\geq 3\,$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 ${\dot {D}}^{1/2}\,$ $f\in L_{x}^{\infty }L_{t}^{2}.$
  • (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
• 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 $L_{x,t}^{4}$ 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 $d\geq 1,X^{d/4-1/2+,1/2+}\,$ 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 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

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

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.