Linear Airy estimates

The following linear estimates for the Airy equation are known:

• If $u\in X^{0,1/2+}$ on R, then
  • u $\in L_{t}^{\infty }L_{x}^{2}$(energy estimate)
  • D_x^{1/4} u is in L^4_t BMO_x (endpoint Strichartz) KnPoVe1993
  • D_x u is in L^\infty_x L^2_t (sharp Kato smoothing effect) KnPoVe1993. Earlier versions of this estimate were obtained in Ka1979b, KrFa1983.
  • D_x^{-1/4} u is in L^4_x L^\infty_t (Maximal function) KnPoVe1993, KnRu1983
  • D_x^{-3/4-} u is in L^2_x L^\infty_t (L^2 maximal function) KnPoVe1993
  • Remark: Further estimates are available by Sobolev, differentiation, Holder, and interpolation. For instance:
    • D_x u is in L^2_{x,t} locally in space Ka1979b - use Kato and Holder (can also be proven directly by integration by parts)
    • u is in L^2_{x,t} locally in time - use energy and Holder
    • D_x^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato
    • D_x^{1/6} u is in L^6_{x,t} - interpolate energy with endpoint Strichartz (or Kato with maximal)
    • D_x^{1/8} u is in L^8_t L^4_x - interpolate energy with endpoint Strichartz. (In particular, D_x^{1/8} u is also in L^4_{x,t}).
    • u is in L^8_{x,t}- use previous and Sobolev in space
    • If u is in X^{0,1/3+}, then u is in L^4_{x,t} Bo1993b - interpolate previous with the trivial identity X^{0,0} = L^2
    • If u is in X^{0,1/4+}, then D_x^{1/2} u is in L^4_x L^2_t Bo1993b - interpolate Kato with X^{0,0} = L^2
• If u is in X^{0,1/2+} on T, then
  • u is in L^\infty_t L^2_x (energy estimate). This is also true in the large period limit.
  • u is in L^4_{x,t} locally in time (in fact one only needs u in X^{0,1/3} for this) Bo1993b.
  • D_x^{-\eps} u is in L^6_{x,t} locally in time. Bo1993b.
  • Remark: there is no smoothing on the circle, so one can never gain regularity.
• If u is in X^{0,1/2} on a circle with large period \lambda, then
  • u is in L^4_{x,t} locally in time, with a bound of \lambda^{0+}.
    • In fact, when u has frequency N, the constant is like \lambda^{0+} (N^{-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. CoKeStTkTa2003.

L^6 conjecture

It is conjectured that if u is a free L^2 solution to the Airy equation on $\mathbb {T}$, then $u$ is in $L_{t,x}^{6}([0,1]\times \mathbb {T} )$ for any $\epsilon >0$. It is known that the claim holds if u is replaced by $\langle \nabla \rangle ^{-\epsilon }u$ Bo1993b.

L^8 conjecture

It is conjectured that if u is a free L^2 solution to the Airy equation on $\mathbb {T}$, then $\langle \nabla \rangle ^{-\epsilon }u$ is in $L_{t,x}^{8}([0,1]\times \mathbb {T} )$ for any $\epsilon >0$. It is known that the epsilon cannot be completely removed Bo1993b.