# Difference between revisions of "Linear Airy estimates"

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* If <math>u \in X^{0,1/2+}</math> on '''R''', then | * If <math>u \in X^{0,1/2+}</math> on '''R''', then | ||

** u <math> \in L^\infty_t L^2_x </math>(energy estimate) | ** u <math> \in L^\infty_t L^2_x </math>(energy estimate) | ||

− | ** <span class="SpellE">D_x</span>^{1/4} u is in L^4_t <span class="SpellE">BMO_x</span> (endpoint <span class="SpellE">Strichartz</span>) [[ | + | ** <span class="SpellE">D_x</span>^{1/4} u is in L^4_t <span class="SpellE">BMO_x</span> (endpoint <span class="SpellE">Strichartz</span>) [[Bibliography#KnPoVe1993|KnPoVe1993]] |

− | ** <span class="SpellE">D_x</span> u is in L^\<span class="SpellE">infty_x</span> L^2_t (sharp Kato smoothing effect) [[ | + | ** <span class="SpellE">D_x</span> u is in L^\<span class="SpellE">infty_x</span> L^2_t (sharp Kato smoothing effect) [[Bibliography#KnPoVe1993|KnPoVe1993]]. Earlier versions of this estimate were obtained in [[Bibliography#Ka1979b|Ka1979b]], [[Bibliography#KrFa1983|KrFa1983]]. |

− | ** <span class="SpellE">D_x</span>^{-1/4} u is in L^4_x L^\<span class="SpellE">infty_t</span> (Maximal function) [[ | + | ** <span class="SpellE">D_x</span>^{-1/4} u is in L^4_x L^\<span class="SpellE">infty_t</span> (Maximal function) [[Bibliography#KnPoVe1993|KnPoVe1993]], [[Bibliography#KnRu1983|KnRu1983]] |

− | ** <span class="SpellE">D_x</span>^{-3/4-} u is in L^2_x L^\<span class="SpellE">infty_t</span> (L^2 maximal function) [[ | + | ** <span class="SpellE">D_x</span>^{-3/4-} u is in L^2_x L^\<span class="SpellE">infty_t</span> (L^2 maximal function) [[Bibliography#KnPoVe1993|KnPoVe1993]] |

** ''Remark''<nowiki>: Further estimates are available by </nowiki><span class="SpellE">Sobolev</span>, differentiation, Holder, and interpolation. For instance: | ** ''Remark''<nowiki>: Further estimates are available by </nowiki><span class="SpellE">Sobolev</span>, differentiation, Holder, and interpolation. For instance: | ||

− | *** <span class="SpellE">D_x</span> u is in L^2_{<span class="SpellE">x,t</span>} locally in space [[ | + | *** <span class="SpellE">D_x</span> u is in L^2_{<span class="SpellE">x,t</span>} locally in space [[Bibliography#Ka1979b|Ka1979b]] - use Kato and Holder (can also be proven directly by integration by parts) |

*** u is in L^2_{<span class="SpellE">x,t</span>} locally in time - use energy and Holder | *** u is in L^2_{<span class="SpellE">x,t</span>} locally in time - use energy and Holder | ||

*** <span class="SpellE">D_x</span>^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato | *** <span class="SpellE">D_x</span>^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato | ||

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*** <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with endpoint <span class="SpellE">Strichartz</span>. (In particular, <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is also in L^4_{<span class="SpellE">x,t</span>}). | *** <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with endpoint <span class="SpellE">Strichartz</span>. (In particular, <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is also in L^4_{<span class="SpellE">x,t</span>}). | ||

*** u is in L^8_{<span class="SpellE">x,t</span>}- use previous and <span class="SpellE">Sobolev</span> in space | *** u is in L^8_{<span class="SpellE">x,t</span>}- use previous and <span class="SpellE">Sobolev</span> in space | ||

− | *** If u is in X^{0,1/3+}, then u is in L^4_{<span class="SpellE">x,t</span>} [[ | + | *** If u is in X^{0,1/3+}, then u is in L^4_{<span class="SpellE">x,t</span>} [[Bibliography#Bo1993b|Bo1993b]] - interpolate previous with the trivial identity X^{0,0} = L^2 |

− | *** If u is in X^{0,1/4+}, then <span class="SpellE">D_x</span>^{1/2} u is in L^4_x L^2_t [[ | + | *** If u is in X^{0,1/4+}, then <span class="SpellE">D_x</span>^{1/2} u is in L^4_x L^2_t [[Bibliography#Bo1993b|Bo1993b]] - interpolate Kato with X^{0,0} = L^2 |

* If u is in X^{0,1/2+} on '''T''', then | * If u is in X^{0,1/2+} on '''T''', then | ||

** <span class="GramE">u</span> is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate). This is also true in the large period case. | ** <span class="GramE">u</span> is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate). This is also true in the large period case. | ||

− | ** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time (in fact one only needs u in X^{0,1/3} for this) [[ | + | ** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time (in fact one only needs u in X^{0,1/3} for this) [[Bibliography#Bo1993b|Bo1993b]]. |

− | ** <span class="SpellE">D_x</span><span class="GramE">^{</span>-\<span class="SpellE">eps</span>} u is in L^6_{<span class="SpellE">x,t</span>} locally in time. [[ | + | ** <span class="SpellE">D_x</span><span class="GramE">^{</span>-\<span class="SpellE">eps</span>} u is in L^6_{<span class="SpellE">x,t</span>} locally in time. [[Bibliography#Bo1993b|Bo1993b]]. It is conjectured that this can be improved to L^8_{<span class="SpellE">x<span class="GramE">,t</span></span>}. |

** ''Remark''<nowiki>: there is no smoothing on the circle, so one can never gain regularity.</nowiki> | ** ''Remark''<nowiki>: there is no smoothing on the circle, so one can never gain regularity.</nowiki> | ||

* If u is in X^{0,1/2} on a circle with large period \lambda, then | * If u is in X^{0,1/2} on a circle with large period \lambda, then | ||

** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time, with a bound of \lambda^{0+}. | ** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time, with a bound of \lambda^{0+}. | ||

− | *** In fact, when u has frequency N, the constant is like \lambda^{0+} (N<span class="GramE">^{</span>-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. [[ | + | *** In fact, when u has frequency N, the constant is like \lambda^{0+} (N<span class="GramE">^{</span>-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. [[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]]. |

[[Category:Airy]] | [[Category:Airy]] | ||

[[Category:Estimates]] | [[Category:Estimates]] |

## Revision as of 16:25, 31 July 2006

The following linear estimates for the Airy equation are known:

- If on
**R**, then- u (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 case.
- 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. It is conjectured that this can be improved to L^8_{x,t}.
*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. CoKeStTkTa-p2.

- u is in L^4_{x,t} locally in time, with a bound of \lambda^{0+}.