PPT Slide

|| ?|| |H s ? 1, s > n/2, tɬ ,

implies

|?(t,x)| ? C(1+ t + |x|)-(n-1)/2 (1+ t - |x|)-1/2

[X , ] = c

METHODS:Generalized Energy, Global Sobolev

S = t ? t + ? i x i ? I

? i j = x i ? j + x j ? i

L i = x i ? t +t ? t

Scaling

Rotations

Boosts

E s [?](t) = ? s k=1 ? X E[X1 ... Xk ?](t)

E s (?)=E s (0)

Define IH s space

Proposition