By Terence Tao

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7) p(2t) as w and t = 0, 0 increasing < Cp(t) constant A function at for all t and some C. (Note: above is an Orlicz function. 6. 8) < 9(2 max(s,t)) < C¢(max(s,t)) < C(¢(S)+¢(t))Now let Orlicz function. 4) follows in the definition of space determined by The space w. The next theorem is due to Rolewicz Cater [1963]. at 0 w immediately from the continuity of the dominated convergence theorem. Condition A norm. and is the Orlicz L ¢ [1959]; see also -30 p has trivial dual Then is non atomic. u Assume that be an Orlicz function.

0 < 5 < 1/2, so that if 7 , IIka0° S [1980], Koosis I2! > 6 e Hm < l -7 with gku < 5 (l < k < n) -53... 13. S Ml(T) ||

0 < r < l 19 )l p dG/Zn (JEJJEHJELSélgiglLE)P (13/21T Il rWel I + < 2 P(i nw|)P( '°) [02 ui r eiel ( +2)9 dO/Zn < c(1 Iw|)P( '°)(1 le)1'(3+2)9. The last estimate can be deduced from the Lemma on p. Duren, since (3+2)p > 1. Thus 1 IIlel < Cl/p(l lw|)l/p a 2 = c /p. 7. If v(z) proof. v is a polynomial, = 1/ o 2n 0 1 p(re )K(z,re 19 )rdrde. n = z . (3+k+l))/k! )/( lnl) The last equality is obtained by noting that -44- 2 J l O (l-r)23r2 +ldr = I ? (1 u)3u du (by induction). nl) Proof. 8. w anp l < 1.

### Fourier transform by Terence Tao

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