Apple Magazine (23 March 2012)

AppleMagazine is a weekly booklet jam-packed with information, iTunes and Apps studies, interviews and unique articles on whatever and every little thing Apple. Apple journal brings a brand new thought of sunshine, clever, cutting edge interpreting for your fingertips; with a world view of Apple and its impact on our lives – be it relaxation actions, relatives or work-collaborative initiatives. Elegantly designed and hugely interactive, Apple journal also will preserve you up-to-date at the most up-to-date weekly information. It’s that straightforward! It’s all approximately Apple and its all over the world tradition impact, multi functional position, and just one faucet away.

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8) runs over all stopping times. Proof. Let f E bmo 1, T be any stopping time. Tidp E(lf- JTIIFT)dp :S llfllbmo,l{r < oo}i. - JTII 11{ T < oo }l- 1< oo. time and F E FT. 5 Another proof of Fefferman's theorem We have L ~~~ If- frldp :S llf- frFihl{rF 57 < oo}l-l :S f3, II E( If - fr IIFr )lloo :S {3. This proves the remaining half of the assertion. The proof is finished. 0 The action of 1-atoms on bmo 1 , or reciprocally the action of bmo 1 on 1-atoms, is very natural and simple, as shown by following lemma.

Martingale. ,... lfml > n}. 11) Proof. Assume {fn} being uniformly integrable. 8) such that C = supn E(rp(lfnl)) < oo. (lfool)dtt :S Ct:n--+ 0. sup lfn I, M f = n n} = {Rn :S m}, {Mf > n} = {Rn < oo}. :O, we get lfn I = 9n +An, /, g E L 1 , Aoo E L 1 . 11) holding. R,.

7) we get E(lfoo I2 1Fo) ~ E(t 2 1Fo), E(lf:'OI 2 ) ~ E(1 2 ). Thus we have + E(Jfool 2 )~ ~ E(E(lf=- fni 2 1Fn+I))~ + E(if=i 2 )~ ~ E(E(i 2 IFn+I))~ + E(lfool 2 )~ ~ 2E(t 2 )~. E(Jfnl 2 )~ ~ E(Jfoo- fnl 2 )~ This shows f E L 2 . ) The index 2 does not play the role in these arguments, so we can consider the space aKp, 1 ~a~ p ~ oo, Write aKoo as BMOa. The following lemma is the main step of our approach to the behavior of S. 11) Proof. 1 Boundedness of M and S, equivalence between M and S 37 Let 1 be any ~me such that E(lf- fn-li 2 1Fn)::; E(I 2 1Fn), for alln.

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