By Jean-Lin Journé (auth.)

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Sample text

_b[ ' ] a ] P-- d x ]-P ' [ f _- _ ] Q Q i [f'b, " mQb I p d x ] g'i 1 1 IQll-~ - mQb I p 'd x ] p ' Q <__llbll ,. _<_Cpllbll. P~ g E L P oc To prove (2), it is enough to show that if Q ~ cube we can find an a = a Q E H I' p is givenj then~ for each such that [~Q~'g-~Q'dXlIIaIIHl, P =< CplfgadxIQ and supp aE Q • We assume, without loss in generality that~ QA[ We define a by g•>%} Ig- ~QIp a=O outside " is a constant chosen so that By the definition of Ig- =QI P' dx , Q , a = C(g,Q) C(g,Q) <%} = QA[ a = ]g-c~QI p ' - I where ~g dx > on Q n [g>c~Q} on Q n [g=

Iii) implies (ii). Let Q be a cube and let aE~Q • so that we want to show ]ITall I <__C . =< ~ , Q • interiors. and s u c h t h a t (That is• and by the above calculation• Qt they are adjacent _g_c lmQ'Ta] <--[QI BMO calculation shows that Im~a-mQ,Ta[ =< ClITalI. <=] C~ . -l]Q] ~Q ITa - m~a]dx + ]m~a - mQ,Ta] + ]mQ,Ta] __c +_c_c <__q_c IQ] = IQ~ <= CIITalI*+ ]QI and Q have a cubes. ) By Now• a standard 51 Putting together these two calculations, we obtain iLx~iLI ~ C , and (ii) is an immed- iate consequence.

That HI~m=HI'P (HI'P)*=BMO imply then that be obtained by using the definition of dense in H I' p . ) We define For a cube Q Hence H I'p let (Hi, m)* in the aforementioned theorem of The Hahn-Banach theorem and the fact for l