A note on an exponential semilinear equation of the fourth by Mugnai D.

By Mugnai D.

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72). + 1 - s~) < IlfH~MO~ and Ek(S~ - s2k_,) < Ilfll2BMO;. )IIBMo;. 57 So, the sequence (s~) resp. 70) with r = 1 and B = Itfll~MO2 resp. 48. 73) was proved by Burkholder [19] and by Garsia [82]. Now we are ready to prove the equivalence between the B M O q spaces (1 < q < oe). This result can be extended to all 0 < q < cr too. , B M O ~ for every 0 < r < er Furthermore, opposed to the Garsia's conjecture we show that the BMO~r spaces (0 < r < oe) are all equivalent to B M 0 2 (see Weisz [194]).

9 T h e dual of H$ (1 < p < oo) was characterized for discrete parameter by Herz [95] and for continuous parameter by Pratelli [147]. We follow the proof belonging to Pratelli. 17). First we prove that the H$ spaces are uniformly convex. 25. If 2 < q < oo then the H i spaces are uniformly convex. Proof. We remark that a Banach space X is uniformly convex if for all e > 0 there exists 6 > 0 such that the properties x,y E X , ]Ix H _< 1, [Iyll < 1 and [Ix - y[[ > e imply IIx + yll -< 2(1 - 6). We shall use the following well-known inequalities.

20): IlfllH; <-- Ilhllg; + Ilg[l"; -< Ilhll~, + Cpllglle, <- Cpllfl[ns. 20) can be proved in the same way for all f E H i if 1 _< p _ 2. 12, the well known Davis's inequality (case p = 1) has already been proved. 20) holds for all f E K0 if p > 2. 24). 23), consequently, Ilfll~,s < 2'/=[[gll~e/~ + 2P/ellfll~;. 26) that p/2 p/2 p/2 p/2 I l f l ] ~ - 2P/2(IIfIIPH; + Cp/22 [[f[[H; ]lfllH s ) < 0. 20) for p > 2 F and for f E K0. 20) has been verified for p = 1 and, moreover, for 1 < p < 0% if f 9 K0.

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