By Ferenc Weisz

The background of martingale concept is going again to the early fifties whilst Doob [57] mentioned the relationship among martingales and analytic services. at the foundation of Burkholder's medical achievements the mar tingale thought can completely good be utilized in advanced research and within the idea of classical Hardy areas. This connection is the most aspect of Durrett's ebook [60]. The martingale conception is additionally good utilized in stochastics and mathematical finance. The theories of the one-parameter martingale and the classical Hardy areas are mentioned exhaustively within the literature (see Garsia [83], Neveu [138], Dellacherie and Meyer [54, 55], lengthy [124], Weisz [216] and Duren [59], Stein [193, 194], Stein and Weiss [192], Lu [125], Uchiyama [205]). the speculation of more-parameter martingales and martingale Hardy areas is investigated in Imkeller [107] and Weisz [216]. this is often the 1st mono graph which considers the idea of more-parameter classical Hardy areas. The equipment of proofs for one and a number of other parameters are en tirely diverse; mostly the theorems acknowledged for a number of parameters are even more tricky to make sure. The so-called atomic decomposition approach that may be utilized either within the one-and more-parameter instances, used to be thought of for martingales through the writer in [216].

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1 } Ig d)' Ig d)', _ Now the interpolation spaces between the Hardy spaces H~ can be identified. 25 110 < {} < 1,0 < Po::; 1 and 0 < q ~ 1 P = 00 then 1- {} Po 10 Proof. Note that H~ = Loo. Let I E H~q and be the non-increasing rearrangement of 10. Set I/O! 24 by gt and ht . ). 41) 36 1. 24 we get that Consequently, < C 10 1 C 9q C 10 1 (lo tQ jO(X)POdXf/PO t(1-9)q/po (~ ~t lot jO(X)PO dX) q/po ~t. 39) we obtain {I ( C9 1I htIIHD )q dt ::; C 10 PO t e 10 t(1-9)q/po jo(t)q dt = t Cll/oll:,q. Furthermore, Substituting u = to< we can see that e {I (t l - 91Igt IlHD)q dt ::; C u(1-9)q/po jo(u)q du = Cll/oll:,q.

In this section the interpolation spaces between the martingale Hardy-Lorentz spaces are identified. The basic definitions and theorems of the interpolation theory in the real method are given without proofs as follows. For the details see Bennett and Sharpley [6] and Bergh and Lofstrom [9]. Suppose that Ao and Al are quasi-normed spaces embedded continuously in a topological vector space A. 5. q ):= (1 00 0 [t -0 dt) I/q K(t, f, Ao, Ar)lq -t < 00 where 0 < e < 1 and 0 < q :s; 00. We use the conventions (Ao, Ado,q = Ao and (Ao, AIh,q = Al for each 0 < q :s; 00.

30) into two parts: L (A) = E(ldnfI21{vk~n»vk+d1{S(f)9k+l}) nENd and L (B) = E(ldnfI21{Vk~n»vk+l}1{S(J»2k+l}). 31) It is easy to show that Idnfl is F n- 1 measurable (n ENd). Since the set {Vk « n t> Vk+l} (B) = is also Fn-l measurable, we obtain L E(ldnfI21{Vk~n»vk+dEn-11{s(J»2k+l}). nENd It follows from the definition of Vk+l that, if Vk+l So (B) ::; %:. 29). Hence there exists a function a k E L2 such that Enak = a~ (n ENd) and (ii) holds. 26 1. 25) we can show that L IJLkl P ::; CpE(S(J)P).