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A Geometrical Characterization of Aumd

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A Geometrical Characterization of Aumd DEMONSTRATIO MATHEMATICA Vol. XXX No 3 1997 Marek Piasecki A GEOMETRICAL CHARACTERIZATION OF AUMV BANACH SPACES VIA SUBHARMONIC FUNCTIONS 1. Introduction In this paper we characterize complex Banach spaces B in which /?-valued analytic martingale difference sequences converge unconditionally(so-called AUMV Banach spaces) via skew-plurisubharmonic functions, defined in this paper by subharmonic functions.We start with recalling well known geometrical characterization [1] for so-called UMV Banach spaces defined to be those in which martingale difference sequences converge unconditionally. THEOREM. A real or complex Banach space B is UMV if and only if there exists a symmetric biconvex function $ : B X B —> [—00,00) such that a) ¥(0,0) > 0; b) 9(x,y) < \\x + y\\ if ||z|| V ||y|| > 1. We shall prove analogical theorem for AL(MV Banach spaces. Note that the class of AUMV Banach spaces is strictly larger than the class of complex UMV Banach spaces and it includes such space as complex X1[0,1] which isn't UMV (LP[0,1] are UMV for p > 1). The theory of UMV Banach spaces was developed by Burkholder, Bourgain, Pisier, Maurey, Davis and the others. The theory of AUMV Banach spaces was built by Garling, Edgar, Kalton, Lindenstrauss, Blower, Pisier and the others. 2. Main definitions 7 Let (íí,^ , (Fn)%L0,P) be a probability space and — {0,12},^",, C .Fn+i C T be a filtration (a nondecreasing sequence of sub-a-fields), B be a a complex Banach space and Mn = B-valued martingale (i.e. dk are strongly measurable relative to Tk and Bochner integrable functions with E{dk+1 I Tk) = 0). DEFINITION 1. A sequence of random variables (vk) is (.F^-predictable if Vk is ^"fc.i-measurable for k = 1,2, 642 M. Piasecki DEFINITION 2. A martingale Mn = Y,k=o dk defined on 0 = [0,1]^ with a natural filtration = a{0\,...0k) (where o(Q\,...0k) stands for the smallest <r-field relative to which functions . .6k are measurable) and product of normalized Lebesgue measures as probability is called analytic martingale if ho = u, do = x 6 B and for each integer k > 1 2 t$k dk = hk-i (91,..., 0k-x)e * , fc = 1,2,... where hk are valued, strongly measurable and bounded. DEFINITION 3. If (£J)f-0 is a deterministic sign sequence of ±1, then a £ martingale Nn = Efc=o kdk is called £;- transform of a martingale Mn = ELo dk- DEFINITION 4. A complex Banach space B is called AUMV (Ana- lytic Unconditional Martingale Differences), if for each p > 1 there exists a constant cp which depends only on p and B such that for every analytic martingale Mn and its every ex- transform Nn we have p E\\Mn\\ < cpE\\Nn\\*. DEFINITION 5. A real-valued function $ : B x B —> [—oo, oo) is said to be skew-plurisubharmonic if for every x,y,u G B the two following functions (defined on a plane C) g±(z) — + uz,y±uz) are subharmonic i.e. g±(z) is upper semi continuous and for all a 6 C, r £ 1Z l \g±(a + re2*t9)d0>g±(a) o (g±(z) stands for either g+(z) = $(2; -f uz, y + uz) or g~(z) = $(2; + uz, y — uz)). EXAMPLE 1. For (x,y) £ B x B define ln wr „\ - / II* + 2/11 for x + y ^ 0 \-oo for x + y = 0. EXAMPLE 2. For (x,y) G B x B and p > 0 define ¥(s,y) = lk + 2/ir Skew-plurisubharmonicity follows from the fact that ln ||a;|| is plurisubhar- ln monic and ||z||p = eP IMI [3], [5]. 2.1. Main theorem. Geometrical characterization We will start with the two well-known facts. Characterization of AUMV Banach spaces 643 PROPOSITION 1 [4, Theorem 2 p.400.]. Suppose that Mn is a B-valued martingale for which In ||M„|| is a submartingale.Then for 0 < p < oo P p E\\M:\\ < eE\\Mn\\ where M* := sup ||M*||. fc=0...n We shall use the above fact for analytic martingales. PROPOSITION 2. [6, Proposition 2.1.2 p. 70] If a real-valued function $ : D C B —• [—00,00) is upper semicontinuous and bounded from above then there exists a sequences of continuous functions (even uniformly continuous) defined on D such that 1. 9k(x) = supyeD{V(y) - fc||s - 2/||}; 2. <supieD$(z); 3. < and lim*-»«, Vk = 9. The proof of this fact for B = C (for complex plane) from the book of Steven Krantz [6] carry over to the case of arbitrary Banach space B by substituting the absolute value with the norm. We shall start the proof of the main theorem with the following lemma. LEMMA 1. Let $ : B x B —• [-00,00) be a continuous function and locally bounded from above [i.e. function bounded from above on bounded sets), 9{x,y) — infM„,Ar„ E$(x + Mn, y + Nn) (where the infimum is taken over all bounded analytic martingales Mn starting from x — 0 and their Si-transforms Nn ). Then $ is a maximal skew-plurisubharmonic function such that 9(x,y) < $(x,y) for (x,y) G B X B. P roof. To prove that g±(z) = iB(x + uz,y±uz) is upper semicontinuous it suffices to show that $ is upper semicontinuous. To this end it suffices to note that for fixed martingales (Mn,Nn) the function h(x,y) = E$(x -f Mn,y+Nn) is upper semicontinuous for an infimum of upper semicontinuous functions is upper semicontinuous. We have indeed, by Fatou's lemma and the local boundness from above for ^ I + Mn,yk + Nn) |< c and if limfc-oo(a;jfc, 2/*) = (x,y), then limfc^oo^Xfc + Mn,yk + Nn) < + Mn,y + Nn). Hence limfc_00/i(xfc, yk) < h(x,y). Therefore g±(z) = ^(x + uz, y±uz) is upper semicontinuous and locally bounded from above. To show that g±(z) is subharmonic we need to prove that for all 7* > 0 and a € ft we have E9(x + u(a + re2n,9),y± u(a + re2™e))d9 > V(x,y). Since x,y,u £ B are arbitrary vectors so we can assume that a = 0 and r = 1. Let, fix e > 0 and 9 6 [0,1], then from definition of 9(x,y) we have analytic martingales where N® is £j transform of M®, satisfying 644 M. Piasecki the following inequality (*) 2m9 2 9 2 9 9 2 6 9 9(x + ue ,y ± ue ™ ) > E${x + ue ™ + M ,y± ue ™ + M n) - e. Since $ is locally bounded from above and for 9 G [0,1] the set Dx^VyU = {x + ue2w'9,y ± lie2*19} C B is bounded for fixed x,y,u G B, then from Proposition 2 we infer that there are continuous functions such that *m+l < m-+lim oo = 9. Since for m £ AT we have > then 2 8 2 ie 2 6 2 6 e Vm{x + ue ™ , y ± ue * ) > E${x + ue ™ + M^y± ue ™ + N n) - e. Now we choose (for fixed 9 G [0,1]) martingales M9,N9 satisfying the above inequalities. We shall show that the following random functions are analytic martingales 2iri9 2 9 9 Mn = ue +M°n,Nn = ±ue ™ + N . Clearly Nn is £j-transform of Mn. Since n can depend on 9 then the main difficulty lies in proving measurability and boundness. Let us fix 9 and let 9e(v) = + ue2™\y ± ue2™>) - E$(x + ue2*,r> + M9,y± ue+ iV®). Since E$(x + ue2lTirt + M9,y ± ue2vt1} + N9) is upper semicontinuous from Fatou's lemma then gg{9) > -e and ge(v) Is lower semicontinuous. Hence there exists an open interval Ig such that 9 £ 1$ and <70(77) > —e for 7? G Ig. Since the set [0,1] is compact then one can choose finitely many disjoint open intervals Igi,...,Ig, covering (0,1] G [0,1], € [0,1],j = 1... s and corresponding martingales (M9 , N9 ),..., (M9', N9') such that the following holds 2 e Mn(9,9u..., 9n) = ue ™ + M* (0i,..., 9n) for 9 G I6j, 2 9 Nn(9,91,...,9n) = ±ue ™ + N* (91,..., 9n) for 9 € I0j. Since we have only finitely many points 9J then one can assume that n does not depend on 9. 6 If M ^ = where hf (9U ... ,9k^) are bounded and strongly measurable then hk(9,9\,.. ,,9h-\) = ue2™9 + 9 h k' (9i,..., 9k-1) for 6 6 Ig. are also bounded and strong measurable. Hence Mn is an analytic martingale and Nn is its transform. After integrating inequality (*) we obtain that for m G AT and e > 0 Characterization of AUM.V Banach spaces 645 1 2 l6 2 l6 \vm(x + ue * ,y±ue * )d9> E$(x + Mn,y + Nn) - £ > 9(x,y)-e. o Since when moo and + ue2ni8, y ± ue2™6) < M then from monotonic convergence theorem we have l \ + ue2™9, y ± ue2™e > 9(x, y) - e. o Since e was arbitrary positive number then $ is skew-plurisubharmonic. To convince oneself that $ < $ it sufRes to take Mn = 0 and Nn = 0. Now we will check that $ is the maximal skew-plurisubharmonic function such that $ < Let $ < $ and $ be skew-plurisubharmonic. Hence E$(x + Mn, y + Nn) > EV(x + Mn,y+ Nn) 1 1 rl n—1 2n 6k 2 fi = S • • • S S + E . • • •, h-1 )e ' + hn(6u..., en)e " -, L 0 0 1-00 fc=lfc=l n—1 2 6 V + E £khk(6!,..., ek_! )e ™ * + Enh^,..., )dfln <¿#1, . ., d0„_i k=1 (by the induction over n).
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