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 $ : 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 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.

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 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). Hence y) > y) as infimum over Mn, Nn. This completes the proof. The proof of the next lemma consists of Proposition 1 and arguments from Burkholder paper [1],

LEMMA 2. Let M stand for family of B-valued martingales Mn = res ec o P t t° filtration (jTn)^=0 such that 1. ||dn|| is Tn-\-measurable; 2. ln||Mn|| is submartingale;

3. for every bounded predictable sequence of real random variables (un) u the martingale Af" = ndn £ M; 4• There is a constant c depending only on M. such that for all martin-

gales Mn € M and its £i-transform Nn we have P(||Mn|| > 1) < c^||iVn||. Then for Mn € M and 6 > 0, /? > 26 + 1, A > 0 one has

P(N* > (3\ , M* < SX) < aP(N* > A),

where M* := sup^o...^ ||Mn||, N* := sup^...^ ||J\Tn||, a = More- over if aft? < 1, p > 0, then

EN*P < fP6Z EM*P ~ 1-afiP 646 M. Piasecki

and

EWNrT < ¥^LE\\Mn\\*.

Proof. Let

fi = inf{n : ||iVn|| > A},

1/ = inf{n : ||iVn|| > /3A},

a = inf{n : \\Mn\\ > S\ or ||d„+i|| > 2£A},

UK =

n n

Mn = J^dfc, Nn - ekdk, fc=0 k=0 n n

Fn = ^~^ukdk, Gn = y^Jkukdk. k=0 fc=0

Since for Mn G M the set {a < k} depends only on variables ...

...,dk and dk is .Ffc_i-measurable then we infer that {a < k} G Fk-i-

Clearly {fi < k} G Tk-\ and {k < u} G Tk-1- Hence the set {fi < k < v A a) G ^jb-i-

It follows from Property 3 that Fn £ M. We shall show that

^llfnll < 46\P(N* > A).

If N* < A, then /i = v - +oo, which yields Fn - 0. Hence -E||-Fn|| =

E {N*>\}\\Fn\V

If a A v = +oo, then \\Mn\\ < 6A. But if a A v < +oo, then

11^11 = 1 J] d*|| = \\MaAv-Mu\\ < 3iA + a = 4a k=it+i

(for HAfo-_i.il < ¿A, ||4,|| < 2£A so ||Ma|| < and \\MaAl/|| < MX and IIM^ < ¿>A|| ,(for n fi A and M* < S A, then fi < v < +oo, CT =

+oo (if Vn ||Mn|| < SX then ||dn+i|| = \\Mn+1 - Mn\\ < 26X). Therefore P(N* > f3X,M* < 6X) < P(fJ. < v < +oo, a = +oo). Observe that if ¡jl < v < +oo, a = +oo, then

||Gn||>A(/?-2tf-l). Characterization of AUMV Banach spaces 647

Indeed under the above assumptions we have

||Gb|| = || J] ekdk k=fi+1

= ||JV„ - tfj > ll^ll - \w\ > /3A - (A - 28X) = X((3 -28-1)

(for ||J\g| = ||iVM_a + e^|| < a|| + ||dj < A + 26X and ||Ar„|| > /3A). Therefore we have that P(N* > ,M* < 8X) < P(p < v < +oo,a = +oo) A(/?-2i-l)).

Since Gn € M we obtain

P(\\Gn\\ > 1) < cE\\Fn\\. Finally

P(!i- > PKM- < IX) < -Pdljp-^njll > ')

£ 5 Atf-M-l)*""-« W-lh)^* > = > which completes the proof of the first part of the lemma. To prove the second part of the lemma let a.(3p < 1 ,p > 0. Then we have

N*p EN*P = /3pE— = 0P J P(N* > (3X)pXp~1dX P o oo < f3p S P(N* > f3X, M" < 6X)pXp~1dX o oo + (3P \ P(N* > /3A,M* > 6X)pXp~1dX o CO < a(3p j P(N* > X)pXp~1dX + /3P J P(M* > 6X)pXp~1dX o o

= a(3pEN*p + pp6~pEM*p. Hence

EN*P < I**6 l EM*P. 1-afiP' 648 M. Piasecki

Now from Proposition 1, for the martingale Mn, we have

|| sup \\Mn\\\\p

E\\Nn\\r < E\\N*\\r < JZ£LE\\MY < E\\Mn\\p, which completes the proof of the lemma. Now we shall proye the following characterization of AUMV Banach space via skew-plurisubharmonic function. THEOREM 1. A complex Banach space B is AUMV iff there exists a function $ : B x B —[—00,00) such that 1. $ is skew-plurisubharmonic; 2. $(z,±x) > $(0,0) > 0; 3. $(z, y) < $(0,0) + \\y\\ for x,y e B; 4. V(x,y) < Hj/II on the set {(x,y) : ||z|| + ||y|| > 1}. Proof. => If B is AUMV then for p = 1 and an analytic martingale Mn and its transform Nn we have [4, Theorem 7]

E\\Mn\\ < c^WNnl Let $$ be the function from Lemma 1 for function y) = ci||y|| —1|®||. Hence from the definition of function $$ we have $$ < $$(a:, ±a:) > 0 and $$(0,0) < $(0,0) = 0. Which implies $$(0,0) = 0. Let

t(l,,) = I C\ + 1 Thus $(0,0) = ^ > 0,W(x,±x) = $(0,0) + ^J;3^ > $(0,0). For all x, y € B we have

Ci + 1 If x,y e B such that ||®|| + \\y\\ > 1 then

V(z,jr)< 1±£iJWjiM < (fi+iM hjii v - a + l - ci + 1 = 11 11 Trivially $ is skew-plurisubharmonic as a linear transform of a skew-pluri- subharmonic function $$. Hence $ satisfies conditions 1, 2, 3 and 4 of the theorem.

Let M be a class of martingales with natural filtration Tn = ..., 9n). Since \\hk(0u..., 9k-i)e2/Kl6k || = ||h^,..., 0fc_i)|| is Tk_x-measurable Characterization of AUMV Banach spaces 649

then M satisfies 1, 2, 3 from Lemma 2. If uk is a real function,

measurable and bounded then functions ukhk are also -measurable and bounded. We will check that if there exists a skew-plurisubharmonic function $ which satisfies the assumptions of Theorem 1. Then M fullfils assumptions of Lemma 2 for c = ^ .

For an analytic martingale Mn and its ^-transform Nn we have

a) ||iVn|| — 9(Mn,Nn) + $(0,0) > 0;

b) *(Af„, JV„) < ||JVn|| for ||Mn|| + ||J\Tn|| > 1. Moreover, since $ is skew-plurisubharmonic then

EV(Mn,Nn) 1 1 rl n —1 27rt8k = J... J ¡«(I + ^M»!,-, )e +hn(61,..., 0n_i )e 0 0 L0 fc=l n—1

k= 1

2 e + enhn(01,...,0n_1)e ™ »)d6n dOi,..., i

> E9(x + Mn^,±x + >,...,> $(x,±:r) > $(0,0) (by the induction over n). Hence c) $(0,0)-£$(M„,iV„) < 0. By the Chebyshev inequality for nonneqative random variable ||iV„|| -

$(M„, Nn) + $(0,0) it follows that

nil^nll > 1) < P(||Mn|| + \\Nn\\ > 1) < P($(M„,iVn) < ||i\g|)

< P(\\Nn\\ - V(Mn,Nn) + $(0,0) > $(0,0))

< + E\\Nn\\ $(0,0) - $(0,0)' Hence Lemma 2 is satisfied with c := ^g-gy. It remains to prove that for all p > 0 there exist 6 > 0 and (3 > 26 + 1 such that

afl> = ^ <1 H $(0,0)(/3 — 2<5 — l) To this end let p > 0, f(p) arbitary positive function, set f3 = 1 + 2f(p)

B and fix arbitrary 6 satisfying 0 < 6 < 2(i+2f(p°y+i(o,o) • y substituting ¡3 = 1 + 2f(p) we obtain

26(l + 2f(P)Y <*(0,0)(f(p)-6). 650 M. Piasecki

= Therefore 0 < 6 < 2(i+2f(p))r+

E\\Nnr <

Hence B is AUMV space, which completes the proof of the theorem.

3. Remarks on definition of AUMV Banach space From the existence of a skew-plurisubharmonic function for AUMV Ba- nach space it follows a direct proof of a theorem stating that in the definition of AUMV space one can take plurisubharmonic martingales Mn (i.e. such that is submartingale for all plurisubharmonic instead of analytic martingales [4], [5]. Now we can state such a theorem in a little more general form. Its proof also implies that in the definition of AUMV space one can take uniformly bounded predictable complex random variables instead of a sign sequences. If those predictable sequences are real and bounded by 1 then one can take the same constant cp,p > 1. We shall start with following simple lemma.

LEMMA 3. Let Xk,yk £ TZ be predictable random variables such that I Xk |51 Vk 1 iP > 0 , cp 6 7Z and dk be random vectors with values in Banach space B. Then the following condition V II ^ MP E£kdk < cpE ' k=0 k=0 for all sequence of predictable random variables (£k) with values in {—1,1} implies

n n I p || l d I k=0X^O** + Vk) k < dpEll ... n dk where 2c„ for p <= (0,1] dp 2 p - 1 2 PCZ for p > 1. Moreover if Zk are constants then Xk,Vk are a^so constants.

Proof. If | Xk |< 1 then Xk = X^i £jk = ±1. If the sequence (xk) is predictable then for j = 1,2,... the sequences Sjk are also pre- Characterization of AUMV Banach spaces 651 dictable. Hence for p > 1 oo n

k=0 P k=0j=l P j=1 k=0 oo n i n

j=l k=0 fc=0

For p e (0,1] we have

OO 71

k=0 k=0 j=1 j=1 fc=0 oo n oo n <£(2_jHlE£^|| -^(E^HE^ J=j=1l fc=0 J = 1 k=0

fc=o By the triangle inequality for p > 1 we have

n n n 1 n Xk k J/fciiA; 2c II +iyk)dk - || E ^ +|E - P II E^l ' A;=0 V k=0 P k=0 P k=0 P while, for p £ (0,1],

II n p II H llp II "

k=o A:=0 k=0

< 2cp 2p HIE* fc=0 Finally, for p > 1, n n e^*+^h r - 2Pcp£i e d*ir» fe=0 k—0 while, for p £ (0,1]

ll p 2c II dk k=0 +^ 2^1^11fc=0 E \

This completes the proof of the lemma.

Theorem 2. If B is a AUMV Banach space and vk is a sequence of complex random variables (Fk) predicatable such that | vk |< 1 then for any 652 M. Piasecki

plurisubharmonic martingale Mn = X)fc=o ^k one has i n p ij n r p

I k=0 fc=0 where ( 2 C™ for pe (0,1] dp = < 2P - 1 p t 2 cp /or p > 1

and cp is a constant from definition of space AUMV.

Proof. If B is AUMV space then for p € (0,oo) there exists cp such that [4, Theorem 7]

E\\An\\*> < cp\\Bn\\>,

where An is an analytic martingale and Bn is its £j-transform. As follows from Lemma 3, one can assume that vk has values in { — 1,1}. p p If = cp||ar|| — ||?/|| then from Lemma 1 the function

¥(a:,y)= inf E${x + An,y + Bn) An,Bn satisfies $(£,±3;) > 0. v Let Nn = X)fc=o kdk and set y) = y — x,x + y). Then is r plurisubharmonic relative to each coordinate and 0) = 4 1(0,x) > 0. Since = ^(x,y) < $(x,y) and since for Tn-\-conditionally we have vn = ±1 and — x,b + y) is plurisubharmonic in coordinates for constant vectors a, b then

p -Mn Mn + Nn E[cp\\Mn|| - ||iVn|n > EV{Mn,Nn) =

. . N —\ - M —1 . Vnd-n d M„_! + N n—1 . T V u . . = EE iT n n n n n 2 + 2 ' 2 + 2 1 '•Fn-1

^ (—-—,—-—~ Mn_i Mn_i + Nn-1j \ >...>«! (vQx 2—'—2—- x x + ;

This completes the proof.

We will now estimate the constant cp from the definition of an AUMV Banach space.

COROLLARY 1. If Mn is an analytic AUMV-valued martingale and Nn its e{-transform, then for p > 0 one has

where $ is a function satisfying conditions of Theorem 1. (ifp > 1 then one can take 18 instead of 19). Characterization of AUMV Banach spaces 653

Proof. From the end of proof of Theorem 1 it follows that if B is AUMV then for f(p) > 0, 0 < 8 < 0,0), 0 = 1 + 2f(p) and a =

*(0,0)03-2,5-1) OIle haS

E\\Nn\\r < J^LeE\\Mn\\'.

Note that for ||u|| = \ we have \\ue2iri6\\ + || ± ue2iri$\\ = 1. Therefore

i i ¥(0,0) <\9(ue2*,9,±ue2"t8)d6 < \\±ue2vi6\\ = -. o Set «(0,0) = 2¥(0,0). Then «(0,0) < 1. Burkholder has calculated [1, end of proof of Lemma 8.3] that if «(0,0)

< 1, /? = 1 + J and 6 = 8(p+^°;°iy2pP+1 then for p > 0 one has f^ < (3u(o o)p ) • P — 1 one can instead of 38. I shall check that for choosen /? and 6 we have a/3p < 1. Indeed, let f(p) — \p and

¥(0,0 )pP = *(0,0)£ t(0,0)l 4(p + + P"+1 2(1 + I)f+i + I < 2(1 + + |

$(0,0)/(p) - 2(l + 2/(p))p + $(0,0)' Hence

for p > 0. Moreover for p > 1 one can take 18 instead of 19, which completes the proof. Observe that form the results of this paper it follows that in the defintion of AUMV Banach space it is enough to consider p = 1 only.

References

[1] D. L. Burkholder, Martingale and Fourier analysis in Banach spaces, Lectures Notes in 1206 (1986), 61-108. [2] D. L. Burkholder, A geometrical characterization of Banach spaces in which mar- tingale difference sequences are unconditional, Ann. Probability 9. No 6 (1981), 997- 1011. [3] G. A. Edgar, Complex Maritngale Convergence. "Lecture Notes in Mathematics" Vol. 1166. Springer-Verlag, New York (1985), 38-59. 654 M. Piasecki

[4] D. J. H. Garling, On martingales with values in a complex Banach space, Math. Proc. Camb. Phil. Soc. (1988) 104. 399. [5] N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math., 84 (1986), 297-324. [6] S. G. Kranz, Functions Theory of Several Complex Variables, PWN Warszawa 1991.

INSTITUTE OF APPLIED MATHEMATICS WARSAW UNIVERSITY ul.Banacha 2 02-097 WARSZAWA, POLAND [email protected]

Received April 22, 1996.