The singular integral characterization of 𝐻^{𝑝} on simple martingales

The singular integral characterization of H' on simple martingales was given by S. Janson. We show that his result cannot be extended to lIP if p ( > 0) is very small. Let Q = (0, 1]. Let F be the a-field of all Borel sets in Q. Let dx be the Lebesgue measure. Then (Q, F, dx) is a probability space. Let d > 3 be an integer. For each integer n > 0, let F,, be the sub-a-field of F generated by ((k )d-n, kd-n], k = 1,...,d'. Set I(kl1..k,k) ((k, -)d-' + +(kn1I)d-1 +(k, -I)d(k I)d-' + +(knI)d'+ knd-n] foreachk . k,iE{1. d). A martingale is a sequence of complex-valued integrable functions {fn})3?0 such that E[fln+ 1 F,1] = , where E I F,,] denotes the conditional expectation with respect to the sub-a-field F,. We writef for {f n)}0. Iff is generated from a function f(X) E L'(Q) by (1I) fn = E[f I Fn] we identify f and f. For a martingalef we define f*(x) = sup Ifn(x). n O For p E (0, oo), f is said to belong to HP if f* 11 p + x, where IIf IIP f*(X) dx It is well known that if p > 1, then HP and LP(Q) can be identified. That is, f* E LP(Q) if and only if there exists a function f(x) E LP(Q) such that (1) and CPjIf*jj IIP1SpII 0, hold. It is also known that if f* E L'(Q), thenf is generated from an L'-function but that the converse is not true. Received by the editors September 9, 1982 and, in revised form, November 30, 1982. 1980 Mathematics Subject Classification. Primary 42B30; Secondary 46J 15. 1 Research partly supported by NSF MCS-82033 19 and Science Research Foundation of Japan. (C)1983 American Mathematical Society 0002-9939/83/0000-1 255/$02.25 617 This content downloaded from 157.55.39.138 on Sun, 26 Jun 2016 05:32:58 UTC All use subject to http://about.jstor.org/terms 618 AKIHITO UCHIYAMA For n > 1, set A fn = f, f, _. Since Afn is F11-measurable, the notation f\n(I(kI, * . .,k,,)) makes sense. Let Z ={ T(X = )kI E~ C : E X k = 0, where (xk )k=d denotes a d-dimensional column vector and C is the set of all complex numbers. Note that (L\f,(I(k ,...,k,_1, k)))d= E V. Let A be a linear operator from V to V. Set (Agn( I(k 1 * ,kn k )))d kI= = A(AfjslI(kj,. . .,k*_ k )))dk1