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Extrapolation and interpolation of quasi-linear operators on martingales. (English) Zbl 0223.60021
Let $$(\Sigma_n)_{n>0}$$ be an increasing sequence of $$\sigma$$-algebras in a probability space. A sequence $$x =(x_n)_{n\geq 1}$$ of integrable random variables (r.v.) forms a martingale difference sequence if $$E(x_n\!\mid\! \Sigma_{n-1})=0$$ for $$\geq 2$$. A sequence $$f = (f_n)_{n\geq 1}$$ is a martingale transform of $$x = (x_n)$$ if $$f_n = \sum_{k=1}^n d_k$$ where $$d_k = v_kx_k$$, $$v_k$$ being a $$\Sigma_{k-1}$$ measurable r.v. If the $$v_k$$’s are such that $$v_kx_k$$’s are integrable then $$f=(f_n)$$ forms a martingale sequence. The generality and utility of martingale transforms lie in the fact that with practically no direct integrability assumptions on $$v_k$$, surprisingly general and profound inequalities can be established for norms of various types for $$f_n$$ or functionals of $$f_n$$. These inequalities in their turn lead to convergence statements concerning $$f_n$$. For $$f_n$$’s which are linear combinations with constant coefficients of a sequence of independent r.v. $$x_n$$’s, the theory was initiated by Marcinkiewicz and Zygmund in the late 1930’s. The inspiration for that work was evidently the inequalities of Paley concerning Walsh and Haar series and the inequalities of Hardy and Littlewood. Indeed the recent rejuvenation of the theory by Burkholder, E. Stein and others (specially in ergodic theory, harmonic analysis and probability theory) still bear deep imprints of the techniques used by the pioneers.
However, the results obtained in this paper attain a degree of generality and precision that would seem to indicate that one important chapter in this theory is complete. Instead of enumerating in detail the theorems of this paper, let us indicate the nature of the results obtained.
Given $$f =(f_n)$$ a martingale transform of $$x=(x_n)$$, form $$f^*= \sup_n | f_n |$$ and $$S (f) =(\sum_{k=1}^\infty d_k^2)^{1/2}$$ where $$f_n = \sum_{k=1}^n d_k$$. If $$f =(f_n)$$ itself is a martingale, it is known that the inequality $\alpha_p\| S(f)\|_p\leq \| f^*\|_p\leq \beta_p \| S(f)\|_p \tag{\text{I}}$ is valid for $$1 < p <\infty$$. Indeed this is an important consequence of the results of D. L. Burkholder [Ann. Math. Stat. 37, 1494–1504 (1966; Zbl 0306.60030)]. The results of the present paper form a considerable extension of (I). Inequalities of type (I) are shown to be true with (1) $$f$$ martingale transforms of $$x$$ where $$x$$ satisfies certain conditions on $$E(| x_k|^\rho \!\mid\! \Sigma_{k-1})$$, (2) $$S$$ replaced by $$T$$, a very general class of quasilinear operators defined an sequences of r.v.’s with values as positive r.v.’s and (3) $$L^p$$ norms replaced by general Orlicz space type norms which include $$L^p$$, $$0 < p < \infty$$. Specialisation of $$T$$ to matrix type operators yields more precise results. Several applications to random walks, Haar and Walsh series, convergence of martingale transforms and to Brownian motion are included. The nature of the conditions imposed on $$x$$ and $$T$$ are elucidated by examples and comments. It is to be noted that the inequality (I) itself (for $$f$$ a martingale) has been shown to be true for $$p=1$$ [B. Davis, Isr. J. Math. 8, 187–190 (1970; Zbl 0211.21902)] and that (I) is false for $$0 < p < 1$$ without some restrictions on $$x$$. The techniques used in the paper combine classical analysis and probability theory. From the former are taken the ideas of interpolation theory and the Rademacher series while the consistent use of stopping and starting times indicates the influence of probabilistic intuition. It is to be hoped that the essential results of this paper would be given easier proofs in the near future.
Show Scanned Page ##### MSC:
 60G46 Martingales and classical analysis 60G44 Martingales with continuous parameter 60G42 Martingales with discrete parameter
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