Inequalities and duality results with respect to two-parameter strong martingales (Q1272185)

The author introduces the spaces \(\text{BMO}_q\) and \(\text{BMO}^-_q\) for two parameter strong martingales. The operator \(\sigma\) called the conditional quadratic variation with respect to \((\mathcal F^-_n =\mathcal F_{n_1,n_2-1} \lor \mathcal F_{n_1-1,n_2}\); \(n=(n_1,n_2)\in {\mathbb N}^2)\) is used. Besides the martingale Hardy spaces \(H^*_p\) and \(H^S_p\), the strong martingale Hardy space \(sH^\sigma_p\) generated by \(\sigma\) is also considered. The author also extends two convexity and concavity theorems due to \textit{D. L. Burkholder, B. J. Davis} and \textit{R. F. Gundy} [in: Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 223-240 (1972; Zbl 0253.60056)], and to \textit{C. Stein} [ibid., 583-602 (1972; Zbl 0278.60026)]. He proves the relations \(sH^S_p\subset sH^\sigma_p\) \((0\leq p\leq 2)\) and \(sH^\sigma_{p}\subset sH^S_p\) \((2\leq p<\infty)\). The author gives the atomic decomposition of \(sH^\sigma_p\) similar to his decomposition of \(H^S_p\) (1990). The following relations \(sH^\sigma_p\subset sH^*_p\), \(sH^S_p\) \((0\leq p\leq 2)\) and \(sH^*_p\), \(sH^S_p\subset sH^\sigma_p\) \((0\leq p<\infty)\) are also given. With the help of a new Davis decomposition for the spaces \(sH^S_p\) and \(sH^*_p\) the Davis's inequality for two-parameter strong martingales is proved. In the last section the following duality theorems are proven: \(sL_p\) and \(sL_q\) \((1<p<\infty,1/p+1/q=1)\) are duals of each other; the dual of \(sH^{at}_1\) is \(s\text{BMO}_q\) \((1<q<\infty,1/q+1/q'=1)\), \(sH^\sigma_1\) to \(s\text{BMO}_2\), respectively. Moreover if the stochastic basis is regular, \(sH^{at}_q\) is equivalent to \(sH^\sigma_1\) \((1<q\leq 2)\) and \(s\text{BMO}_q\) is equivalent to \(s\text{BMO}_2\) \((2\leq q'<\infty)\) as well. Finally the author provides some inequalities between \(s\text{BMO}_p\) and \(sL_q\) norms.