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

Abstract

In the mar t ingale theory usually the following three mart ingale Hardy spaces are considered: the spaces Hp resp. Hp s resp. H i (0 < p < oc) generated by the maximal funct ion f* resp. by the quadrat ic variation S(f) resp. by s(f) called the condit ional quadrat ic variation wi th respect to (Svn_]). For one-parameter mart ingales it was proved by Davis ([9], p = 1) and by BURKHOLDER and GUNDY ([4], [6], 1 < p < oc) tha t H~ is equivalent to H i . Fur thermore , GARSlA [11] and HERZ [13], [14] proved tha t the duals of H~', H~ and H i (1 < p < oc) are BMO~, BM02 and H~ (1/p + 1/q = 1), respectively. The equivalence of the BMOq (1 < q < ~ ) spaces was shown also by GARSlA [11] and HERZ [13]. We recall tha t the BMO~ and BMOq norms are defined by [[f[[BMO; := sup [[(En[f En-lf]q)l/ql[~ hEN and IlfllBMoq := sup I I (En l f Enflq)l/qll~, hEN respectively, where En denotes the condit ional expectat ion operator with respect to ~-n. Some of these results are known for two-parameter mart ingales, too. The B u r k h o l d e r G u n d y inequality was proved by METRAUX [20]. In case the stochastic basis is regular, Davis' inequality was verified by BROSSARD [2], [3], while for general stochastic basis, it is yet unknown. Up to this time the B~/IOq spaces have been introduced only (see BERNARD [i], WEISZ [29]), nevertheless, the BMO~ spaces have not been considered as yet. The