Hardy spaces of 2-parameter Brownian martingales (Q1076420)

We investigate 2-parameter Brownian martingales and give some applications to Hardy spaces on the torus. In order to describe our results we prepare notations: Let \(T^ 2\) be the distinguished boundary of the bidisc. Let \({\mathcal H}^ 1(T^ 2)\) (resp. \(h^ 1(T^ 2))\) be the Hardy space on \(T^ 2\) defined by Hilbert transforms and double Hilbert transform (resp. radial maximal functions). Let \((\Omega,F,P;(F_{st})_{s,t};(z^ 1_ s)_ s,(z^ 2_ t)_ t)\) be the product of two Brownian spaces of dimension 2 with some conditions (see section 1 of the paper). Let \(K^ p\) (resp. BMO) be the Hardy space (resp. BMO-space) of \((F_{st})\)-martingales which admit 2-parameter Ito integral representations. (For detail, see section 1 of the paper). Put \[ BMO(T^ 2) = \{f\in L^ 2(T^ 2) : \quad (E[f(z^ 1_{\tau (1)},z^ 2_{\tau (2)})| F_{st}])_{s,t}\in BMO\}, \] where \(\tau(j) = \inf \{t>0:\quad | z^ j_ t| =1\}\) \((j=1,2)\) and E[\(\cdot | \cdot]\) is the conditional expectation. Our main result is the following: Theorem. \({\mathcal H}^ 1(T^ 2)\) (resp. \(BMO(T^ 2))\) is isomorphic to a closed complemented subspace of \(K^ 1\) (resp. BMO). As corollaries of the theorem we have the following: Corollary (Gundy and Stein) [see \textit{R. F. Gundy}, Ecole d'été de probabilités de Saint-Flour VIII-1978, Lect. Notes Math. 774, 251-334 (1980; Zbl 0427.60046)]: \({\mathcal H}^ 1(T^ 2)\) is isomorphic to \(h^ 1(T^ 2)\). Corollary [\textit{H. Sato}, Caractérisation par les transformations de Riesz de la classe de Hardy \(H^ 1\) de fonctions bi-harmoniques sur \({\mathbb{R}}_+^{m+1}\times {\mathbb{R}}_+^{n+1}\). Thése de doctrat 3e cycle, Grenoble (1979)]: The dual of \({\mathcal H}^ 1(T^ 2)\) is isomorphic to \(BMO(T^ 2)\). In our paper we also study Hardy spaces of 2-parameter holomorphic martingales. Correction on p. 364 of the paper: The definition of radial maximal functions n(f) should read: \[ n(f)(e^{i\theta},e^{i\phi}) = \sup \{| PP[f](r_1e^{i\theta}, r_ 2e^{i\phi})| \;:\;0\leq r_ j<1, \quad j=1,2\}. \] This error is a slip of typewriting. Hence this change does not affect any other statement of the paper.