Bounded double square functions (Q1069206)

We extend some recent work of S. Y. Chang, J. M. Wilson, and T. Wolff to the bidisc. For \(f\in L^ 1_{loc}({\mathbb{R}}^ 2)\), we determine the sharp order of local integrability obtained when the square function of f is in \(L^{\infty}\). The Calderòn-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector- valued form of an inequality for dyadic martingales that yields the sharp dependence on p of \(C_ p\) in \(\| f\|_ p\leq C_ p\| Sf\|_ p\).