On an inequality of Varopoulos for 2-parameter Brownian martingales (Q1820504)

In this paper the inequality of \textit{N. Th. Varopoulos} [Bull. Sci. Math., II. Ser. 105, 181-224 (1981; Zbl 0472.32009)] for bi-Brownian martingales is extended. More precisely, the author shows that the inequality \[ \| (\int^{\infty}_{0}\int^{\infty}_{0}| \nabla_ 1X_{st}|^ 2| \nabla_ 2Y_{st}|^ 2dsdt)^{1/2}\|_ r\leq C_{p,q}\| \sup_{s,t}| X_{st}| \|_ p\| \sup_{s,t}| Y_{st}| \|_ q \] holds for \(0<p<\infty\), \(0<q\leq \infty\) and \(r^{-1}=p^{-1}+q^{- 1}\). Here \(\nabla_ 1X\) and \(\nabla_ 2Y\) denote the stochastic gradients of the bi-Brownian martingales X and Y \((\nabla_ i\) contains the stochastic derivatives with respect to the i-th Brownian motion, \(i=1,2)\) and \(C_{p,q}\) is a constant depending on p and q only. As an application of this result the author gives an extension of a result due to \textit{E. M. Stein} [ibid. 103, 449-461 (1979; Zbl 0468.42012)] on a variant of the area integral for the bi-disc.