A stochastic Carleson embedding theorem with constant $e$ and the vector of Riesz transforms

We prove a continuous-time Carleson embedding theorem with constant $e$ for a system of square-integrable continuous martingales whose quadratic covariations mimic the generalised Cauchy–Riemann relations. The argument is based on a Bellman function and a multidimensional Itô formula. As an application we transfer the estimate to the upper half-space via the Gundy–Varopoulos representation and obtain a Carleson embedding, still with constant $e$, for the vector consisting of a function and its Riesz transforms in arbitrary dimension.