Optimal distributional estimates of the multiple Hilbert transform

In this paper, we study optimal distributional estimates for the multiple Hilbert transform. We obtain pointwise upper and lower distributional estimates of the multiple Hilbert transform in terms of the $d$-fold composition of the Calder\'{o}n operator with itself. This extends the fundamental results by A. P. Calder\'{o}n, D. Boyd, and Ch. Fefferman for arbitrary $d\in\mathbb N.$