Sharp inequalities for the dyadic square function in the BMO setting

We introduce a method for the simultaneous study of a BMO function φ and its dyadic square function S(φ) that can yield sharp norm inequalities between the two. One of the applications is the sharp bound for the p-th moment of S(φ), 0<p<∞, which in turn implies the square-exponential integrability of the square function. We also present sharp refinements of these inequalities in the more restrictive case when φ is assumed to be bounded.