Square function inequalities for monotone bases in $L^1$

The paper contains the description of a novel method of handling general sharp square-function inequalities for monotone bases and contractive projections in L1. The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong L1 → L1 and a weak-type L1 → L1,∞ estimate for square functions.