Square function inequalities for monotone bases in L1

Abstract

We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in $L^1$. 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 $L^1\to L^1$ and a weak-type $L^1\to L^{1,\infty }$ estimate for square functions.