Chanillo–Wheeden Inequalities for 0 < p ⩽ 1

For f ∈ S , a tempered distribution, let S ( f ) be a real‐variable variant of the Lusin area function of f and let G ( f ) be the Fefferman–Stein ‘grand’ maximal function of f . Let M denote the usual Hardy–Littlewood maximal operator. We prove that for all 0 < p ⩽ 1 there is a C ( p, d ) < ∞ such that∫ S p ( f ) V d x ⩽   C ( p , d ) ∫| G ( f ) | p M V   d xfor all f and for all non‐negative weights V .