The Bilinear Maximal Function Maps into L P for 2=3 < P 1

The bilinear maximal operator de ned below maps L L into L provided 1 < p; q <1, 1=p+ 1=q = 1=r and 2=3 < r 1. Mfg(x) = sup t>0 1 2t Z t t jf(x+ y)g(x y)j dy In particular Mfg is integrable(!) if f and, g are square integrable, answering a conjecture posed by Alberto Calder on. 1 Principal Results In 1964 Alberto Calder on de ned the maximal operator Mfg(x) = sup t>0 1 2t Z t t jf(x y)g(x y)j dy; 6= 1 which has come to be known as the bilinear maximal function. He raised the striking conjecture that Mfg is integrable if f and g are square integrable. A proof of this and more is provided in this paper. 1.1. Theorem. Let 1 < p; q < 1 and set 1=r = 1=p+ 1=q. If 2=3 < r 1 then M maps L L into L. Now, if r > 1 the bilinear maximal function maps into L, as follows from an application of Holder's inequality in the y variable. Thus the interest is in the case 2=3 < r 1. That r can be less than one is intriguing and unexpected. Our proof forsakes the maximal function for the maximal truncations of singular integrals. Let K(y) be a singular integral kernel. Thus K(y) is a distribution on R which for y 6= 0 satis es jK(y)j C jyj (1.2) d dy K(y) C jyj2 ; (1.3)