The bilinear maximal functions map into L p for 2 / 3 < p ≤ 1

The bilinear maximal operator defined below maps Lp × Lq into Lr provided 1 < p, q < ∞, 1/p + 1/q = 1/r and 2/3 < r ≤ 1. Mfg(x) = sup t>0 1 2t ∫ t −t |f(x+ y)g(x− y)| dy. In particular Mfg is integrable if f and g are square integrable, answering a conjecture posed by Alberto Calderón. 1. Principal results In 1964 Alberto Calderón defined a family of maximal operators by Mfg(x) = sup t>0 1 2t ∫ t −t |f(x− αy)g(x − y)| dy, α 6= 0, 1 which have come to be known as bisublinear maximal functions. 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 α 6= 0, 1, and let 1 < p, q < ∞ and set 1/r = 1/p + 1/q. If 2/3 < r ≤ 1 then M extends to a bounded map from Lp × Lq into Lr. Now, if r > 1, M maps into Lr, as follows from an application of Hölder’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. ∗This work has been supported by an NSF grant, DMS-9706884.