Abstract
The bilinear maximal operator defined below maps $L^p \times L^q$ into $L^r$ provided $1 < p,q < \infty$, $1/p + 1/q = 1/r$ and $2/3 < r \le 1$.
\[
Mfg(x) = \sup_{t>0} \frac{1}{2t} \int_{-t}^{t} |f(x+y)g(x-y)| dy.
\]
In particular $Mfg$ is integrable if $f$ and $g$ are square integrable, answering a conjecture posed by Alberto Calderón.
