  The bilinear maximal operator defined below maps $L^p\times L^q$ into $L^r$
provided $1<p,q<\zI$, $1/p+1/q=1/r$ and $2/3<r\le1$. $$
Mfg(x)=\sup_{t>0}\frac1{2t}\int_{-t}^t\abs{f(x+y)g(x-y)} dy.$$ In particular
$Mfg$ is integrable\thinspace if $f$ and $g$ are square integrable, answering a
conjecture posed by Alberto Calder\'on.
