  We study a function space $JN_p$ based on a condition introduced by John and
Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq
L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our
first main result is the construction of a function that belongs to $JN_p$ but
not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove
that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our
second main result describes $JN_p$ as the dual of a new Hardy kind of space
$HK_{p'}$.
