  We discuss certain facts involving a continuous local martingale $N$ and its
supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic
functions is proposed. This yields an important family of martingales, the
usefulness of which is demonstrated, by means of examples involving the
Skorokhod embedding problem, bounds on the law of the supremum, or the local
time at 0, of a martingale with a fixed terminal distribution, or yet in some
Brownian penalization problems. In particular we obtain new bounds on the law
of the local time at 0, which involve the excess wealth order.
