The history of martingale theory goes back to the early fifties when
Doob [57] pointed out the connection between martingales and analytic
functions. On the basis of Burkholder's scientific achievements the mar-
tingale theory can perfectly well be applied in complex analysis and in
the theory of classical Hardy spaces. This connection is the main point
of Durrett's book [60]. The martingale theory can also be well applied
in stochastics and mathematical finance. The theories of the
one-parameter martingale and the classical Hardy spaces are discussed
exhaustively in the literature (see Garsia [83], Neveu [138],
Dellacherie and Meyer [54, 55], Long [124], Weisz [216] and Duren
[59], Stein [193, 194], Stein and Weiss [192], Lu [125],
Uchiyama [205]). The theory of more-parameter martingales and
martingale Hardy spaces is investigated in Imkeller [107] and Weisz
[216]. This is the first mono- graph which considers the theory of
more-parameter classical Hardy spaces. The methods of proofs for one and
several parameters are en- tirely different; in most cases the theorems
stated for several parameters are much more difficult to verify. The
so-called atomic decomposition method that can be applied both in the
one-and more-parameter cases, was considered for martingales by the
author in [216].