Summability of Multi-Dimensional Fourier Series and Hardy Spaces

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 Burkholders 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 Durretts 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].