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**Martingale Hardy spaces and their applications in Fourier analysis.**
*(English)*
Zbl 0796.60049

Lecture Notes in Mathematics. 1568. Berlin: Springer-Verlag. viii, 217 p. (1994).

This book is devoted to some extensions of classical martingale Hardy spaces and their applications to Vilenkin-Fourier series. In Chapter 1 the main definitions about martingale spaces and Vilenkin orthonormed systems are summarized. Chapter 2 displays the one-parameter theory; atomic decompositions are used to give duality theorems and martingale inequalities, and martingale transforms are extended into conjugate martingale transforms with matrix operators. Chapter 3 deals with two- parameter martingales, and the method of atomic decompositions is adapted to this case; some results of the one-parameter case are still true, others require regularity of the stochastic base as additional assumption. In Chapter 4 tree martingales are introduced, that is to say martingales with a so-called tree stochastic basis; since the partial sums of the Vilenkin-Fourier series can be expressed as martingale transforms of a tree martingale, it is possible to prove \(L_ p\)- convergence of Vilenkin-Fourier series. Chapter 5 deals briefly with interpolation between martingale Hardy-Lorentz spaces and in Chapter 6 Hardy type inequalities are presented for Walsh-Fourier and Vilenkin- Fourier coefficients. Coming shortly after the recent book by R. Long [Martingale spaces and inequalities (1993; Zbl 0783.60047)], the monograph under review attests that martingale spaces are always a topical subject. After a comprehensive exposition of the classical one- parameter case with emphasis on atomic decompositions, it focuses on two- parameter and tree martingales, and on this topic many new results from the author are gathered and brought to light.

Reviewer: D.Lepingle (Orléans)

### MSC:

60G42 | Martingales with discrete parameter |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G46 | Martingales and classical analysis |