Martingale Hardy Spaces and their Applications in Fourier Analysis

On the maximal operator of Walsh-Kaczmarz-Fejér means

“…, n ∈ N) a martingale with respect to (F n , n ∈ N) (for details see, e. g. [19], [20]). The maximal function of a martingale f is defined by…”

Boundedness of vector-valued martingale transforms on extreme points and applciations

“…* < 2 m } and X* e U, we have lim m^0O (/ v -) n = / " and ||(/" 1 '-) B ||B < 2n2~m almost surely. Define, for each k such that P({v t^o o})^0, = 2 k 3P({v k jL oo}) 1^ and a k = ^' ( ( /^O n -(/"On); in other cases define Mi = 0. a\ = 0-The proof follows now the lines of the scalar case, see [17].…”

“…In particular, if we take \x n = inf x A. B as a consequence of the following well known result due to Davis (see [17]) in the scalar-valued case. The proof in the vector-valued setting is straightforward.…”

“…The following lemma is a slight generalization of the scalar-valued case, see [17]. since, by using that {v m >j} = (A.…”

“…In the scalar-valued case, the following facts are well known, see [11] and [17]. Their proofs go straightforward over the Banach-valued case.…”