Abstract.It is shown that partial sums of Walsh-KaczmarzFourier series of functions in the Orlicz class L(log+ L)2 converge a.e. The proof utilizes an estimate of P. Sjölin on the partial sums of the usual Walsh-Fourier series.

ABSTRACT. Paley proved that Walsh-Fourier series converges in I?(1 < p < °°). We generalize Paley's result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.

Abstract.It is shown that the double sequence {Xm"} with X"" = 1 if n < m and 0 otherwise is an Lp multiplier for the Walsh system in two dimensions only if p = 2. This result is then used to show that the one-dimensional trigonometric system and the Walsh system are nonequivalent bases of the Banach space Lp[0, 1], and hence have different Lp multipliers, 1 < p < oo, p # 2.

Abstract. Let k be a positive integer and 1 < p < oo. It is shown that if T is a multiplier operator on Lp of the line with weight | x \kp~\ then Tf equals a constant times/almost everywhere. This does not extend to the periodic case since m(j) = l/j,j ¥* 0, is a multiplier sequence for Lp of the circle with weight |x|*''_l. A necessary and sufficient condition is derived for a sequence m(j) to be a multiplier on L2 of the circle with weight | x |2/t~'.

A characterization is obtained for weight functions v for which the Hardy-Littlcwood maximal operator is bounded from IJ'(R", wdx) to IJ'(R", vd.x) for some nontrivial »'. In this note we obtain a necessary and sufficient condition on weight functions v s* 0 such that the Hardy-Littlewood maximal operator is bounded from LP(R", wdx) to LP(R", vdx) for some w < oo a.e. This answers a question posed by B. Muckenhoupt in [3]. The problem of characterizing all weight functions w > 0 for which there are nontrivial «'s was solved independently by J. L. Rubio de Francia