Adam Osękowski
(2014)
Bulletin of the Polish Academy of Sciences. Mathematics
Let be the Haar system on [0,1]. We show that for any vectors from a separable Hilbert space and any , k = 0,1,2,..., we have the sharp inequality
, n = 0,1,2,...,
where W([0,1]) is the weak- space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound
,
where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.