Abstract. Let B denote the Beurling-Ahlfors transform defined on L p (C),

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Let {d k } k≥0 be a complex martingale difference in L p [0, 1], where 1 < p < ∞, and {ε k } k≥0 a sequence in {±1}. We obtain the following generalization of Burkholder's famous result. If τ ∈ [− 1 2 , 1 2 ] and n ∈ Z+ then n k=0 ε k τ d k L p ([0,1],C 2 )

Given martingales W and Z such that W is differentially subordinate to Z, Burkholder obtained the sharp martingale inequality E|W | p ≤ where X = (X 1 , X 2 ), Y = (Y 1 , Y 2 ). By definition, Y is said to be differentially subordinate to X or to be a martingale transform of X. If, for 1 < p < ∞, we have E|X(t)| p < ∞, then the Burkholder-Davis-Gundy and Doob inequalities (see [39]) imply that E|Y (t)| p < ∞ and there exists a universal constant C p such that Y (t) p ≤ C p X(t) p . We use the notation X p = X(t) p = (E|X(t)| p ) 1/p . An evident problem then is to find the best constant C p .Burkholder solved this problem completely in a series of papers in the 1980's, see in particular [9] and [11]. He proved thatHere B t is 3-dimensional Brownian motion, τ is its exit time from R 3 + , and the conditional expectation E [Y τ |B τ = x] is the average value of Y τ over paths that exit at x. This then implies (essentially)

Abstract. The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operatorFor the maximal operator M , the corresponding result is