| Given a sequence of martingale differences, Burkholder found the sharp constant for the Lp-norm of the corresponding martingale transform. We are able to determine the sharp L p-norm of small "quadratic perturbations" of the martingale transform in Lp. By "quadratic perturbation" of the martingale transform we mean the L p norm of the square root of the squares of the martingale transform and the original martingale (with small constant). The problem of perturbation of martingale transform appears naturally if one wants to estimate the linear combination of Riesz transforms (as, for example, in the case of Ahlfors--Beurling operator). Let {dk} k≥0 be a complex martingale difference in Lp[0, 1], where 1 < p < infinity and {epsilonk}k ≥0 a sequence of signs. We obtain the following generalization of Burkholder's famous result. If n ∈ Z+ , then we have the following estimate k=0n 3k,t dkLp 0,1,C 2≤ p*-12 +t21 2 k=0ndk Lp0,1 ,C, with ((p* -- 1)2 + tau2)½ being the sharp constant in the estimate for 1 < p < 2 and tau2 ≤ 12p-1 or 2 ≤ p < infinity and tau ∈ R , where p* -- 1 = max{p -- 1, 1p-1 }. This result is significant not only because it is a generalization of Burkholder's famous result from the 1980's, but also because we can apply to obtaining the exact operator norm of a certain singular integral operator.;Let R1 and R2 be the planar Riesz transforms. We compute the Lp-operator norm of a quadratic perturbation of R21-R22 as R21 -R22,tI LpC,C →LpC,C2 = p*-12+t 212 , for 1 < p < 2 and tau2 ≤ 12p-1 or 2 ≤ p < infinity and tau ∈ R . To obtain the lower bound estimate of, what we are calling a quadratic perturbation of R21-R22 , we discuss a new approach of constructing laminates (a special type of probability measure on matrices) to approximate the Riesz transforms. |
