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Perturbation Of Burkholder's Martingale Transform And Monge-Ampère Equation, Nicholas Boros, Prabhu Janakiraman, Alexander Volberg
Perturbation Of Burkholder's Martingale Transform And Monge-Ampère Equation, Nicholas Boros, Prabhu Janakiraman, Alexander Volberg
Faculty Scholarship – Mathematics
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 Lp-norm of a small "quadratic perturbations"
of the martingale transform in Lp. By "quadratic perturbation" of the martin-
gale transform we mean the Lp 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).