Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Abstract

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)

The L^p-norms of the Beurling–Ahlfors transform on radial functions

“…While the lower bound B L p (C)→L p (C) ≥ p * −1 was already known to Lehto [14], the question about the opposite estimate remains open. Most results rely on the ideas of Burkholder and the Bellman function technique [7,19,4,11,2,8], with the current best being B L p (C)→L p (C) ≤ 1.575(p * − 1) due to Bañuelos and Janakiraman [2] (see also [8] for an asymptotically better estimate as p → ∞).…”

A method of rotations for Lévy multipliers

“…The methods of [8] were refined in [7] taking advantage of the fact that the martingales arising in the representation of the Beurling-Ahlfors transform have certain orthogonality properties to produce the bound B p ≤ 1.575(p * − 1) which is, as of now, the best known bound valid for all 1 < p < ∞. In [11], this bound was improved to B p ≤ 1.4(p * − 1) for 1000 < p < ∞.…”

On the Π-operator in Clifford analysis

“…Here we could use the idea of estimates via Martingales (see [3] and references therein). But for the purpose of this paper it is enough to apply the idea of Petermichl, Slavin and Wick [21] of using the Bellman function technique.…”