A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales

Abstract

Abstract. If (dn) n≥0 is a martingale difference sequence, (εn) n≥0 a sequence of numbers in {1, −1}, and n a positive integer, thenHere αp denotes the best constant. If 1 ≤ p ≤ 2, then αp = 2/Γ(p + 1) as was shown by Burkholder. We show here that αp = p p−1 /2 for the case p > 2, and that p p−1 /2 is also the best constant in the analogous inequality for two martingales M and N indexed by [0, ∞), right continuous with limits from the left, adapted to the same filtration, and such that [M, M ]t −[N, N ]t is nonnegative and nondecreasing in t. In Section 7, we prove a similar inequality for harmonic functions.

Sharp maximal Lp-bounds for continuous martingales and their differential subordinates

“…In the boundary case p = 1, the above moment inequality breaks down, but, as a substitution, there are certain weak-type and logarithmic estimates; see [8], [21] and [22]. There is also a corresponding maximal L 1 bound, which will be important for our considerations below.…”

“…This natural method has been applied in many cases: see e.g. [8,10,22]. The second method is more abstract and indirect in nature: one assumes that an estimate under investigation holds with some constant and then proves the existence of an associated special function, enjoying appropriate size and regularity conditions.…”

Sharp inequalities for Riesz transforms

“…t≥0 is nonnegative and nondecreasing as a function of t. Differential subordination of Y to X implies many interesting inequalities which have found applications in many areas of mathematics; see e.g. [2]- [6], [16]- [18], [22] and [23].…”

Sharp Localized Inequalities for Fourier Multipliers

“…The literature is too vast to review it here; we refer the interested reader to the papers [3]- [11], [17] and the references therein. Furthermore, the above theorem has been extended in many directions; consult, for instance, [10], [12], [13], [14], [18], [29] and [33].…”