Adam Osȩkowski

Survey Article: Bellman function method and sharp inequalities for martingales

The Bellman function method is an efficient device which enables relating certain types of estimates arising in probability and harmonic analysis to the existence of the associated special function satisfying appropriate majorization and concavity. This technique has gained considerable interest in recent years and led to many interesting results concerning the boundedness of wide classes of singular integrals, Fourier multipliers, maximal functions and other related objects. The objective of this survey is to describe the Bellman function approach to certain classical results for martingale transforms. We present the detailed study of the weak-type and moment estimates, and develop some arguments which allow us to simplify and extend the statements, originally proven by Burkholder and others.

BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

Let μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and c p , q , and C p , q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for c p , q and C p , q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝ n with respect to a general product measure.

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

LetM and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on Rd. (i) We prove that for any 0 < p < ∞, any weight w on Rd and any measurable f on Rd, we have Fefferman-Stein-type estimate ||G(f)||Lp(w) ≤ e||f ||Lp(Mw). For each p the constant e1/p is the best possible. (ii) We show that for any weight w on Rd and any measurable f on Rd, ∫ Rd G(f)1/Mwwdx ≤ e ∫

Sharp estimates for holomorphic functions on the unit ball of ℂn

The purpose of the paper is to establish sharp estimates for the Hilbert transform and the Riesz projection on the unit sphere of . The proof rests on the existence of certain superharmonic functions on satisfying appropriate majorization conditions.

Sharp Inequalities for the Haar System and Fourier Multipliers

A classical result of Paley and Marcinkiewicz asserts that the Haar system on forms an unconditional basis of provided . That is, if denotes the projection onto the subspace generated by ( is an arbitrary subset of ), then for some universal constant depending only on . The purpose of this paper is to study related restricted weak-type bounds for the projections . Specifically, for any we identify the best constant such that for every and any Borel subset of . In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.