The Bellman functions and two-weight inequalities for Haar multipliers

Abstract

We are going to give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.1991 Mathematics Subject Classification. 42B20, 42A50, 47B35.

Weighted square function inequalities

“…We establish the inequality (1.5) with the use of the so-called Bellman function method, a powerful technique used widely in analysis and probability (cf. [17], [18], and the paper [10] already cited above). The sharpness of the exponent 1/2 is shown with the use of appropriate examples.…”

The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure

“…There are also some interesting results about two weights. We refer the reader to the work in [6,24,[26][27][28]. In [26,27], Pérez provided a sufficient condition on weights ω, ν to ensure the boundedness of the general maximal functions M ℘ including the boundedness of M n dx from L p (ω) to L q (ν).…”

The Bellman functions for a certain two-weight inequality: A case study

“…. The answer is "Yes", and this was proved in [NTV1] for the family T (a) as a whole (this means that the Sawyer type conditions are fulfilled uniformly in a if and only if all operators T (a) are bounded uniformly). In the following statement σ means an arbitrary sequence of signs.…”

“…Throughout, conditions of this kind will be called the Sawyer type conditions. In the paper [NTV1], the main aim was to present a uniform approach to recovering many of Sawyer's results (at least in L 2 ) and to obtaining at least one new theorem of Sawyer type: a necessary and sufficient condition for the boundedness of Haar multipliers. As far as we know, it was the first theorem of Sawyer type for singular integral operators: all earlier results concerned operators with positive kernels only.…”

“…As far as we know, it was the first theorem of Sawyer type for singular integral operators: all earlier results concerned operators with positive kernels only. The Bellman function method was used in [NTV1]; actually, the 1995 preprint version of that paper was the place where the harmonic analysis Bellman function appeared for the first time -except, of course, the famous Burkholder paper [B], which was implicitly related to the Bellman function technique, but employed another language.…”