Title: "Haar multipliers, Bellman functions and more."

Abstract: In this talk we will revisit known results about Haar
   multipliers and variants.
    The Haar multipliers are like pseudodifferential operators where
   trigonometric functions have been replaced by Haar functions:
      \[Tf(x) = \sum_{I\in D}\sigma (x,I)<f,h_I>h_I(x)\],
   here $D$ denotes the dyadic intervals, $\{h_I\}$ the Haar functions,
   $\sigma (x,I)$ is the symbol.  The symbols we are interested in
   are of the form $\sigma (x,I)= (w(x)/m_Iw)^t$, where $w$ is a weight,
   and $m_Iw$ denotes the average on the interval $I$. The necessary and
   suficient conditions for boundedness in $L^p$ of these operators
   are known. In particular for $t=1$ they reduce to $w$ belonging to
   the Reverse H\"older p class, when $t=-1/2$ to the $A_p$ class.
    We are interested in finding sharp bounds in term of the $A_p$
   and $RH_p$ characteristic of the weights. The techniques that
   have given fantastic results along those lines are the Bellman
   functions of Nazarov, Treil and Volberg. This is closely related to the
   problem of finding sharp bounds for the dyadic square function on
   weighted spaces. Recently S. Petermichl found the sharp bound for the
   Hilbert transform on $L^2(w)$, the proof reduces to checking some
   estimates on operators closely related to the Haar multipliers.