What is left of weighted singular integrals estimates (or Monge and Ampère meet Itô and Bollobas)?
In the last 20 years there was a huge progress in finding the sharp estimates of weighted singular integrals. This progress was motivated by problems in such various areas as regularity of stochastic processes and borderline regularity of Beltrami equation and certain elliptic equations in divergence form. However, there is one singular operator and exactly one weighted space, where the sharp estimates are still unknown. This is a dyadic square (or Lusin) function operator.
We will show some progress in this remaining sharp estimate and relate it to certain non-linear PDE and its discrete form.