In recent years a significant progress has been made in the theory of weighted norm inequalities with matrix weights. Original motivation comes from the theory of stationary stochastic processes (angle between "past" and "future") and goes back to Wiener.
For scalar stationary processes the problem reduces to weighted norm inequalities with scalar weights, and the answer is given by the famous Helson--Szego and Hunt--Muckenhoupt--Wheeden theorems. For multivariate processes the matrix-valued weights appear naturally, and the answer was obtained in in in our joint paper with A. Volberg paper about matrix Muckenhoupt condition (A_{2}).
Later it turns out that many interesting and difficult problems in harmonic analysis, operator theory and probability are closely related to Muckenhoupt weights and their matrix generalizations.
New methods were developed, which turn out to be very helpful in different areas of harmonic analysis.
Among the topics we are going to discuss:
- Weighted norm inequalities with matrix weights for singular integrals;
- Matrix Muckenhoupt weights, geometric theory of weights, difference
with scalar theory;
- Completely regular stationary processes;
- Bellman function and its applications.