S. Treil and A. Volberg
In the paper we give an alternative proof of the fact that the vector Muckenhoupt condition,
is necessary and sufficient for the boundedness of Hilbert Transform in with matrix weight (for the original proof see our paper Wavelets and the angle between past and future ).
The main technical tool we are using here is a matrix version of the Littlewood--Paley type inequality that gives an equivalent norm in the weighted space in terms of weighted norm of the derivative of the harmonic extension see Theorem 4.2 in the paper. The scalar version was developed by us earlier in the paper A simple proof of the Hunt - Muckenhoupt - Wheeden theorem.
This equivalent norm inequality can be viewed as a continuous analogue of the wavelet type decomposition (Haar system is a Riesz basis in ) that was used by us in Wavelets and the angle between past and future . But in this case the continuous ``system of coefficients'' (derivatives of harmonic extension) is over-determined, so it is more appropriate to call it continuous frame decomposition.
Although the main result about boundedness of the Riesz Projection is already known, we feel that the technique we use is of independent interest, and deserves separate consideration. In a sense the main result is Theorem 4.2 in the text about continuous frame decomposition, and boundedness of the Riesz Projection serves as an illustration of usefulness of this theorem.