Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator

In this paper we consider a new approach to weighted norm inequalities. This approach is based on weighted embedding theorems of Carleson type. When $p=2$ the boundedness of an embedding operator follows from a technical trick (the Vinogradov–Senichkin test) which amounts to “doubling” the kernel of this operator. We show how this approach enables us to prove the Hunt–Muckenhoupt–Wheeden and Sawyer theorems. We also formulate a necessary and sufficient condition for vector weighted boundedness of the Hubert transform (the matrix $A_2$-condition), which we have obtained using this approach.