The Hilbert transform has a linear bound in the $A_{ 2}$ characteristic on weighted $L^{ 2}$, \begin{ equation*} \left\Vert H\right\Vert _{ L^{ 2}(w)\rightarrow L^{ 2}(w)}\lesssim \left[ w \right] _{ A_{ 2}}, \end{ equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{ w^{ \frac{ 1}{ 2}}}\mathcal{ S }M_{ w^{ -\frac{ 1}{ 2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{ equation*} M_{ f}=P_{ f}^{ -}+P_{ f}^{ 0}+P_{ f}^{ +}. \end{ equation*} The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.
Categories: math.CA math.CV