Weighted norm inequalities for the Dirichlet transform

The Dirichlet transform is defined for suitable functions by(Df)(x):=1π∫−∞∞sin(x−y)x−yf(y)dy,x∈R. We show that for 10Cp>0 independent of f  , if and only if there exists K>0K>0 such that∫Iw(x)dx(∫Iw(x)−1p−1dx)p−1⩽K|I|p for all intervals I⊂RI⊂R with length |I|⩾π|I|⩾π.