Weighted norm inequalities for the Dirichlet transform

The Dirichlet transform is defined for suitable functions by ( D f ) ( x ) : = 1 π ∫ − ∞ ∞ sin ( x − y ) x − y f ( y ) d y , x ∈ R . We show that for 1 < p < ∞ and nonnegative w ( x ) on R ∫ R | ( D f ) ( x ) | p w ( x ) d x ⩽ C p p ∫ R | f ( y ) | p w ( y ) d y , with C p > 0 independent of f, if and only if there exists K > 0 such that ∫ I w ( x ) d x ( ∫ I w ( x ) − 1 p − 1 d x ) p − 1 ⩽ K | I | p for all intervals I ⊂ R with length | I | ⩾ π .