Weighted norm inequalities for singular integral operators

Abstract

For a Calderón-Zygmund singular integral operator T, we show that the following weighted inequality holds Z Rn|T f(y)|p w(y)dy ≤ C Z Rn |f(y)| p M[p]+1w(y)dy, where Mk is the Hardy-Littlewood maximal operator M iterated k times, and [p] is the integer part of p. Moreover, the result is sharp since it does not hold for M[p]. We also give the following endpoint result: w({y ∈ R n : |T f(y)| > λ}) ≤ C λ Z Rn |f(y)| M2w(y)dy.