Weighted Norm Inequalities for Singular Integral Operators

For a Calderón‐Zygmund singular integral operator T , we show that the following weighted inequality holds∫ R n| T f ( y ) | p w ( y ) dy ⩽ C ∫ R n| f ( y ) | pM [ p ] + 1w ( y ) dy ,where M k 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 endpoints results: w ( { y ∈ R n : | Tf ( y ) | > λ } ) ⩽ C λ ∫ R n| f ( y ) | M 2 w ( y ) dy , and∫ R n| T f ( y ) | w ( y ) dy ⩽ C ∥ f ∥ H 1 ( M w ), where H 1 (μ) is the atomic Hardy space with respect to μ.