A rearranged good 𝜆 inequality

Abstract

Let T f Tf be a maximal Calderón-Zygmund singular integral, M f Mf the Hardy-Littlewood maximal function, and w w an A ∞ {A_\infty } weight. We replace the “good λ \lambda ” inequality \[ w ( { x : T f ( x ) > 2 λ and M f ( x ) ≤ ε λ } ) ≤ C ( ε ) w ( { x : T f ( x ) > λ } ) w\left ( {\{ x:\,Tf(x) > 2\lambda \,{\text {and}}\,Mf(x) \leq \varepsilon \lambda \} } \right ) \leq C(\varepsilon )w\left ( {\{ x:\,Tf(x) > \lambda \} } \right ) \] by the rearrangement inequality \[ ( T f ) w ∗ ( t ) ≤ C ( M f ) w ∗ ( t / 2 ) + ( T f ) w ∗ ( 2 t ) (Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t) \] and show that it gives better estimates for T f Tf . In particular, we obtain best possible weighted L p {L^p} bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for ( T f ) ∗ {(Tf)^ \ast } .