A rearranged good 𝜆 inequality

Bagby, Richard J.; Kurtz, Douglas S.

doi:10.1090/S0002-9947-1986-0814913-7

A rearranged good $\lambda$ inequality
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by Richard J. Bagby and Douglas S. Kurtz PDF

Trans. Amer. Math. Soc. 293 (1986), 71-81 Request permission

Abstract:

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

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