From weak type weighted inequality to pointwise estimate for the decreasing rearrangement

Abstract: We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form $$ (Tf)^*(t) \leq C\left( \frac 1t\int_0^tf^*(s)\,ds + \int_t^\infty \left( 1 + \log\frac st \right)^{- 1}\varphi \left(1 + \log\frac st \right) f^*(s)\,\frac{ds}s \right), $$ where $\varphi$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator.