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 $$(Tf{)}^{\ast }(t)\le C\left(\frac{1}{t}{\int }_0^t{f}^{\ast }(s)ds+{\int }_t^{\infty }{\left(1+\log \frac{s}{t}\right)}^{-1}\ddot{I}†\left(1+\log \frac{s}{t}\right)f^{\ast }(s)\frac{ds}{s}\right),$$ where $\ddot{I}†$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator.

23 pages