From Weak Type Weighted Inequality to Pointwise Estimate for the Decreasing Rearrangement

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. In particular, our main estimate is of the form $$\begin{aligned} (Tf)^*(t) \le C\left( \frac{1}{t}\int _0^tf^*(s)\,ds + \int _t^\infty \left( 1 + \log \frac{s}{t} \right) ^{- 1}\varphi \left( 1 + \log \frac{s}{t} \right) f^*(s)\,\frac{ds}{s} \right) , \end{aligned}$$ where $$\varphi $$ is determined by the Muckenhoupt $$A_p-$$ weight norm behavior of the operator.