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)≤C(1/t ∫ t0 f∗(s)ds+∫∞t(1+log s/t)−1 φ(1+log s/t) f∗(s)ds/s), where φ is determined by the Muckenhoupt Ap-weight norm behaviour of the operator.