The author considers the operators \[ \widetilde Tf(t)= {1\over\sqrt{w(t)}} T(f(r)\sqrt{w(r)})(t), \] where \(T\) is the Hardy-Littlewood maximal function, the Hilbert transform or the Carleson operator. It has been relevant, in questions of harmonic analysis on noncompact rank one symmetric spaces, the boundedness of \(\widetilde T\) from \(L^p_w({\mathcal A})\) to \(L^p_w({\mathcal A})+ L^2_w({\mathcal A})\), \(10\) and \(t\in{\mathcal A}= [1,\infty)\) [\textit{E. Prestini}, Proc. Am. Math. Soc. 124, No. 4, 1171-1175 (1996; Zbl 0847.42006)]. The author extends the result in the above paper to a larger class of weights that include for instance \(\exp(\cdots\exp(t)\cdots)\) with \(t\in{\mathcal A}= [1,\infty)\). The case \({\mathcal A}= (0,1]\) is also studied.