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In 1966, using the weak type (2, 2) of C∗, Carleson [2] proved his celebrated theorem on almost everywhere convergence of the Fourier series of the functions in L([−π, π]). Following that, Hunt [7] extended Carleson’s theorem to L([−π, π]) for 1 < p < ∞. In 1970, Sjölin [11] studied an analogue of the Carleson operator C∗ on R. Suppose that K is an appropriate Calderón-Zygmund kernel. Then the Carleson type maximal operator S∗ on R is defined by

Weighted $L^{p}$ boundedness of Carleson type maximal operators