Weak type bounds for rough maximal singular integrals near L1

Abstract

In this paper it is shown that for Ω∈Llog⁡L(Sd−1), the rough maximal singular integral operator TΩ⁎ is of weak type Llog⁡log⁡L(Rd). Furthermore, for w∈A1 and Ω∈L∞(Sd−1), it is shown that TΩ⁎ is of weak type Llog⁡log⁡L(w) with weight dependence [w]A1[w]A∞log⁡([w]A∞+1), which is same as the best known constant for the singular integral TΩ.