The maximal Beurling transform associated with squares

It is known that the improved Cotlar's inequality B*f(z) ≤ CM(Bf)(z), z ∈ C, holds for the Beurling transform B, the maximal Beurling transform B*f(z) = supε>0 |∫|w|>ε f(z-w) 1/w2 dw|, z ∈ C, and the Hardy-Littlewood maximal operator M. In this note we consider the maximal Beurling transform associated with squares, namely, B*Sf(z) = supε>0 ∫w∉Q(0, ε) f(z-w) 1/w2 dw|, z ∈ C, Q(0, ε) being the square with sides parallel to the coordinate axis of side length ε. We prove that B*Sf(z) ≤ CM2(Bf)(z), z ∈ C, where M2 = M.M is the iteration of the Hardy-Littlewood maximal operator, and that M2 cannot be replaced by M.