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)=ε>0sup∫∣w∣>εf(z−w)w21dw, z∈C, and the Hardy--Littlewood maximal operator M. In this note we consider
the maximal Beurling transform associated with squares, namely,
BS∗f(z)=ε>0sup∫w∈/Q(0,ε)f(z−w)w21dw, z∈C,
Q(0,ε) being the square with sides parallel to the coordinate axis
of side length ε. We prove that BS∗f(z)≤CM2(Bf)(z),
z∈C, where M2=M∘M is the iteration of the
Hardy--Littlewood maximal operator, and M2 cannot be replaced by M.Comment: 3 figure