It is known that the improved Cotlar's inequality Bβf(z)β€CM(Bf)(z),
zβC, holds for the Beurling transform B, the maximal Beurling
transform Bβf(z)=Ξ΅>0supβββ«β£wβ£>Ξ΅βf(zβw)w21βdwβ, zβC, and the Hardy--Littlewood maximal operator M. In this note we consider
the maximal Beurling transform associated with squares, namely,
BSββf(z)=Ξ΅>0supβββ«wβ/Q(0,Ξ΅)βf(zβw)w21βdwβ, zβC,
Q(0,Ξ΅) being the square with sides parallel to the coordinate axis
of side length Ξ΅. We prove that BSββf(z)β€CM2(Bf)(z),
zβC, where M2=MβM is the iteration of the
Hardy--Littlewood maximal operator, and M2 cannot be replaced by M.Comment: 3 figure