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)= sup_ε>0∣∫_|w|>εf(z−w)1/w²dw∣, 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)=sup_ε>0∣∫_w∉Q(0,ε) f(z−w)1/w²dw∣∣∣, 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 M² cannot be replaced by M. [ABSTRACT FROM AUTHOR]