On $A_p$-$A_{\mathrm{\infty }}$ type estimates for square functions

We prove strong-type $A_p$-$A_{\mathrm{\infty }}$ estimate for square functions, improving on the $A_p$ bound due to Lerner. Entropy bounds, in the recent innovation of Treil-Volberg, are then proved. The techniques of proof include parallel stopping cubes, pigeon-hole arguments, and the approach to entropy bounds of Lacey–Spencer.