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

We prove strong-type $A_p$-$A_\infty$ 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.

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