Mixed $A_p$-$A_r$ inequalities for classical singular integrals and Littlewood-Paley operators

Abstract: We prove mixed $A_p$-$A_r$ inequalities for several basic singular integrals, Littlewood-Paley operators, and the vector-valued maximal function. Our key point is that $r$ can be taken arbitrary big. Hence such inequalities are close in spirit to those obtained recently in the works by T. Hyt\"onen and C. P\'erez, and M. Lacey. On one hand, the "$A_p$-$A_{\infty}$" constant in these works involves two independent suprema. On the other hand, the "$A_p$-$A_r$" constant in our estimates involves a joint supremum but of a bigger expression. We show in simple examples that both such constants are incomparable. This leads to a natural conjecture that the estimates of both types can be further improved.