Weak and strong $A_p$-$A_{\infty }$ estimates for square functions and related operators

Abstract

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w{]}_{A_p}^{1/p}[w{]}_{A_{\infty }}^{1/2-1/p}\lesssim [w{]}_{A_p}^{1/2}$ for the weak type norm of square functions on $L^p(w)$ for $p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.

12 pages, typos corrected. Accepted for publication in Proc. Amer. Math. Soc

References in corpus (1)

• A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators

Cited by in corpus (6)

• Sparse domination of sharp variational truncations
• $A_p$-$A_{\infty }$ estimates for multilinear maximal and sparse operators
• $A_1$ theory of weights for rough homogeneous singular integrals and commutators
• A quantitative approach to weighted Carleson Condition
• Intrinsic square functions with arbitrary aperture
• Weighted $L^p$ bounds for the Marcinkiewicz integral