From the Classical Fefferman-Kenig-Pipher $$A_{\infty }$$ Inequality to Its Matrix Version and Beyond

Abstract The main goal of this article is the celebration of Jill Pipher, who was postdoctoral mentor of the author and continued to be her mentor. We review the Fefferman-Kenig-Pipher $$A_{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> inequality for scalar weights using the heat extension. We transition from a Bellman function proof in the heat setting to its dyadic scalar version. From here, we transition to the matrix analog of this inequality. The matrix result itself is implicit in papers by Treil and Volberg. In this text we present a more classical looking Bellman proof with derivative estimates resulting in the dyadic convexity estimate. We also prove via Bellman function an unrelated general embedding sum whose scalar version has been used in sharp paraproduct estimates. In the matrix case this inequality holds in the sense of operators and it appears to be new.