On Kenig's Class $AE$

In this note, we explore Kenig's class $AE$ of analytic extensions. It was introduced in his PhD Thesis. Later, a Smirnov-type condition was added by Jerison and Kenig. We re-examine this class and prove that, with this addition, the $H^1$-condition in the original definition becomes unnecessary. In order to accomplish this, we establish an equivalence between a stronger $H^1$-integrability condition and the Smirnov-type condition. To do so, we state in closed form some ideas of an article of Hunt, Muckenhoupt and Wheeden. As a consequence, we generalize in a direct manner some results of Carro, Naibo and Ortiz-Caraballo needed to study the Neumann problem with $A_{\infty}$-measures on the boundary.