Extrapolation via Sawyer-type inequalities

We present a multi-variable extension of Rubio de Francia's restricted weak-type extrapolation theory that does not involve Rubio de Francia's iteration algorithm; instead, we rely on the following Sawyer-type inequality for the weighted Hardy-Littlewood maximal operator Mᵤ: Mᵤ (fv) v ₋^₁, (uv) Cₔ, ₕ f ₋℉ (ₔₕ), u, \, uv A_. Our approach can be adapted to recover weak-type A ₏ extrapolation schemes, including an endpoint result that falls outside the classical theory. Among the applications of our work, we highlight extending outside the Banach range the well-known equivalence between restricted weak-type and weak-type for characteristic functions, and obtaining mixed and restricted weak-type bounds with A₏^ R weights for relevant families of multi-variable operators, addressing the lack in the literature of these types of estimates. We also reveal several standalone properties of the class A₏^ R.