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 $M_u$: $${\parallel \frac{M_u(fv)}{v}\parallel }_{L^{1,\mathrm{\infty }}(uv)}\le C_{u,v}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(uv)},u,uv\in A_{\mathrm{\infty }}\mathrm{.}$$ Our approach can be adapted to recover weak-type $A_{\vec{P}}$ 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 $A_p^{\mathcal{R}}$ 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 $A_p^{\mathcal{R}}$.