From A1 to $A_{\infty }$: New Mixed Inequalities for Certain Maximal Operators

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in Berra (Proc. Am. Math. Soc. 147(10), 4259–4273, 2019). Concretely, given r ≥ 1, u ∈ A1, \(v^{r}\in A_{\infty }\) and function Φ with certain properties, have that inequality $$ uv^{r}\left( \left\{x\in \mathbb{R}^{n}: \frac{M_{\Phi}(fv)(x)}{ v(x)}>t\right\}\right)\leq C{\int}_{\mathbb{R}^{n}}{\Phi}\left( \frac{|f(x)|}{t}\right)u(x)v^{r}(x) dx holds every positive t. As application, furthermore exhibe the generalized fractional operator Mγ,Φ, where 0 < γ n is \(L\log L\) type.