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

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Abstract

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, uA1, \(v^{r}\in A_{\infty }\) and a Young function Φ with certain properties, we 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 for every positive t. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator Mγ, where 0 < γ < n and Φ is a Young function of \(L\log L\) type.

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Correspondence to Fabio Berra.

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The author was supported by CONICET and UNL

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Berra, F. From A1 to \(A_{\infty }\): New Mixed Inequalities for Certain Maximal Operators. Potential Anal 57, 1–27 (2022). https://doi.org/10.1007/s11118-021-09903-6

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