论文标题
从$ a_1 $到$ a_ \ infty $：某些最大运营商的新混合不平等现象
From $A_1$ to $A_\infty$: New mixed inequalities for certain maximal operators
论文作者
论文摘要
在本文中，我们证明了与年轻功能相关的最大运算符的混合不平等，这是对\ cite {berra}中建立的猜想的改进。具体而言，给定$ r \ geq 1 $，$ u \ in a_1 $，$ v^r \ in A_ \ infty $和某些属性的年轻函数$φ$，我们有这种不平等 \ [uv^r \ left（\ left \ {x \ in \ mathbb {r}^n：\ frac {m_或c \ int _ {\ mathbb {r}^n}φ\ left（\ frac {| f（x）|} {t} {t} \ right）u（x）v^r（x）\，dx \，dx \]持有每个正$ t $。相关运算符$ \ frac {m_φ（fv）（x）} {m_φv（x）} $似乎是一个足够的扩展，当$ v^r \ in A_ \ infty $中，因为当我们假设我们在a_1 $中假设$ v^r \ in a_1 $，我们可以替换$ v $ $ v $ pred $ v $ pret a $ m_ pret a $ m_ m m mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm m m imm m m_ \ cite {berra-carena-pradolini（mn）}。 作为应用程序，我们进一步展示并证明了通用分数最大运算符$ m_ {γ，φ} $的混合不平等，其中$ 0 <γ<n $和$φ$是$ l \ log l log l $ type的年轻函数。
In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in \cite{Berra}. Concretely, given $r\geq 1$, $u\in A_1$, $v^r\in A_\infty$ and a Young function $Φ$ with certain properties, we have that inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{M_Φv(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. The involved operator $\frac{M_Φ(fv)(x)}{M_Φv(x)}$ seems to be an adequate extension when $v^r\in A_\infty$, since when we assume $v^r\in A_1$ we can replace $M_Φv$ by $v$, yielding a mixed inequality for $M_Φ$ proved in \cite{Berra-Carena-Pradolini(MN)}. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator $M_{γ,Φ}$, where $0<γ<n$ and $Φ$ is a Young function of $L\log L$ type.
