arXiv 2108.092962021-08-20math.CA

含径向函数的混合不等式的更好界及其应用

Better bounds on mixed inequalities involving radial functions and applications

Fabio Berra

备注：: 18 pages

摘要：

当函数$v$是一个不能局部可积的径向幂函数时，我们证明了广义极大算子$M_Φ$的混合不等式。具体地，令$u$为权重，$v(x)=|x|^β$具有$β<-n$和$r\geq 1$。如果$Φ$是具有某些性质的Young函数，则不等式\[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\]对每个$t>0$和每个有界函数都成立。这改进了在\cite{BCP-M}中证明的类似混合估计。作为应用，我们给出了广义分数次极大算子$M_{γ,Φ}$的混合估计，其中$0<γ<n$和$Φ$是$L\log L$型的.一个特殊的情况下，涉及分数极大算子$M_γ$允许获得一个类似的估计分数积分算子$I_γ$通过外推的结果。此外，我们还给出了奇异积分Calderón-Zygmund算子和$I_γ$算子的算子的混合估计，两者都带有Lipschitz符号。

英文摘要：

We prove mixed inequalities for the generalized maximal operator $M_Φ$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^β$ with $β<-n$ and $r\geq 1$. If $Φ$ is a Young function with certain properties, then the inequality \[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\] holds for every $t>0$ and every bounded function. This improves a similar mixed estimate proved in \cite{BCP-M}. As an application, we give mixed estimates for the generalized fractional maximal operator $M_{γ,Φ}$, where $0<γ<n$ and $Φ$ is of $L\log L$ type. A special case involving the fractional maximal operator $M_γ$ allows to obtain a similar estimate for the fractional integral operator $I_γ$ through an extrapolation result. Furthermore, we also give mixed estimates for commutators of singular integral Calderón-Zygmund operators and of $I_γ$, both with Lipschitz symbol.

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