arXiv 2003.041672020-03math.FA

Lorentz空间的Sawyer型不等式

Sawyer-type inequalities for Lorentz spaces

Carlos Pérez, Eduard Roure Perdices

DOI：: 10.1007/s00208-021-02240-4

摘要：

Hardy-Littlewood极大算子满足经典的Sawyer型估计$$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^{1}(u)}, $$，其中$u\in A_1$和$uv\in A_{\infty}$ .我们证明了一个新的扩展，这一结果的一般限制弱型的情况下。也就是说，对于$p>1$，$u\in A_p^{\mathcal R}$和$uv^p \in A_\infty$，$$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{p,\infty}(uv^p)} \leq C_{u,v} \Vert f \Vert_{L^{p,1}(u)}. $$，我们从这些估计中导出了Hardy-Littlewood极大算子的$m$ -重积的新的加权限制弱型界和Sawyer型不等式.我们还提出了一种创新的技术，使我们能够转移这样的估计到一个大类的多变量算子，包括$m$线性Calderón-Zygmund运营商，避免了$A_\infty$外推定理，并产生许多估计，还没有出现在文献中。特别是，我们得到了一个新的表征$A_p^{\mathcal R}$。进一步，我们引入了刻画多重（次）线性极大算子$\mathcal M$的限制弱型上界的权类，记为$A_{\vec P}^{\mathcal R}$，建立了稀疏算子和m-线性Calderón-Zygmund算子的类似上界，并研究了这类算子和权的相应的多元Sawyer型不等式.我们的结果结合了混合限制弱型范数不等式、$A_p^{\mathcal R}$和$A_{\vec P}^{\mathcal R}$权以及Lorentz空间.

英文摘要：

The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^{1}(u)}, $$ where $u\in A_1$ and $uv\in A_{\infty}$. We prove a novel extension of this result to the general restricted weak type case. That is, for $p>1$, $u\in A_p^{\mathcal R}$, and $uv^p \in A_\infty$, $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{p,\infty}(uv^p)} \leq C_{u,v} \Vert f \Vert_{L^{p,1}(u)}. $$ From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the $m$-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including $m$-linear Calderón-Zygmund operators, avoiding the $A_\infty$ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $A_p^{\mathcal R}$. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $\mathcal M$, denoted by $A_{\vec P}^{\mathcal R}$, establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $A_p^{\mathcal R}$ and $A_{\vec P}^{\mathcal R}$ weights, and Lorentz spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文