论文标题
索耶型的洛伦兹空间的不平等现象
Sawyer-type inequalities for Lorentz spaces
论文作者
论文摘要
坚硬的小木最大操作员满足经典的锯型估计$$ \ left \ weft \ vert \ frac {mf} {v} {v} \ right \ right \ vert_ {l^{1，\ infty}（\ infty}（uftty}（uv）（uv）（uv）} \ leq c_ { $ u \ in a_1 $和$ uv \ in _ {\ infty} $中的$ u \。我们证明了这一结果的新型扩展到一般受限的弱类案例。也就是说，对于$ p> 1 $，$ u \ a_p^{\ mathcal r} $，以及a_ \ infty $，$ uv^p \ in a_ \ infty $，$$ \ left \ welet \ vert \ frac {mf} {v} {v} {v} \ right \ right \ right \ vert_ { \ vert f \ vert_ {l^{p，1}（u）}。 $$从这些估计值中，我们推断出新的加权限制性弱类型界限和锯齿类型的不平等现象，用于Hardy-Little Woods Maximal Maxal Operators的$ M $折叠产品。我们还提出了一种创新技术，该技术使我们能够将这些估计值转移到大量的多变量运营商中，包括$ M $lineareareareareareareareArearcalderón-Zygmund运营商，避免了$ a_ \ iffty $ undty $外推理，并产生许多以前从未出现过的估计值。特别是，我们获得了$ a_p^{\ Mathcal r} $的新特征。此外，我们介绍了特征多（子）线性最大运算符$ \ Mathcal m $的限制弱类型界限的重量类别此类操作员和权重的锯木型不平等。我们的结果结合了混合限制的弱类型规范不等式，$ a_p^{\ Mathcal r} $和$ a _ {\ vec p}^{\ Mathcal r} $ weights $ strace和lorentz Space。
The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \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.
