论文标题
通过螺旋方法估计的混合符号估计
Mixed-norm estimates via the helicoidal method
论文作者
论文摘要
我们证明了$ \ rr r^d $的多线性操作员的多个矢量值和混合声 - 估计值，更准确地说，对于沿$ k $尺寸的空间，与符号单数相关的多线性操作员$ t_k $，用于$ k $二维的空间，以及用于Hardy-little Wiles-Little Wood-Little Wood-little Wood-Liles-Liles-litter Woodsmimal Maximal Maximal Maximal Maximal Maximal Maximal Maximal Maximal Maximal功能。当尺寸$ d \ geq 2 $时，输入函数不一定在$ l^p（\ rr r^d）$中，而是可以是混合 - 摩尔空间的元素$ l^{p_1} _ {x_1} \ ldots ldots ldots ldots l^{p_d} _ {p_d} _ {x_d} $。 这样的结果会带来有趣的后果，尤其是在涉及$ l^\ infty $空间时。其中，我们提到了奇异积分的混合 - 织机 - 丝毫型型不平等，以及与某些合理符号相关的多线性操作员的界限。我们还提出了不容易受到各向同性续订的操作员的例子，这些示例仅满足``纯粹的混合估计估计''而没有经典的$ l^p $估计。 依赖于螺旋性方法暗示的先前估计值，我们还证明了（非混合标记）对通用奇异的brascamp-lieb型不平等的估计值。
We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy ``purely mixed-norm estimates" and no classical $L^p$ estimates. Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.
