论文标题
初始$ l^2 \ times \ cdots \ times l^2 $多线性操作员的边界
Initial $L^2\times\cdots\times L^2 $ bounds for multilinear operators
论文作者
论文摘要
$ l^p $有限的卷积运算符理论是基于最初的$ l^2 \ to l^2 $估算的估算值。鉴于在这种情况下，Plancherel的身份不可用，多线性操作员的相应理论缺乏如此简单的初始估计，而到目前为止，尚不清楚自然的初始估计可能是什么。在这项工作中，我们准确地实现了这个目标，即获得初始的$ l^2 \ times \ cdots \ times l^2 \至l^{2/m} $估算$ m $ linear乘数运算符的通用构建块。我们将此结果应用于推导多线性粗糙奇异积分，Hörmander型的乘数以及其衍生物满足定性估计值的乘数的类似边界。
The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogous bounds for multilinear rough singular integrals, multipliers of Hörmander type, and multipliers whose derivatives satisfy qualitative estimates.
