Multilinear Fourier multipliers with minimal Sobolev regularity, II
Multilinear Fourier multipliers with minimal Sobolev regularity, II
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DOI：
10.2969/jmsj/06920529
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发表时间：
2017-04
影响因子：
0.7
通讯作者：
L. Grafakos;Akihiko Miyachi;H. Nguyen;Naohito Tomita
中科院分区：
文献类型：
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作者：
L. Grafakos;Akihiko Miyachi;H. Nguyen;Naohito Tomita
给出了多线性傅立叶算子在符号局部为Sobolev空间的Hardy空间或Lebesgue型空间上有界性的刻画。设H(R)表示0<；Q≤1时的Hardy空间和1<；Q≤∞时的勒贝格空间L(R)。利用算子符号的局部L-索博列夫空间估计，得到了当1/p=1/p1+···+1/Pm时，m-线性傅立叶乘子算子从h1×···×Hm到L有界的最优条件。我们的条件提供了Calderon和Torchinsky[1]的线性结果以及Miyachi和Tomita[17]的双线性结果的多线性相似。推广到一般的m在技术上和组合上都要复杂得多；符号所需的最优Soblev空间光滑性取决于哈代-勒贝格指数，并且在由[0，−)中的M2M∞1+1点的构形形成的各种凸单纯形上是恒定的。
We provide characterizations for boundedness of multilinear Fourier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let H(R) denote the Hardy space when 0 < q ≤ 1 and the Lebesgue space L(R) when 1 < q ≤ ∞. We find optimal conditions on m-linear Fourier multiplier operators to be bounded from H1 × · · ·×Hm to L when 1/p = 1/p1 + · · ·+ 1/pm in terms of local L-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderon and Torchinsky [1] and of the bilinear results of Miyachi and Tomita [17]. The extension to general m is significantly more complicated both technically and combinatorially; the optimal Sobolev space smoothness required of the symbol depends on the Hardy-Lebesgue exponents and is constant on various convex simplices formed by configurations of m2m−1 + 1 points in [0,∞).