Characterization of multilinear multipliers in terms of Sobolev space regularity. (English) Zbl 1470.42017
Summary: We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in \(L^r\)-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case \(1<r\le 2\) and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case \(r>2\) cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous results of L. Grafakos et al. [Can. J. Math. 65, No. 2, 299–330 (2013; Zbl 1275.42016); J. Math. Soc. Japan 69, No. 2, 529–562 (2017; Zbl 1372.42007)], L. Grafakos and H. van Nguyen [Colloq. Math. 144, No. 1, 1–30 (2016; Zbl 1339.42012)], and A. Miyachi and N. Tomita [Rev. Mat. Iberoam. 29, No. 2, 495–530 (2013; Zbl 1275.42017)], who only considered the case \(r=2\).
MSC:
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B30 | \(H^p\)-spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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