The Hörmander multiplier theorem. III: The complete bilinear case via interpolation. (English) Zbl 1426.42012
Summary: We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space \(L^r_s(\mathbb{R}^{2n})\), \(2\le r<\infty\), \(rs>2n\), uniformly over all annuli. More precisely, given such a symbol with smoothness index \(s\), we find the largest open set of indices \((1/p_1,1/p_2)\) for which we have boundedness for the associated bilinear multiplier operator from \(L^{p_1}(\mathbb{R}^{ n})\times L^{p_2} (\mathbb{R}^{ n})\) to \(L^p(\mathbb{R}^{ n})\) when \(1/p=1/p_1+1/p_2\), \(1<p_1\), \(p_2<\infty\).
For Parts I and II, see [L. Grafakos et al., Ill. J. Math. 61, No. 1–2, 25–35 (2017; Zbl 1395.42025)] and [L. Grafakos et al., Math. Z. 289, No. 3–4, 875–887 (2018; Zbl 1405.42014)].
For Parts I and II, see [L. Grafakos et al., Ill. J. Math. 61, No. 1–2, 25–35 (2017; Zbl 1395.42025)] and [L. Grafakos et al., Math. Z. 289, No. 3–4, 875–887 (2018; Zbl 1405.42014)].
MSC:
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42B30 | \(H^p\)-spaces |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
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