Multilinear Fourier multipliers with minimal Sobolev regularity, II

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  • Multilinear Fourier multipliers with minimal Sobolev regularity(2)

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<p>We provide characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let Hq(ℝn) denote the Hardy space when 0 < q ≤ 1 and the Lebesgue space Lq(ℝn) when 1 < q ≤ ∞. We find optimal conditions on m-linear Fourier multiplier operators to be bounded from Hp1 × … × Hpm to Lp when 1/p = 1/p1 + … + 1/pm in terms of local L2-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón 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,∞)m.</p>

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