JournalsrmiVol. 36, No. 1pp. 99–158

LpL^p estimates for semi-degenerate simplex multipliers

  • Robert Kesler

    Santa Monica, USA
$L^p$ estimates for semi-degenerate simplex multipliers cover
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Abstract

Muscalu, Tao, and Thiele prove LpL^p estimates for the "Biest" operator defined on Schwartz functions by the map

C1,1,1:(f1,f2,f3)ξ1<ξ2<ξ3[j=13f^j(ξj)e2πixξj]dξC^{1,1,1}: (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < \xi_3} \Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j } \Big] \,d \vec{\xi}

via a time-frequency argument that produces bounds for all multipliers with non-degenerate trilinear simplex symbols. In this article we prove LpL^p estimates for a pair of simplex multipliers defined on Schwartz functions by the maps \begin{align*} C^{1,1,-2}:\ & (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < -{\xi_3}/{2}}\Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j } \Big] \,d \vec{\xi} \\ C^{1,1,1,-2}:\ & (f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 < \xi_2 < \xi_3 < -{\xi_4}/{2}} \Big[\prod_{j=1}^4 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j} \Big] \,d \vec{\xi} \end{align*} for which the non-degeneracy condition fails. Our argument combines the standard 2\ell^2-based energy with an 1\ell^1-based energy in order to enable summability over various size parameters. As a consequence, we obtain that C1,1,2C^{1,1,-2} maps into LpL^p for all 1/2<p<1/2 < p < \infty and C1,1,1,2C^{1,1,1,-2} maps into LpL^p for all 1/3<p<1/3 < p < \infty. Both target LpL^p ranges are shown to be sharp.

Cite this article

Robert Kesler, LpL^p estimates for semi-degenerate simplex multipliers. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 99–158

DOI 10.4171/RMI/1123