Mathematical Research Letters

Volume 27 (2020)

Number 2

Improved bounds for the bilinear spherical maximal operators

Pages: 397 – 434



Yaryong Heo (Department of Mathematics, Korea University, Seoul, South Korea)

Sunggeum Hong (Department of Mathematics, Chosun University, Gwangju, South Korea)

Chan Woo Yang (Department of Mathematics, Korea University, Seoul, South Korea)


In this paper we study the bilinear multiplier operator of the form\begin{align*}&H^t(f,g)(x)=\int_{\mathbb{R}^d}\!\int_{\mathbb{R}^d} m(t\xi,t\eta)\,\mathrm{e}^{2 \pi \mathrm{i} t|(\xi,\eta)|}\, \widehat{f}(\xi)\, \widehat{g}(\eta)\, \mathrm{e}^{2 \pi \mathrm{i}x(\xi +\eta)}\,d \xi d \eta, \\&1 \le t \le 2\end{align*}where $m$ satisfies the Marcinkiewicz–Mikhlin–Hörmander’s derivative conditions. And by obtaining some estimates for $H^t$, we establish the $L^{p_1}(\mathbb{R}^d) \times L^{p_2}(\mathbb{R}^d) \rightarrow L^p(\mathbb{R}^d)$ estimates for the bi(sub)-linear spherical maximal operators\[\mathcal{M}(f,g)(x)=\sup_{t>0} \left| \int_{\mathbb{S}^{2d-1}} f(x-ty)\, g(x-tz)\, d \sigma_{2d}(y,z)\right|\]

which was considered by Barrionevo et al in [1], here $\sigma_{2d}$ denotes the surface measure on the unit sphere $\mathbb{S}^{2d-1}$. In order to investigate $\mathcal{M}$ we use the asymptotic expansion of the Fourier transform of the surface measure $\sigma_{2d}$ and study the related bilinear multiplier operator $H^t(f,g)$. To treat the bad behavior of the term $\mathrm{e}^{2 \pi \mathrm{i} t|(\xi,\eta)|}$ in $H^t$, we rewrite $\mathrm{e}^{2 \pi \mathrm{i} t|(\xi,\eta)|}$ as the summation of $\mathrm{e}^{2 \pi \mathrm{i}t\sqrt{N^2+|\eta|^2}} a_N(t\xi,t\eta)$’s where $N$’s are positive integers, $a_N(\xi,\eta)$ satisfies the Marcinkiewicz–Mikhlin–Hörmander condition in $\eta$, and $\mbox{supp}(a_N(\cdot, \eta)) \subset \{\xi : N\le |\xi|<N+1\}$. By using these decompositions, we significantly improve the results of Barrionevo et al in [1].

The authors’ research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology NRF-2018R1D1A1B07042871, NRF-2017R1A2B4002316, and NRF-2016R1D1A1B01014575.

Received 11 April 2018

Accepted 7 October 2019

Published 8 June 2020