Contents Online

# Mathematical Research Letters

## Volume 27 (2020)

### Number 2

### Improved bounds for the bilinear spherical maximal operators

Pages: 397 – 434

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n2.a4

#### Authors

#### Abstract

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