Improved bounds for the bilinear spherical maximal operators

In this paper we study the bilinear multiplier operator of the form H^t(f, g)(x) = ^Z_Rd^Z_Rd m(tξ, tη) e²πit|(ξ,η)^| f^b(ξ) gb(η) e²πix(ξ+η⁾ dξdη, 1 ≤ t ≤ 2 where m satisfies the Marcinkiewicz-Mikhlin-Hörmander's derivative conditions. And by obtaining some estimates for H^t, we establish the Lp¹(R^d) × Lp²(R^d) → L^p(R^d) estimates for the bi(sub)linear spherical maximal operators M(f, g)(x) = sup _t>0^Z_S2d−₁ f(x − ty) g(x − tz) dσ_2d(y, z) which was considered by Barrionevo et al in [1], here σ_2d denotes the surface measure on the unit sphere S^2d−1. In order to investigate M we use the asymptotic expansion of the Fourier transform of the surface measure σ_2d and study the related bilinear multiplier operator H^t(f, g). To treat the bad behavior of the term e²πit|(ξ,η)^| in H^t, we rewrite e²πit|(ξ,η)^| as the summation of e²πit√^N2+|η^|2a_N(tξ, tη)'s where N's are positive integers, a_N(ξ, η) satisfies the Marcinkiewicz-Mikhlin-Hörmander condition in η, and supp(a_N(·, η)) ⊂ {ξ: N ≤ |ξ| < N + 1}. By using these decompositions, we significantly improve the results of Barrionevo et al in [1].