Improved bounds for the bilinear spherical maximal operators

Abstract

In this paper we study the bilinear multiplier operator of the form H-t(f,g)(x) = integral(Rd)integral(Rd) m(t xi,t eta)e(2 pi it vertical bar(xi,eta)vertical bar)(f) over cap(xi)(g) over cap(eta) e(2 pi ix(xi+eta))d xi d eta, 1 <= t <= 2 where m satisfies the Marcinkiewicz-Mikhlin-Hormander's derivative conditions. And by obtaining some estimates for H-t , we establish the L-P1 ( R-d ) x L-P2 ( R-d ) -> L-P( R-d) estimates for the bi(sub)- linear spherical maximal operators M (f,g)(x) = sup(t>0 )vertical bar integral(S2d-1 )f (x - ty) g(x - tz) d sigma(2d)(y, z)vertical bar which was considered by Barrionevo et al in [1], here sigma(2d) denotes the surface measure on the unit sphere S2d-1 . In order to investigate M we use the asymptotic expansion of the Fourier transform of the surface measure sigma(2)(d) and study the related bilinear multiplier operator H-t (f, g). To treat the bad behavior of the term e(2 pi it vertical bar(xi,eta)vertical bar) in H-t , we rewrite e(2)(pi it vertical bar(xi,eta)vertical bar) as the summation of e2 pi it root n(2)+vertical bar eta vertical bar(2)a(N)(t xi,t eta)'s where N's are positive integers, a(N)(xi,eta) satisfies the Marcinkiewicz-Mikhlin-Hormander condition in eta, and supp(a(N) (., eta)) subset of{xi : N <= vertical bar xi vertical bar<N+ 1}. By using these decompositions, we significantly improve the results of Barrionevo et al in [1].