MAXIMAL FUNCTIONS OF MULTILINEAR MULTIPLIERS

Let m(j) be Fourier multipliers on R(2d) that satisfy vertical bar partial derivative(alpha)m(j)(xi(1), xi(2))vertical bar <= A(alpha)(vertical bar xi(1)vertical bar + vertical bar xi(2)vertical bar)(-vertical bar alpha vertical bar) for sufficiently large alpha uniformly in j, for j = 1,2, ..., N. We study the maximal operator of two variables m(f, g)(x) = sup(1 <= j <= N) vertical bar T(mj)(f, g)(x)vertical bar, where T(mj) are the associated bilinear operators T(mj)(f, g)(x) = integral(R2d) m(xi(1), xi(2))(f) over cap(xi(1))(g) over cap(xi(2))e(2 pi i(xi 1+xi 2).x)d xi(1)d xi(2). We prove that m maps L(p1) (R(d)) x L(p2) (Rd) to L(p)(R(d)) with norm at most a constant multiple root log(N + 2). We also provide an example to indicate the sharpness of this result.