ENDPOINT BOUNDS FOR MULTILINEAR FRACTIONAL INTEGRALS

Abstract

We prove that the multilinear fractional integral operator I-alpha(f(1), ... , f(k))(x) = integral(Rn) f(1)(x-theta(1y)) ... f(k)(x-theta(ky))vertical bar y vertical bar(alpha-n)dy, where theta(j), j = 1, ... , k are distinct and nonzero, (due to Grafakos [G]) has the endpoint weak-type boundedness into L-r,L-infinity when r = n/2n-alpha. Hence, we obtain by the multilinear interpolation theorem that I-alpha is bounded into L-r for all r > n/2n-alpha. Moreover, We also prove that I-alpha is not bounded into L-r for any r < n/2n-alpha under some conditions on theta(j)'s. Similarly, we show that the multilinear Hilbert transform H(f, g, h(1) ,... , h(k))(x) = p.v. integral f(x + t) g(x - t) Pi(k)(j=1) h(j) (x - theta(j)t)dt/t, where theta(j) not equal +/- 1 are distinct and nonzero, is not bounded into L-r for any r < 1/2 under some conditions on theta(j)'s.