Bilinear fractional integral operators on Morrey spaces

We prove a plethora of the boundedness property of Adams type for bilinear fractional integral operators of the form B-alpha(f, g) (x) = integral(Rn) f(x -y) g(x +y)/vertical bar y vertical bar(n-alpha) dy, 0 < alpha < n. For 1 < t <= s < infinity, we prove the non -weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For 0 < t <= s < infinity and 0 < t <= 1, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights (nu, <(omega)over right arrow>) satisfy [nu, (omega) over right arrow](t, (q) over right arrow /a) (r,as) = sup (Q,Q'is an element of DQ subset of Q) (vertical bar Q vertical bar/vertical bar Q'vertical bar) (1-s/as) vertical bar Q'vertical bar 1/r (integral Q (nu t/1-t)) (1-t/t) Pi(2)(i=1)(integral(Q') (omega i-(qi/a)t))(1/(qi/a)') < infinity, 0 < t < s <1 and [nu, (omega) over right arrow](t, (q) over right arrow /a) (r,as) = sup (Q,Q'is an element of DQ subset of Q) (vertical bar Q vertical bar/vertical bar Q'vertical bar) (1-s/as) vertical bar Q'vertical bar 1/r (integral Q (nu t/1-t)) (1-t/t) Pi(2)(i=1)(integral(Q') (omega i-(qi/a)'))(1/(qi/a)') < infinity, s >= 1 where parallel to nu parallel to(L infinity(Q)) = sup(Q) nu when t = 1, a, r, s, t and <(q)over right arrow>, satisfy proper conditions. As some applications we formulate a bilinear version of the Olsen inequality, the Fetterman Stein type dual inequality and the Stein Weiss inequality on Money spaces for fractional integrals.