ON THE EXISTENCE OF LINEAR AND BILINEAR MULTIPLIERS ON LORENTZ SPACES

First we show that any translation invariant bounded linear operator from L(p,t)(R) the Lorentz space on R to L(p,s)(R) (1 < p < infinity, 1 <= s < t < infinity) is trivial, whose result improves Blozinski's result [4]. Next let phi be a bounded continuous function on R(2), and T(phi)(f,g)(x) = integral integral phi(xi,eta)(f) over cap(xi)(g) over cap(eta)e(i,x(xi+eta))d xi d eta the bilinear operator on Lorentz spaces. Then, we prove that the bounded bilinear operator T(phi) is trivial in some cases of Lorentz spaces.