Dual spaces to Orlicz-Lorentz spaces

For an Orlicz function co and a decreasing weight omega, two intrinsic exact descriptions are presented for the norm in the Kothe dual of the Orlicz-Lorentz function space Lambda(phi,omega) or the sequence space lambda(phi,omega) equipped with either the Luxemburg or Amemiya norms. The first description is via the modular inf{integral phi(*)(f*/vertical bar g vertical bar)vertical bar g vertical bar : g< w}, where f * is the decreasing rearrangement of f, < denotes submajorization, and phi(*) is the complementary function to phi. The second description is in terms of the modular integral(1) phi co.((f *) /w)w, where (f*)(0) is Halperin's level function of f* with respect to w. That these two descriptions are equiva ent results from the identity inf{integral psi(f*/vertical bar g vertical bar)vertical bar g vertical bar : g < w} = integral(I) psi((f*)(0)/omega) omega, valid for any measurable function f and any Orlicz function 0. An analogous identity and dual representations are also presented for sequence spaces.