Benedetto, John J.Heinig, HansFourier transform norm inequalities, ||f^||_(q,u) <= C||f^||_(p, v'). are proved for measure weights MU on moment subspaces of L{^P AND {SUB V}}V(R^n).Density theorems are established to extend the inequalities to all of L{^P and {SUB V}}(R^n). In both cases the conditions for validity are computable. For n > 2,MU and v are radial, and the results are applied to prove spherical restriction theorems which include power weights v(t) = |t|^ALPHA,n/(p' - 1) < ALPHA < (p' + n)/(p' - 1).en-USTechnical Report