The diagonal mapping in mixed norm spaces

For any holomorphic function F in the unit polydisc U-n of C-n, we consider its restriction to the diagonal, i.e., the function in the unit disc U of C defined by DF(z) = F(z,..., z), and prove that the diagonal mapping D maps the mixed norm space HPp,q,alpha(U-n) of the polydisc onto the mixed norm space H-p,H-q,H-\alpha\+(p/q+1)(n-1) (U) of the unit disc for any 0 < p < infinity and 0 < q < infinity.