Derivatives of Inner Functions in Weighted Mixed Norm Spaces

For 0 < p, q < infinity, we characterize those radial weights omega satisfying a two-sided doubling condition for which the asymptotic equation parallel to Theta'parallel to(q)(A omega p,q) = integral(1)(0) M-p(q)(r, Theta') omega (r)dr (sic) integral(1)(0)(integral(2 pi)(0)(1 - vertical bar Theta(re(i theta))vertical bar/1 - r)(p) d(theta))(q/p) omega(r)dr is valid for all inner functions Theta. As a consequence of this result, we obtain a sharp condition which guarantees that the only inner functions whose derivative belongs to the weighted mixed norm space A(omega)(p,q). are Blaschke products. Moreover, a condition which implies that the only inner functions whose derivative belongs to A(omega)(p,q) are finite Blaschke products is proved.