Composition Operators fronm Weak to Strong Vector-Valued Hardy Spaces

Let phi be an analytic map from the unit disk into itself, 1<p<2 and 1 <= q <= p. It is shown that the composition operator C-phi(f)=f circle phi is bounded from H-weak(p)(D,L-q(mu)) into H-p(D,L-q(mu)) if and only if C-phi is a 2-summing operator from H-p(D) into H-p(D). Here H-weak(p)(D,X) and H-p(D,X) are the weak and strong formulation of X-valued Hardy spaces on the unit disc.