A simple proof of a theorem of Jean Bourgain

We give a simple proof of Bourgain's disc algebra version of Grothendieck's theorem, i.e. that every operator on the disc algebra with values in $L_1$ or $L_2$ is 2-absolutely summing and hence extends to an operator defined on the whole of $C$. This implies Bourgain's result that $L_1/H^1$ is of cotype 2. We also prove more generally that $L_r/H^r$ is of cotype 2 for $0<r< 1$.