Recognizing Q(p,0) functions per Dirichlet space structure

Under p is an element of (0,x) and Mobius map sigmaw(z) = (w-z)/(1-wz), a h olomorphic function on the unit disk Delta is said to be of Q(p,0) class if lim(/w/-->1) Ep(f,w) = 0,. where E-p(f,w) = integral (Delta)/f'(z)/(2)[1-/sigma (w)(z)/(2)](p)dm(z). and where dm means the element of the Lebesgue area measure on Delta. In pa rticular, Q(p,0) = B-D, the little Bloch space for all p is an element of ( 1,infinity), Q(1.0) = VMOA and Q(p,0) contains D, the Dirichlet space. Moti vated by the linear structure of D, this paper is devoted to: first show th at Q(p,)0 is a Mobius invariant space in the sense of Arazy-Fisher-Peetre; secondly identify Q(p,0) with the closed ball of Q(p,0): and finally invest iage the semigroups of the composition operators on Q(p,0).