ABSTRACT. There has been a great deal of work done in recent years on weighted Bergman spaces A p α on the unit ball B n of C n , where 0 < p < ∞ and α > −1. We extend this study in a very natural way to the case where α is any real number and 0 < p ≤ ∞. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space H 2 , and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.

Some subspaces and subsets of BMOA and UBC are defined by using similar area integral conditions to the definitions of BMOA and UBC. These subspaces and subsets axe shown to be separate and included in each other. AMS Classification: 30D50 Δ Q,,ο = {/:/ analytic in Δ and Jim Jf Δ Qjf = ^ / : / meromorphic in Δ and sup J J (/*(z)) 2 g r {z,a) dx dy < oo j> and Q* 0 = I / : / meromorphic in Δ and ^lim J J ( / * ( z ) ) 2 < / p ( z , a ) dx dy = 0 Typeset by .AjuS-TfeX Brought to you by | University of Pennsylvania Authenticated Download Date | 6/14/15 4:46 PMWe can choose a pseudohyperbolic disk U((io, . <) such that U(ao,s) Π Δ \ U(0,r) φ 0 and «! 6 U(a 0 ,s) η Δ \ U(0,r). ThenBy (2.1) we find a constant C depending only on do, s and s', s < s', such that the following inequality holds IJ \1'{ζ)\ 2 9 '(ζ,α 0 )άχά ν <ϋ> j j \f'(z)\ 2 g"(z,a l )dxdy < C" < 00. (2.5) Δ\('(α0,ν) Δ\(/(α0,»') At the same time, ff \f'(z)\ 2 gP(z,a 0 )dxdy< max \f'(z)\ 2 [[ (log τ-^-λ' dxdy < 00. (2.6) J J Ι6ί/(α",5') J J \z-a 0 \' Brought to you by | University of Pennsylvania Authenticated Download Date | 6/14/15 4:46 PM 104 Aulaskari et al Combining (2.5) and (2.6) we get J J\f'(z)\ 2 g p (z,ao)dxdy < oo, Δ which contradicts (2.4). Case 2. Next suppose that, for all α € U(0, r), J J \f'(z)\ 2 g"(z,a)dxdy

The main results are conditions on g such that the Volterra type operator Jg(f)(z) = z 0 f (ζ)g (ζ) dζ, is bounded or compact on BMOA. We also point out certain information when Jg is considered as an operator on a general space X of analytic functions on the disc.

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