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Since Gk C G, the comparison principle for the hyperbolic metric yields that pa ::; pak. 12), since po(O,w)::; log3. Hence 00 Since 1-lakl::; 2exp(-po(O,ak))::; 2exp(-k), it follows that { exp( -ppa(w,z)) ~ . (1 -lzl2)2a dm(z) ~ exp(-J(p- 2a + 2)) ~ 1 Jo f;:o 22 2. Composite Embedding for any w E Go. Next suppose w E Sk. kSj exp (- PPc(w,z)) ~ . J exp( -J(p- 2a + 2)) ~ 1, Z J:ftk and { exp (- PPc(w, z)) dm(z) (1 -lzl 2 ) 2a Jsk -< flak! 1 -lakl dr-< 1, - Jo (1- r) 2a - since a E (0, 1] and exp (- ppc(w, z)) ~ 1.

1, random series are similar to lacunary series in case of QP, p E [0, 1): they are very well behaved if the coefficients are weightedly square-summable and very badly behaved if not. 1 fails, as shown by Sledd and Stegenga [113). 1 is false, see Duren [51) and Sledd [112]. 2 is analogous to that of Theorem 3(b) of Cohran-Shapiro-Ullrich [42). 4. Modified Carleson Measures In this chapter, we show that QP can be equivalently characterized by means of a modified Carleson measure. In the subsequent three sections, this geometric characterization is used to compare QP with the class of mean Lipschitz functions as well as the Besov space (as one of representatives of the conformally invariant classes of holomorphic functions), and to discuss the mean growth of the derivatives of functions in QP.

Modified Carleson Measures ~ IIIJ-tlllc P inf ( 1- ~wl 11 - wzl 2 zES(I) )P IJ-ti(S(J)) t IJ-ti(S(J)) IIIP ' which implies IIJ-tllcp ::5 IIIJ-tlllcp < oo. Conversely, assume that J-t is a p-Carleson measure, that is, IIJ-tllcP If wE D(O, 3/4), then < oo. I'I(D):; lll'llc,. If wED\ D(O, 3/4), then we put En= {zED: lz- w/lwll < 2n(1-lwl)} and hence get IJ-ti(En) ::5 IIJ-tllcP2nP(1 -lwi)P for n EN. We also have 1 - lwl2 11 - wzl 2 and so for n ~ 1 and Eo= 1-lwl2 1_ _J __ - 1- ' ZE lwl' E t, 0, _J I1 - wz 12 -' 1 22n ( 1 - lw I), Z E En\ En-l· Consequently, 00 ::5 IIJ-tllcP L 2-np, n=l that is to say, IIIJ-tlllcp < oo.

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