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

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Abstract. It is proved that for α ∈ (0, 1), Qα(R n ), not only as an intermediate space of W 1,n (R n ) and BM O(R n ) but also as a homothetic variant of, is isomorphic to a quadratic Morrey space under fractional differentiation. At the same time, the dot product ∇·(Qα(R n )) n is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system in R 1+n + = (0, ∞) × R n .