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Distances from Bloch functions to some Möbius invariant spaces. (English) Zbl 1147.30024

Let H(𝒟) be the space of all analytic functions on the unit disk 𝒟. For a𝒟, let ϕ a (z)=(z-a)/(1-a ¯z) and let g(z,a)=log(1/|ϕ a (z)|) be the Green function of 𝒟 with logarithmic singularity at a. Let 0<p<,-2<q<,0<s<,-1<q+s<, and let fH(𝒟). Then fF(p,q,s) if

f p,q,s p =sup a𝒟 𝒟 |f ' (z)| p (1-|z| 2 ) q g s (z,a)dA(z)<,

and fF 0 (p,q,s), if

lim |a|1 𝒟 |f ' (z)| p (1-|z| 2 ) q g s (z,a)dA(z)=0,

where dA(z)=dxdy/π is the Lebesgue area measure. The Bloch space consists of all fH(D) such that

f =sup z𝒟 |f ' (z)|(1-|z| 2 )<

and the little Bloch space 0 is the space of functions fH(D) for which |f ' (z)|(1-|z| 2 )0 as |z|1. It is known that for s>1, F(p,p-2,s)= and F 0 (p,p-2,s)= 0 . Further, F(2,0,s)=Q s and F 0 (2,0,s)=Q s,0 , which are called Q s -, and Q s,0 -spaces, respectively. For s=1, we have F(2,0,1)=Q 1 =BMOA and F 0 (2,0,1)=Q 1,0 =VMOA. We know that, for 0s<, F(p,p-2,s) and F 0 (p,p-2,s) are Möbius invariant function spaces, and for 0s<1, F(p,p-2,s) and F 0 (p,p-2,s) are subspaces of BMOA and VMOA, respectively. For 0<s<, a positive measure μ defined on 𝒟 is an s-Carleson measure provided μ(S(I))=𝒪((|I| s )) for all subarcs I of 𝒟, where |I| denotes the arc length of I and S(I) denotes the Carleson box based on I. For a subspace X of , the distance from a function f to the space X is denoted by dist (f,X).

The well-known distance formula, known as Jones’ theorem, gives the distance when X=BMOA. The author’s interesting main theorem generalizes this theorem to the case where X=F(p,p-2,s) in the following way: Let 0<s1,1p<,0t<, and let f. Then the following quantities are equivalent:

dist (f,F(p,p-2,s));

inf{ε:χ Ω ε (f) dA(z) (1-|z| 2 ) 2-s is an s-Carleson measure};

inf{ε:sup a𝒟 Ω ε (f) |f ' (z)| t (1-|z| 2 ) t-2 (1-|ϕ a (z)| 2 ) s dA(z)<};

inf{ε:sup a𝒟 Ω ε (f) |f ' (z)| t (1-|z| 2 ) t-2 g s (z,a)dA(z)<}, where Ω ε (f)={z𝒟:|f ' (z)|(1-|z| 2 )ε}.

The corresponding “little oh”-theorem, where instead of F(p,p-2,s) there is F 0 (p,p-2,s) 0 , is also proved, and several corollaries are represented.


MSC:
30D45Bloch functions, normal functions, normal families
30D50Blaschke products, etc. (MSC2000)