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

Let \(H(\mathcal D)\) be the space of all analytic functions on the unit disk \(\mathcal D\). For \(a\in\mathcal D\), let \(\varphi_a(z)=(z-a)/(1-\bar az)\) and let \(g(z,a)=\log(1/| \varphi_a(z)| )\) be the Green function of \(\mathcal D\) with logarithmic singularity at \(a\). Let \(0<p<\infty, -2<q<\infty, 0<s<\infty, -1<q+s<\infty\), and let \(f\in H(\mathcal D)\). Then \(f\in F(p,q,s)\) if
\[ \| f\|_{p,q,s}^p=\sup_{a\in\mathcal D}\int_{\mathcal D}| f'(z)| ^p(1-| z| ^2)^qg^s(z,a)\,dA(z)<\infty, \]
and \(f\in F_0(p,q,s)\), if
\[ \lim_{| a| \to1}\int_{\mathcal D}| f'(z)| ^p(1-| z| ^2)^qg^s(z,a)\,dA(z)=0, \]
where \(dA(z)=dxdy/\pi\) is the Lebesgue area measure. The Bloch space \(\mathcal B\) consists of all \(f\in H(D)\) such that
\[ \| f\|_{\mathcal B}=\sup_{z\in\mathcal D}| f'(z)| (1-| z| ^2)<\infty \]
and the little Bloch space \(\mathcal B_0\) is the space of functions \(f\in H(D)\) for which \(| f'(z)| (1-| z| ^2)\to0\) as \(| z| \to1\). It is known that for \(s>1\), \(F(p,p-2,s)=\mathcal B\) and \(F_0(p,p-2,s)=\mathcal B_0\). 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 \(F(2,0,1)=Q_1= \text{BMOA}\) and \(F_0(2,0,1)=Q_{1,0}= \text{VMOA}\). We know that, for \(0\leq s<\infty\), \(F(p,p-2,s)\) and \(F_0(p,p-2,s)\) are Möbius invariant function spaces, and for \(0\leq s<1\), \(F(p,p-2,s)\) and \(F_0(p,p-2,s)\) are subspaces of BMOA and VMOA, respectively. For \(0<s<\infty\), a positive measure \(\mu\) defined on \(\mathcal D\) is an \(s\)-Carleson measure provided \(\mu(S(I))=\mathcal O((| I| ^s))\) for all subarcs \(I\) of \(\partial\mathcal D\), where \(| I| \) denotes the arc length of \(I\) and \(S(I)\) denotes the Carleson box based on \(I\). For a subspace \(X\) of \(\mathcal B\), the distance from a function \(f\in\mathcal B\) to the space \(X\) is denoted by \(\text{dist}_{\mathcal B}(f,X)\).
The well-known distance formula, known as Jones’ theorem, gives the distance when \(X=\text{BMOA}\). The author’s interesting main theorem generalizes this theorem to the case where \(X=F(p,p-2,s)\) in the following way: Let \(0<s\leq1, 1\leq p<\infty, 0\leq t<\infty\), and let \(f\in\mathcal B\). Then the following quantities are equivalent:
(A)
\(\text{dist}_{\mathcal B}(f,F(p,p-2,s))\);
(B)
\(\inf\{\varepsilon:\,\chi_{\Omega_{\varepsilon}(f)}\frac{dA(z)}{(1-| z| ^2)^{2-s}}\) is an \(s\)-Carleson measure};
(C)
\(\inf\{\varepsilon:\,\sup_{a\in\mathcal D}\int_{\Omega_{\varepsilon}(f)}| f'(z)| ^t(1-| z| ^2)^{t-2}(1-| \varphi_a(z)| ^2)^s\,dA(z)<\infty\)};
(D)
\(\inf\{\varepsilon:\,\sup_{a\in\mathcal D}\int_{\Omega_{\varepsilon}(f)}| f'(z)| ^t(1-| z| ^2)^{t-2}g^s(z,a)\,dA(z)<\infty\)}, where \(\Omega_{\varepsilon}(f)=\{z\in\mathcal D:\,| f'(z)| (1-| z| ^2)\geq\varepsilon\}\).
The corresponding “little oh”-theorem, where instead of \(F(p,p-2,s)\subset\mathcal B\) there is \(F_0(p,p-2,s)\subset\mathcal B_0\), is also proved, and several corollaries are represented.

MSC:

30D45 Normal functions of one complex variable, normal families
30D50 Blaschke products, etc. (MSC2000)