Zhao, Ruhan Distances from Bloch functions to some Möbius invariant spaces. (English) Zbl 1147.30024 Ann. Acad. Sci. Fenn., Math. 33, No. 1, 303-313 (2008). 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. Reviewer: Rauno Aulaskari (Joensuu) Cited in 3 ReviewsCited in 64 Documents MSC: 30D45 Normal functions of one complex variable, normal families 30D50 Blaschke products, etc. (MSC2000) Keywords:\(F(p; q; s)\) spaces; \(Q_p\) spaces; Carleson measures; distance × Cite Format Result Cite Review PDF Full Text: EuDML