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

Let $H\left(𝒟\right)$ be the space of all analytic functions on the unit disk $𝒟$. For $a\in 𝒟$, let ${\varphi }_{a}\left(z\right)=\left(z-a\right)/\left(1-\overline{a}z\right)$ and let $g\left(z,a\right)=log\left(1/|{\varphi }_{a}\left(z\right)|\right)$ be the Green function of $𝒟$ with logarithmic singularity at $a$. Let $0, and let $f\in H\left(𝒟\right)$. Then $f\in F\left(p,q,s\right)$ if

${\parallel f\parallel }_{p,q,s}^{p}=\underset{a\in 𝒟}{sup}{\int }_{𝒟}|{f}^{\text{'}}{\left(z\right)|}^{p}{\left(1-|z|}^{2}{\right)}^{q}{g}^{s}\left(z,a\right)\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty ,$

and $f\in {F}_{0}\left(p,q,s\right)$, if

$\underset{|a|\to 1}{lim}{\int }_{𝒟}|{f}^{\text{'}}{\left(z\right)|}^{p}{\left(1-|z|}^{2}{\right)}^{q}{g}^{s}\left(z,a\right)\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)=0,$

where $dA\left(z\right)=dxdy/\pi$ is the Lebesgue area measure. The Bloch space $ℬ$ consists of all $f\in H\left(D\right)$ such that

${\parallel f\parallel }_{ℬ}=\underset{z\in 𝒟}{sup}|{f}^{\text{'}}\left(z\right)|\left(1-{|z|}^{2}\right)<\infty$

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

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 $X=F\left(p,p-2,s\right)$ in the following way: Let $0, and let $f\in ℬ$. Then the following quantities are equivalent:

${\text{dist}}_{ℬ}\left(f,F\left(p,p-2,s\right)\right)$;

$inf\left\{\epsilon :\phantom{\rule{0.166667em}{0ex}}{\chi }_{{{\Omega }}_{\epsilon }\left(f\right)}\frac{dA\left(z\right)}{{\left(1-|z|}^{2}{\right)}^{2-s}}$ is an $s$-Carleson measure};

$inf\left\{\epsilon :\phantom{\rule{0.166667em}{0ex}}{sup}_{a\in 𝒟}{\int }_{{{\Omega }}_{\epsilon }\left(f\right)}|{f}^{\text{'}}{\left(z\right)|}^{t}{\left(1-|z|}^{2}{\right)}^{t-2}\left(1-|{\varphi }_{a}\left(z\right){{|}^{2}\right)}^{s}\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty$};

$inf\left\{\epsilon :\phantom{\rule{0.166667em}{0ex}}{sup}_{a\in 𝒟}{\int }_{{{\Omega }}_{\epsilon }\left(f\right)}|{f}^{\text{'}}{\left(z\right)|}^{t}{\left(1-|z|}^{2}{\right)}^{t-2}{g}^{s}\left(z,a\right)\phantom{\rule{0.166667em}{0ex}}dA\left(z\right)<\infty$}, where ${{\Omega }}_{\epsilon }\left(f\right)=\left\{z\in 𝒟:\phantom{\rule{0.166667em}{0ex}}|{f}^{\text{'}}\left(z\right)|\left(1-{|z|}^{2}\right)\ge \epsilon \right\}$.

The corresponding “little oh”-theorem, where instead of $F\left(p,p-2,s\right)\subset ℬ$ there is ${F}_{0}\left(p,p-2,s\right)\subset {ℬ}_{0}$, is also proved, and several corollaries are represented.

##### MSC:
 30D45 Bloch functions, normal functions, normal families 30D50 Blaschke products, etc. (MSC2000)