© Wuhan University 2023

0 Introduction

In this paper, a unit disc $\left\{z:\left|z\right|<\mathrm{1}\right\}$ is denoted by D. The $\mathrm{M}\mathrm{ö}\mathrm{b}\mathrm{i}\mathrm{u}\mathrm{s}$ transformation of D is defined by:

${\phi }_a(z)=\frac{a-z}{\mathrm{1}-\overline{a}z},a\in D$

Note that

$\beta (z,w)=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{1}+\rho (z,w)}{\mathrm{1}-\rho (z,w)}$

is the hyperbolic metric for any $z,w\in D$, where $\rho (z,w)=\left|{\phi }_z(w)\right|$ is the pseudohyperbolic distance for any $z,w\in D$. For $\mathrm{0}<p<\mathrm{\infty }$, an analytic function $f$ belongs to the space $Q_p$ if

${\Vert f\Vert }_{Q_p}^{\mathrm{2}}=\underset{a\in D}{\mathrm{s}\mathrm{u}\mathrm{p}}{\underset{D}{\int }\left|f\text{'}(z)\right|}^{\mathrm{2}}{\left(\mathrm{1}-{\left|{\phi }_a(z)\right|}^{\mathrm{2}}\right)}^p\mathrm{d}A(z)<\mathrm{\infty }$(1)

where dA(z) is an area measure on D normalized, which makes $\underset{D}{\int }\mathrm{d}A\left(z\right)=\mathrm{1}$.

Equipped with the norm $\left|f\left(\mathrm{0}\right)\right|+{\Vert f\Vert }_{Q_p}$, the space $Q_p$ is Banach. Generally, $Q_p$ is the Bloch space if $p>\mathrm{1}$. If $p=\mathrm{1}$,$Q_{\mathrm{1}}$ coincides with $\mathrm{B}\mathrm{M}\mathrm{O}\mathrm{A}$, analytic function of bounded mean oscillation. $Q_{\mathrm{0}}$ is just the Dirichlet space. Regarding the theory of $Q_p$ spaces, readers may refer to Refs. [1-4].

An analytic function $f$ belongs to the Bloch space, denoted by $B$, if

${\Vert f\Vert }_B=\underset{z\in D}{\mathrm{s}\mathrm{u}\mathrm{p}}(\mathrm{1}-{\left|z\right|}^{\mathrm{2}})\left|f\text{'}(z)\right|<\mathrm{\infty }$

The space $H^{\mathrm{\infty }}$ consists of all bounded analytic functions $f$ on D with:

$\underset{z\in D}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|f(z)\right|<\mathrm{\infty }$

For a subarc $I\subset \partial D$, $\theta$ acts as the midpoint of $I$ and denotes the Carleson box:

$S(I)=\left\{z\in D:\mathrm{1}-\left|I\right|<\left|z\right|<\mathrm{1},\left|\theta -\mathrm{a}\mathrm{r}\mathrm{g}z\right|<\frac{\left|I\right|}{\mathrm{2}}\right\}$

for $\left|I\right|⩽\mathrm{1}$ and $S(I)=D$ for $\left|I\right|>\mathrm{1}$. A positive measure $\mu$ is a $p$-Carleson measure if

${\Vert \mu \Vert }_p=\underset{I\subset \partial D}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\mu (S(I))}{{\left|I\right|}^p}<\mathrm{\infty }$

where $\left|I\right|$ denotes the arc length of $I$. An analytic function $f\in Q_p$ if and only if the positive measure ${\left|f\text{'}(z)\right|}^{\mathrm{2}}(\mathrm{1}-{\left|z\right|}^{\mathrm{2}}{)}^p\mathrm{d}A(z)$ is a p-Carleson measure.

The sequence space $\mathrm{S}\mathrm{C}{\mathrm{M}}_p$ consists of all complex numbers $\left\{{\lambda }_j\right\}$, so that ${{\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{\infty }}}\left(\mathrm{1}-{\left|z_j\right|}^{\mathrm{2}}\right)}^p{\left|{\lambda }_j\right|}^{\mathrm{2}}{\delta }_{z_j}$ is a p-Carleson measure, where ${\delta }_z$ denotes the unit point-mass measure at $z\in D$ and ${\left\{z_j\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}\subset D$.

A sequence ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ is called an interpolating sequence for $Q_p\bigcap H^{\mathrm{\infty }}$ if, for each bounded sequence ${\left\{{\lambda }_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of complex values, there exists an $f\in Q_p\bigcap H^{\mathrm{\infty }}$ such that $f(z_n)={\lambda }_n$ for all n.

A sequence ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ in D is separated if $\underset{m\ne n}{\mathrm{i}\mathrm{n}\mathrm{f}}\rho (z_n,z_m)>\mathrm{0}$.

Usually ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ is an interpolating sequence for $H^{\mathrm{\infty }}$ if and only if ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ in D is separated and $\sum _n\left(\mathrm{1}-{\left|z_n\right|}^{\mathrm{2}}\right){\delta }_{z_j}$ is a Carleson measure for $H^{\mathrm{\infty }}$. See Ref.[5] for interpolating sequence in $H^{\mathrm{\infty }}$. Readers can refer to Refs.[6,7] about the Hardy and Bergman space theory. See Ref.[8] for more interpolating sequences. Sundberg solved the interpolating question for BMOA in Ref.[9]. A necessary and sufficient condition is obtained for the interpolating sequence in the Bloch space ^[10]. Pascuas characterized the interpolating sequence in the Bloch space by the p-Carleson measure, as details in Ref. [11]. Ref. [12] gave a characterization of the interpolating sequence for $Q_p\bigcap H^{\mathrm{\infty }}$ and $H^{\mathrm{\infty }}\bigcap Q_{p,\mathrm{0}}$. The main result is listed as follows:

Theorem 1^[12] Let $p\in (\mathrm{0,1})$. A sequence ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of points in the unit disc is an interpolating sequence for $Q_p\bigcap H^{\mathrm{\infty }}$ if and only if ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ in D is separated and ${\sum _n\left(\mathrm{1}-{\left|z_n\right|}^{\mathrm{2}}\right)}^p{\delta }_{z_n}(z)$ is a p-Carleson measure.

In the following analysis, $f\prec g$ (for two functions $f$ and $g$) if there is a constant C such that $f⩽Cg$. $f\approx g$ (that is, $f$ is comparable with $g$) whenever $g\prec f\prec g$.

1 Interpolating Sequence in ${\mathit{Q}}_{\mathit{p}}\bigcap {\mathit{H}}^{\mathbf{\infty }}$

If the sequence $\left\{z_n\right\}$ is an interpolating sequence for the Bloch space, then

$\beta (z_n,z_m)⩾C\beta (z_n,\mathrm{0})$(2)

The aforementioned condition represents the separation criterion. It is subsequently reformulated into the following condition

$\mathrm{1}-\rho (z_n,z_m)⩽C{(\mathrm{1}-\left|z_n\right|)}^{\lambda }$(3)

where $\lambda$ is the constant from the separation condition (2), coinciding with the one used in Ref.[13]. Readers can refer to Refs.[8] and [13] about the separation condition (3). The condition (2) holds, then $\left\{z_n\right\}$ is separated.

To a point z, a region $V_z$ is defined by

$V_z=\left\{w:w\in D,\left|w-z^{\mathrm{*}}\right|⩽{\left(\mathrm{1}-\left|z\right|\right)}^{\beta }\right\}$(4)

where $z^{\mathrm{*}}$ denotes the radial projection $\frac{z}{\left|z\right|}$ of $z$ and $\mathrm{0}<\beta <\mathrm{1}$. If the two regions $V_{z_n}$, $V_{z_m}$ intersect and $\left|z_n\right|>\left|z_m\right|$, then $(\mathrm{1}-\left|z_n\right|)⩽{(\mathrm{1}-\left|z_m\right|)}^{\eta }$ can be obtained and $z_m$ is outside of $V_{z_n}$. The constants $\beta$($\mathrm{0}<\beta <\mathrm{1}$) and $\eta (\eta >\mathrm{1})$ are chosen so that $\mathrm{1}<\eta <\frac{\mathrm{2}\beta -\mathrm{1}}{\mathrm{1}-\lambda }$ and $\eta \beta >\mathrm{1}$. Those implications of the separation condition can be found in Ref. [13].

This paper chooses a constant $\rho$ so that $\mathrm{1}>\rho >\beta$. The constant $\rho$ will be needed to define the support of the function $g$ in the following Lemma 1. The paper constructs a function living essentially in a region $V_z$, with reasonable estimates of how it behaves for all points.

Lemma 1   Let $s>-\mathrm{1}$ and $\mathrm{0}<p<\mathrm{2}$. For any given point $b\in D$, there exists a function $g_b$ so that:

$f_b\left(w\right)=\underset{D}{\int }\frac{g_b(u){(\mathrm{1}-\left|u\right|)}^s}{{(\mathrm{1}-\overline{u}w)}^{\mathrm{1}+s}}\mathrm{d}A(u)$

satisfies $f_b(b)$=1, and for points in $V_b$, the value is estimated by:

$f_b\left(w\right)=c(\gamma (w))+C{(\mathrm{1}-\left|b\right|)}^{\mathrm{1}-\rho }$

Here $\gamma =\gamma (w)$ is defined by $\left|w-b^{\mathrm{*}}\right|={\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\rho }$ and

$c(\gamma (w))=\{\begin{array}{l}\mathrm{0},\gamma <\rho ,\\ {\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\gamma },\rho ⩽\gamma ⩽\mathrm{1},\\ \mathrm{1},\gamma >\mathrm{1}.\end{array}$

For points outside of $V_b$, the bound of $f$ is obtained by:

$\left|f_b\left(w\right)\right|⩽C{\left(\mathrm{1}-\left|b\right|\right)}^p$(5)

Further,

${\underset{D}{\int }{\left|g_b(u)\right|}^{\mathrm{2}}\left(\mathrm{1}-\left|u\right|\right)}^p\mathrm{d}A(u)⩽C{\left(\mathrm{1}-\left|b\right|\right)}^p$(6)

Proof   In this proof, a technique that is borrowed from Ref.[8] has been adapted to enhance its efficiency. The relation defining $g_b$ is as follows

$\frac{{(\mathrm{1}-\left|\zeta \right|)}^s}{{\left(\mathrm{1}-\overline{\zeta }b\right)}^{\mathrm{1}+s}}{g}_b\left(\zeta \right)=K\cdot \frac{\mathrm{1}-\left|b\right|}{{\left|\zeta -b^{\mathrm{*}}\right|}^{\mathrm{3}}}$

where $\zeta$ lives in the annulus

$E_b=\left\{\zeta :(\mathrm{1}-\left|b\right|)⩽\left|\zeta -b^{\mathrm{*}}\right|⩽{(\mathrm{1}-\left|b\right|)}^{\rho }\right\}$(7)

and is further restricted to a cone with the vertex in $b^{\mathrm{*}}$ and a fixed small aperture. For all others $\zeta$, $g_b$ is taken to be zero. There is the following equation:

$\begin{array}{l}\underset{D}{\int }\frac{{(\mathrm{1}-\left|\zeta \right|)}^s}{{(\mathrm{1}-\overline{\zeta }b)}^{\mathrm{1}+s}}{g}_b\left(\zeta \right)\mathrm{d}A\left(\zeta \right)\\ =K(\mathrm{1}-\left|b\right|){\displaystyle \underset{\left(\mathrm{1}-\left|b\right|\right)}{\overset{{(\mathrm{1}-\left|b\right|)}^{\rho }}{\int }}}\frac{\mathrm{1}}{r^{\mathrm{2}}}\mathrm{d}r=K\left(\mathrm{1}-{(\mathrm{1}-\left|b\right|)}^{\mathrm{1}-\rho }\right)\end{array}$

where $K$ is chosen so that $f_b(b)=\mathrm{1}$. Observing the defining of $g_b$, then we get:

$\left|g_b\left(\zeta \right)\right|⩽C\frac{\mathrm{1}-\left|b\right|}{{\left|\zeta -b^{\mathrm{*}}\right|}^{\mathrm{2}}}$

The function $f_b\left(w\right)$ is to be estimated. Let us suppose that $w$ is in $V_b$. Let:

$E_{\mathrm{1}}=\left\{\zeta :\left|\zeta -b^{\mathrm{*}}\right|⩽{\left(\mathrm{1}-\left|b\right|\right)}^{\gamma }\right\}$

and

$E_{\mathrm{2}}=\left\{\zeta :\left|\zeta -b^{\mathrm{*}}\right|>{\left(\mathrm{1}-\left|b\right|\right)}^{\gamma }\right\}$

The contributions on $E_{\mathrm{1}}$ and $E_{\mathrm{2}}$ are considered. For any $\zeta \in E_{\mathrm{1}}$, there is $\left|\mathrm{1}-\overline{\zeta }w\right|⩾C{\left(\mathrm{1}-\left|b\right|\right)}^{\gamma }$ by $\left|w-b^{\mathrm{*}}\right|={(\mathrm{1}-\left|b\right|)}^{\gamma }$. Then:

$\begin{array}{l}\left|\underset{E_{\mathrm{1}}}{\int }\frac{{(\mathrm{1}-\left|\zeta \right|)}^s}{{(\mathrm{1}-\overline{\zeta }w)}^{\mathrm{1}+s}}{g}_b(\zeta )\mathrm{d}A\left(\zeta \right)\right|\\ ⩽C{\left(\mathrm{1}-\left|b\right|\right)}^{-\gamma (\mathrm{1}+s)}\underset{E_{\mathrm{1}}}{\int }\left|g_b\left(\zeta \right)\right|\left(\mathrm{1}-{\left|\zeta \right|}^s\right)\mathrm{d}A\left(\zeta \right)\\ ⩽C{\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\gamma }.\end{array}$

For $E_{\mathrm{2}}$, after a calculation,

$\begin{array}{l}\underset{E_{\mathrm{2}}}{\int }\frac{{(\mathrm{1}-\left|\zeta \right|)}^s}{{\left(\mathrm{1}-\overline{\zeta }b\right)}^{\mathrm{1}+s}}{g}_b\left(\zeta \right)\mathrm{d}A\left(\zeta \right)\\ ={\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\gamma }-{\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\rho }.\end{array}$

Since

$\left|{\left(\mathrm{1}-\overline{\zeta }w\right)}^{-\mathrm{1}-s}-{\left(\mathrm{1}-\overline{\zeta }b\right)}^{-\mathrm{1}-s}\right|⩽C\left|b-w\right|\cdot {\left(\mathrm{1}-\left|\zeta \right|\right)}^{-\mathrm{2}-s},$

and

$\left|b-w\right|⩽\left|b-b^{\mathrm{*}}\right|+\left|w-b^{\mathrm{*}}\right|⩽\mathrm{2}{(\mathrm{1}-\left|b\right|)}^{\gamma },$

then

$\left|\underset{E_{\mathrm{2}}}{\int }\frac{{\left(\mathrm{1}-\left|\zeta \right|\right)}^s}{{\left(\mathrm{1}-\overline{\zeta }w\right)}^{\mathrm{1}+s}}g\left(\zeta \right)\mathrm{d}A\left(\zeta \right)-\underset{E_{\mathrm{2}}}{\int }\frac{{\left(\mathrm{1}-\left|\zeta \right|\right)}^s}{{\left(\mathrm{1}-\overline{\zeta }b\right)}^{\mathrm{1}+s}}g\left(\zeta \right)\mathrm{d}A\left(\zeta \right)\right|$

$\begin{array}{l}⩽C\left|b-w\right|\cdot \underset{E_{\mathrm{2}}}{\int }\frac{\left|g\left(\zeta \right)\right|}{{\left(\mathrm{1}-\left|\zeta \right|\right)}^{\mathrm{2}}}\mathrm{d}A\left(\zeta \right)\\ ⩽C{\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\gamma }+O{\left(\mathrm{1}-\left|b\right|\right)}^{\mathrm{1}-\rho }.\end{array}$

For a point $w$ outside of $V_b$, $\left|\mathrm{1}-\overline{w}\zeta \right|⩾{\left(\mathrm{1}-\left|b\right|\right)}^{\beta }$ holds when $\zeta$ belongs to the support of $g_b$. Thus,

$\left|f_b(w)\right|⩽{\left(\mathrm{1}-\left|b\right|\right)}^{(\mathrm{1}+s)(\rho -\beta )}.$

$p=(\mathrm{1}+s)(\rho -\beta )$ for sufficiently large s can be obtained so that the condition (5) holds. Finally, we can get (6) by the direct calculation.

An operator $T_s$ is defined as follows:

$T_s\left(g\right)\left(z\right)=\underset{D}{\int }\frac{{\left(\mathrm{1}-{\left|w\right|}^{\mathrm{2}}\right)}^s}{{\left(\mathrm{1}-\overline{w}z\right)}^{\mathrm{2}+s}}\overline{w}g\left(w\right)\mathrm{d}A\left(w\right)$(8)

where $g$ is a measurable function on D. The following Lemma 2 is Lemma 3.1.2 in Ref.[2].

Lemma 2   Let $p>\mathrm{0}$. If ${\left|g\left(z\right)\right|}^{\mathrm{2}}{\left(\mathrm{1}-{\left|z\right|}^{\mathrm{2}}\right)}^p\mathrm{d}A\left(z\right)$ is a p-Carleson measure on D, then

${\left|T_s\left(g\right)\left(z\right)\right|}^{\mathrm{2}}{\left(\mathrm{1}-{\left|z\right|}^{\mathrm{2}}\right)}^p\mathrm{d}A\left(z\right)$

is also a p-Carleson measure on D.

A finite number of points can be added to an interpolating sequence, rendering it interpolating. This fact is employed in Ref. [13] to derive the implications of the separation condition.

Lemma 3   Let ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ be a sequence in D so that ${\sum _n\left(\mathrm{1}-\left|z_n\right|\right)}^p<\mathrm{\infty }$. When (3) holds and $\left\{{\lambda }_n\right\}\in l^{\mathrm{\infty }}$, it can be founded by ${\left\{a_j\right\}}_{j=\mathrm{1}}^{\mathrm{\infty }}$, and $\delta \in (\mathrm{0,1})$ so that $f(z)=\sum a_j{f}_{z_j}$ approximates it in the sense

${\Vert f(z_n)-{\lambda }_n\Vert }_{l^{\mathrm{\infty }}}⩽\delta {\Vert {\lambda }_n\Vert }_{l^{\mathrm{\infty }}}$

The coefficients $a_i$ as well as ${\Vert f\Vert }_{H^{\mathrm{\infty }}}$ are bounded by $C\cdot {\Vert {\lambda }_n\Vert }_{l^{\mathrm{\infty }}}$. Corresponding to $z_n$, $f_{z_n}$ is the function in Lemma 1.

Proof   Without loss of generality, it is supposed ${\Vert {\lambda }_n\Vert }_{l^{\mathrm{\infty }}}$=1. The points are ordered in sequence by their distance to the boundary. To a point $z_{\mathrm{1}}$, an increasing chain of regions $V_{z_{\mathrm{1}}}\subset V_{z_{\mathrm{2}}}\subset \cdots \subset V_{z_k}$ is chosen at each step, selecting the smallest region strictly containing all the previous ones. ${\beta }_j=c({\gamma }_j)$ is defined in Lemma 1, where ${\gamma }_j$ is given by

$\left|z_{j-\mathrm{1}}-z_j^{\mathrm{*}}\right|={\left(\mathrm{1}-\left|z_j\right|\right)}^{{\gamma }_j}.$

The coefficients $a_{\mathrm{2}},\cdots ,a_k$ corresponding $z_{\mathrm{2}},\cdots ,z_k$ are already defined. An induction is assumed that $\left|{\displaystyle \sum _{\mathrm{3}}^n}{\beta }_i{a}_i\right|⩽\mathrm{1}$. The coefficient $a_{\mathrm{1}}$ corresponding to $z_{\mathrm{1}}$ is determined by the equation:

$a_{\mathrm{1}}={\lambda }_{\mathrm{1}}-{\displaystyle \sum _{\mathrm{2}}^n}{a}_i{\beta }_i$

Note that

$\begin{array}{l}{\displaystyle \sum _{\mathrm{2}}^n}{\beta }_i{a}_i={\beta }_{\mathrm{2}}\left(a_{\mathrm{2}}+{\displaystyle \sum _{\mathrm{3}}^n}{\beta }_i{a}_i\right)+\left(\mathrm{1}-{\beta }_{\mathrm{2}}\right){\displaystyle \sum _{\mathrm{3}}^n}{\beta }_i{a}_i\\ ={\beta }_{\mathrm{2}}{\lambda }_{\mathrm{2}}+\left(\mathrm{1}-{\beta }_{\mathrm{2}}\right){\displaystyle \sum _{\mathrm{3}}^n}{\beta }_i{a}_i.\end{array}$

Since $a_{\mathrm{2}}$ is defined in the same way as $a_{\mathrm{1}}$, $\left|{\displaystyle \sum _{\mathrm{2}}^n}{\beta }_i{a}_i\right|⩽\mathrm{1}$ can be obtained. The above estimate gives $\left|a_{\mathrm{1}}\right|⩽\mathrm{2}$. The asserted properties of $f$ is checked. A point $z_{\mathrm{1}}$ is fixed, and a notation is kept as in the construction above. Here:

$f(z_{\mathrm{1}})-{\lambda }_{\mathrm{1}}={\displaystyle \sum _{i=\mathrm{2}}^n}{a}_i(f_{z_i}(z_{\mathrm{1}})-{\beta }_i)+\sum _{z_i\ne z_{\mathrm{1}},\cdots ,z_j}{\lambda }_j{f}_{z_i}(z_{\mathrm{1}})$(9)

The first term of the right of (9) should be considered first. Since

$\begin{array}{l}\left|z_{\mathrm{1}}-z_{i-\mathrm{1}}\right|⩽\left|z_{\mathrm{1}}-z_{i-\mathrm{1}}^{\mathrm{*}}\right|+\left|z_{i-\mathrm{1}}-z_{i-\mathrm{1}}^{\mathrm{*}}\right|\\ ⩽{(\mathrm{1}-\left|z_i\right|)}^{\eta \beta }+{(\mathrm{1}-\left|z_i\right|)}^{\eta },\end{array}$

then

$\left|f_{z_i}(z_{\mathrm{1}})-{\beta }_i\right|⩽C{\left(\mathrm{1}-\left|z_i\right|\right)}^{\mathrm{1}-\rho }.$

By repeatedly applying the separation condition, there comes:

$\left(\mathrm{1}-\left|z_i\right|\right)⩽{\left(\mathrm{1}-\left|z_n\right|\right)}^{{\eta }^{n-i}}$

Then

$\begin{array}{l}{\displaystyle \sum _{\mathrm{1}}^n}\left|a_i(f_{z_i}(z_{\mathrm{1}})-\beta )\right|⩽{{\displaystyle \sum _{\mathrm{1}}^n}\left|a_i\right|\left(\mathrm{1}-\left|z_i\right|\right)}^{\mathrm{1}-\rho }\\ ⩽{\displaystyle \sum _{\mathrm{1}}^n}{\left(\mathrm{1}-\left|z_i\right|\right)}^{\mathrm{1}-\rho }\\ ⩽{\displaystyle \sum _{\mathrm{1}}^n}{\left(\mathrm{1}-\left|z_n\right|\right)}^{\left(\mathrm{1}-\rho \right){\eta }^{n-i}}\\ ⩽C{\left(\mathrm{1}-\left|z_n\right|\right)}^{\mathrm{1}-\rho }.\end{array}$

Now, let us look to the second term of the right of (9). If $z_{\mathrm{1}}$ is not in $V_{z_i}$, by Lemma 1 the estimate is made as follows:

$\left|f_{z_i}(z_{\mathrm{1}})\right|⩽\left|{\lambda }_i\right|{\left(\mathrm{1}-\left|z_i\right|\right)}^p.$

If $z_{\mathrm{1}}$ is in $V_{z_i}$, $V_{z_j}$ is founded in the chain not contained in $V_{z_i}$ so that $\left|z_j\right|\ge \left|z_i\right|.$ Since

$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(V_{z_j})⩽\mathrm{2}{\left(\mathrm{1}-{\left|z_j\right|}^{\mathrm{2}}\right)}^{\eta \beta }⩽\mathrm{2}{\left(\mathrm{1}-\left|z_i\right|\right)}^{\eta \beta }⩽\mathrm{2}\left(\mathrm{1}-\left|z_j\right|\right),$

then $\left|z_{\mathrm{1}}-z_i^{\mathrm{*}}\right|⩾C{\left(\mathrm{1}-\left|z_i\right|\right)}^{\beta }$. This gives the estimate

$\left|f_{z_i}(z_{\mathrm{1}})\right|⩽C{\left(\mathrm{1}-\left|z_i\right|\right)}^p.$

The above estimates show that

$\begin{array}{l}\left|f(z_{\mathrm{1}})-{\lambda }_{\mathrm{1}}\right|\\ \le {\displaystyle \sum _{\mathrm{1}}^n}\left|a_i\right|\left|f_{z_i}(z_{\mathrm{1}})-{\beta }_i\right|+\sum _{z_i\ne z_{\mathrm{1}},\cdots ,z_j}\left|{\lambda }_j{f}_{z_i}\left(z_{\mathrm{1}}\right)\right|\\ \le {\displaystyle \sum _{\mathrm{1}}^n}\left|a_i\right|{\left(\mathrm{1}-\left|z_i\right|\right)}^{\mathrm{1}-\rho }+\sum _{z_i\ne z_{\mathrm{1}},\cdots ,z_j}\left|{\lambda }_j\right|{\left(\mathrm{1}-\left|z_j\right|\right)}^p\\ \le {\displaystyle \sum _{\mathrm{1}}^n}{\left(\mathrm{1}-\left|z_i\right|\right)}^{\mathrm{1}-\rho }+\left(\sum _{z_i\ne z_{\mathrm{1}},\cdots ,z_j}{\left(\mathrm{1}-\left|z_j\right|\right)}^p\right).\end{array}$

Note: the last term is finite by the p-Carleson measure condition. The left expression can be made smaller than a $\delta <\mathrm{1}$ by removing finitely many points from the sequence.

Now, let us prove $f\in H^{\mathrm{\infty }}$. If z is not contained in $V_{z_i}$, then $\left|f(z)\right|⩽C$ is easy to get. If z is included in some region $V_{z_{\mathrm{1}}}$, it is assumed that $V_{z_{\mathrm{1}}}$ is the smallest, and $\left|f(z)\right|$ is bounded by applying the estimates on $f_{z_i}(z_{\mathrm{1}})$.

This paper points out that the following result is essential Theorem 1. Here, a new method is used to construct the function sequence $f_j\in Q_p\bigcap H^{\mathrm{\infty }}$ such that $f=\sum _j{a}_j{f}_{z_j}\in Q_p\bigcap H^{\mathrm{\infty }}$.

Theorem 2   A sequence ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ is an interpolating sequence for $Q_p\bigcap H^{\mathrm{\infty }}$ if and only if the condition (3) holds and ${\sum _n\left(\mathrm{1}-{\left|z_n\right|}^{\mathrm{2}}\right)}^p{\delta }_{z_n}\left(z\right)$ is a p-Carleson measure. Furthermore, there is an analytic function $f=\sum _j{a}_j{f}_{z_j}\in Q_p\bigcap H^{\mathrm{\infty }}$ such that $f(z_n)={\lambda }_n,n=\mathrm{1,2},\cdots$, for any sequence $\left\{{\lambda }_n\right\}\in l^{\mathrm{\infty }}$.

Proof   Note the condition (3) holds. By repeatedly applying Lemma 3 we can find a function $f=\sum _j{a}_j{f}_{z_j}\in H^{\mathrm{\infty }}$ with $f(z_j)={\lambda }_j$ satisfying $\left|a_j\right|⩽C{\Vert {\lambda }_j\Vert }_{l^{\mathrm{\infty }}}$. Since each $f_{z_j}$ comes from a $g_{z_j}$ as in Lemma 1, $f$ is given by $g=\sum _j{a}_j{g}_{z_j}$ and

$f\text{'}\left(w\right)=T_{s}g\left(w\right)=\underset{D}{\int }\frac{\overline{u}{(\mathrm{1}-\left|u\right|)}^s}{{(\mathrm{1}-\overline{u}w)}^{\mathrm{2}+s}}g\left(u\right)\mathrm{d}A\left(u\right)$

As a consequence of the separation condition, the support of $g_i$ is disjoint. Also, if the support of $g_i$ intersects a Carleson box $S(I)$, then $z_i$ is contained in $S(\mathrm{2}I)$. Just for this condition, a relation can be obtained:

$\begin{array}{l}\underset{S(I)}{\int }{\left|\sum _j{a}_j{g}_{z_j}\right|}^{\mathrm{2}}{\left(\mathrm{1}-\left|z\right|\right)}^p\mathrm{d}A\left(z\right)\\ ⩽{\sum _j\left|a_j\right|}^{\mathrm{2}}{\underset{S\left(\mathrm{2}I\right)}{\int }\left|g_{z_j}(z)\right|}^{\mathrm{2}}{\left(\mathrm{1}-\left|z\right|\right)}^p\mathrm{d}A(z)\\ ⩽C{\sum _{z_j\in S(\mathrm{2}I)\ne \mathrm{\varnothing }}\left(\mathrm{1}-\left|z_j\right|\right)}^p⩽C{\left|I\right|}^p.\end{array}$

Then $f=\sum _j{a}_j{f}_{z_j}\in Q_p\bigcap H^{\mathrm{\infty }}$, by Lemma 2.

Conversely, $\sum _n\left(\mathrm{1}-{\left|z_n\right|}^{\mathrm{2}}\right){\delta }_{z_n}\left(z\right)$ is a p-Carleson measure by Theorem 1. Since $\left\{z_n\right\}$ is also an interpolating sequence for Bloch space, the condition (3) holds by (2).

Remark 1   The analytic function $f$ in Theorem 1 is not unique. It is assumed that $f=\sum _j{a}_j{f}_{z_j}\in Q_p\bigcap H^{\mathrm{\infty }}$ such that $f(z_n)={\lambda }_n,n=\mathrm{1,2},\cdots$ for any sequence $\left\{{\lambda }_n\right\}\in l^{\mathrm{\infty }}$, causing the following relation,

$\underset{a\in D}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{\left|f(z_n)\right|}^{\mathrm{2}}{\left(\mathrm{1}-{\left|{\phi }_a(z_n)\right|}^{\mathrm{2}}\right)}^p<\mathrm{\infty }$

Since the inner function

$B(z)={\displaystyle \prod _{n=\mathrm{1}}^{\mathrm{\infty }}}\frac{\left|z_n\right|}{z_n}\frac{z_n-z}{\mathrm{1}-\overline{z_n}z}\in Q_p$

by Theorem 5.2.1 in Ref.[1], then $fB\in Q_p$ by Lemma 1 of Ref.[14]. $h(z)=f(z)(\mathrm{1}+B(z))$ is defined. Then $h(z_n)={\lambda }_n,n=\mathrm{1,2},\cdots ,$ and $h\in Q_p\bigcap H^{\mathrm{\infty }}$.

2 Interpolating Sequence in $\mathit{F}\left(\mathit{p},\mathit{q},\mathit{s}\right)\bigcap {\mathit{H}}^{\mathit{\infty }}$

For $\mathrm{0}<p<\mathrm{\infty }$,$\mathrm{0}<s<\mathrm{\infty }$,an analytic function $f$ belongs to the space $F(p,p-\mathrm{2},s)$ if

$\underset{a\in D}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{D}{\int }{\left|f\text{'}(z)\right|}^p{\left(\mathrm{1}-{\left|z\right|}^{\mathrm{2}}\right)}^{p-\mathrm{2}}{\left(\mathrm{1}-{\left|{\phi }_a(z)\right|}^{\mathrm{2}}\right)}^s\mathrm{d}A(z)<\mathrm{\infty }$(10)

We know in this paper that the space $F(p,p-\mathrm{2},s)$ is a subspace of $B$ by Corollary 2.8 of Ref.[15]. An analytic function $f\in F(p,p-\mathrm{2},s)$ if and only if the positive measure ${\left|g\text{'}\left(z\right)\right|}^p{\left(\mathrm{1}-{\left|z\right|}^{\mathrm{2}}\right)}^{p-\mathrm{2}+s}\mathrm{d}A\left(z\right)$ is an s-Carleson measure. $F(p,p-\mathrm{2},s)$ belongs to a general of function space $F(p,q,s)$. See Ref.[15].

A sequence $\left\{z_n\right\}$ in the unit disc is an interpolating sequence for $F(p,p-\mathrm{2},s)\bigcap H^{\mathrm{\infty }}$ if each bounded sequence $\left\{{\lambda }_n\right\}$ of complex values; there exists an $f\in F(p,p-\mathrm{2},s)\bigcap H^{\mathrm{\infty }}$ so that $f(z_n)={\lambda }_n$ for all n. Yuan and Tong gave a characterization of the interpolating sequences for $F(p,p-\mathrm{2},s)\bigcap H^{\mathrm{\infty }}$ in Ref. [16].

Lemma 4^[16] Let $\mathrm{1}<p<\mathrm{\infty },\mathrm{0}<s<\mathrm{1}.$ Then ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$$\subset D$ is an interpolating sequence for $F(p,p-\mathrm{2},s)\bigcap H^{\mathrm{\infty }}$ if and only if ${\left\{z_n\right\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ is a separated sequence and ${\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{\infty }}}{\left(\mathrm{1}-{\left|z_n\right|}^{\mathrm{2}}\right)}^p{\delta }_{z_n}$ is an s-Carleson measure.