On Balayaga and B-Balayage Operators

Abstract

Here, we consider the balayage operator in the setting of $H^p$ spaces and its Bergman space version (B-balayage) introduced by Wulan et al. (Complex Var Ellipt Equ 59(12):1775–1782, 2014), and extend some known results on these operators.

1 Introduction

Let ${\mathbb {D}}$ denote the unit disk $\{z:{\mathbb {C}}:|z|<1\}$ and ${\mathbb {T}}$ the unit circle. For $0<p<\infty$, the Hardy space $H^p$ consists of all functions f which are holomorphic on ${\mathbb {D}}$ and satisfy

$$\begin{aligned} \Vert f\Vert _{H^p}=\sup _{0<r<1}\left\{ \frac{1}{2\pi }\int _0^{2\pi }|f(r\mathrm{e}^{it})|^p {\text {d}}t\right\} ^\frac{1}{p}<\infty . \end{aligned}$$

It is known that each function $f\in H^p$ has the radial limit $f(\mathrm{e}^{it})=\lim _{r\rightarrow 1^-}f(r\mathrm{e}^{it})$ a.e. on ${\mathbb {T}}$ and $f(\mathrm{e}^{it})\in L^p({\mathbb {T}})$.

For $\phi \in L^1({\mathbb {T}})$, we say that $\phi \in {\text {BMO}}({\mathbb {T}})$ if

$$\begin{aligned} \Vert \phi \Vert _*=\sup _{I\subset {\mathbb {T}}}\frac{1}{|I|}\int _I |\phi (\mathrm{e}^{it})-\phi _I|{\text {d}}t<\infty , \end{aligned}$$

where I denotes any arc of ${\mathbb {T}}$, |I| is its arc length and

$$\begin{aligned} \phi _I=\frac{1}{|I|}\int _I\phi (\mathrm{e}^{it}){\text {d}}t. \end{aligned}$$

In [7], the authors have recently considered Campanato spaces ${\mathcal {L}}^{p,\lambda }({\mathbb {T}})$ defined as follows. For $\lambda \ge 0$ and $1\le p<\infty$, the space ${\mathcal {L}}^{p,\lambda }({\mathbb {T}})$ consists of all functions $\phi \in L^p({\mathbb {T}})$ for which

$$\begin{aligned} \sup _{I\subset {\mathbb {T}}}\frac{1}{|I|^\lambda }\int _I |\phi (\mathrm{e}^{it})-\phi _I|^p{\text {d}}t<\infty . \end{aligned}$$

We note that $\mathrm{BMO}({\mathbb {T}})={\mathcal {L}}^{p,1},\ 1 \le p<\infty ,$ (see [3, pp. 222-235]).

For a finite positive Borel measure $\mu$ on ${\mathbb {D}}$, the function

$$\begin{aligned} S_\mu (\mathrm{e}^{it})=\int _{\mathbb {D}} \frac{1-|z|^2}{|1-z\mathrm{e}^{-it}|^2}\mathrm{d}\mu (z), \end{aligned}$$

(1)

is called the balayage of$\mu$. It follows from Fubini’s theorem that $S_\mu (\mathrm{e}^{it})\in L^1({\mathbb {T}})$ (see [3, p. 229]).

If I is an arc of ${\mathbb {T}}$, the Carleson square S(I) is defined as

$$\begin{aligned} S(I)=\left\{ r\mathrm{e}^{it}:\mathrm{e}^{it}\in I, 1-\frac{|I|}{2\pi }\le r<1\right\} . \end{aligned}$$

A positive Borel measure $\mu$ is called an s-Carleson measure, $0<s<\infty$, if there exists a positive constant $C=C(\mu )$, such that

$$\begin{aligned} \mu (S(I))\le C(\mu )|I|^s,\quad \text {for any arc } I\subset \mathbb T. \end{aligned}$$

A 1-Carleson measure is simply called a Carleson measure. In [1], Carleson proved that if $\mu$ is a positive Borel measure in ${\mathbb {D}}$, then, for $0<p<\infty$, $H^p\subset L^p(\mathrm{d}\mu )$ if and only if $\mu$ is a Carleson measure.

It has been proved in [3, p. 229] that if $\mu$ is the Carleson measure, then $S_\mu$ belongs to $\mathrm{BMO}({\mathbb {T}})$. However, the Carleson property of measure $\mu$ is not a necessary condition for $S_\mu$ being a $\mathrm{BMO}({\mathbb {T}})$ function [5].

In the next section, we obtain an extension of the result mentioned above. More precisely, we prove that if $\mu$ is an s-Carleson measure, $0<s\le 1$, then $S_\mu$ belongs to ${\mathcal {L}}^{1,s}$.

In [6], H. Wulan, J. Yang, and K. Zhu introduced the Bergman space version of the balayage operator on the unit disk that was called B-balayage. The B-balayage of a finite complex measure $\mu$ on ${\mathbb {D}}$ is given by

$$\begin{aligned} G_\mu (z)=\int _{\mathbb {D}} \frac{(1-|w|^2)^2}{|1-{{\bar{z}}}w|^4} \mathrm{d}\mu (w), \quad z\in {\mathbb {D}}. \end{aligned}$$

It has been proved in [6] that if $\mu$ is a 2-Carleson measure, then there exists a constant $C>0$, such that

$$\begin{aligned} |G_{\mu }(z)-G_{\mu }(w)|\le C \beta (z,w),\quad z,w\in \mathbb D, \end{aligned}$$

(2)

where $\beta$ is the hyperbolic metric on ${\mathbb {D}}$. Here, applying a similar idea to that used in the proof of this result, we prove the following theorem.

Theorem 1

Assume that $1< p<\infty$ and $\mu$ is a positive Borel measure on ${\mathbb {D}}$. If $\mu$ is a 2p-Carleson measure, then there exists a positive constant $C=C(p)$, such that

$$\begin{aligned} |G_{\mu }(z)-G_{\mu }(w)|\le C\,\left( \beta (z,w)\right) ^{\frac{1}{p}} \end{aligned}$$

for all $z,w\in {\mathbb {D}}$.

Actually, this theorem is a special case of a more general theorem stated in Sect. 3.

Here, C will denote a positive constant which can vary from line to line.

2 Balayage Operators and Campanato Spaces ${\mathcal {L}}^{1,s}$

We start with the following result.

Theorem 2

If $\mu$ is an s-Carleson measure, $0<s\le 1$, $S_\mu$ is given by (1) and $0\le \gamma <1$, then there exists a positive constant C, such that for any $I\subset {\mathbb {T}}$:

$$\begin{aligned} \frac{1}{|I|^{1+s-\gamma }}\int _I\int _I\frac{|S_\mu (\mathrm{e}^{i\theta })-S_\mu (\mathrm{e}^{i\varphi })|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \le C. \end{aligned}$$

Proof

Without loss of generality, we can assume that $|I|<1$.

Let, for $z\in {\mathbb {D}}$ and $\theta \in {\mathbb {R}}$:

$$\begin{aligned} P_z(\theta )=\frac{1-|z|^2}{|1-z\mathrm{e}^{-i\theta }|^2}=\text {Re}\left( \frac{1+z\mathrm{e}^{-i\theta }}{1-z\mathrm{e}^{-i\theta }}\right) \end{aligned}$$

be the Poisson kernel for the disk ${\mathbb {D}}$. By the Fubini theorem:

$$\begin{aligned}&\int _I\int _I\frac{|S_\mu (\mathrm{e}^{i\theta })-S_\mu (\mathrm{e}^{i\varphi })|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \le \int _I\int _I\int _{\mathbb D}\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\mathrm{d}\mu (z)\,\mathrm {d}\theta \,\mathrm{d}\varphi \ \nonumber \\&\qquad =\int _{\mathbb D}\int _I\int _I\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \ \mathrm{d}\mu (z). \end{aligned}$$

(3)

For a subarc I of ${\mathbb {T}}$, let $2^nI,\ n\in {\mathbb {N}}$ denote the subarc of ${\mathbb {T}}$ with the same center as I and the length $2^n|I|$.

In view of the equality

$$\begin{aligned} \int _0^{2\pi }P_z(\theta )\mathrm{d}\theta =2\pi , \end{aligned}$$

we have

$$\begin{aligned} \int _I\int _I \frac{P_z(\theta )}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi= & {} \int _IP_z(\theta )\int _I \frac{\mathrm{d}\varphi }{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \le C|I|^{1-\gamma }. \end{aligned}$$

Consequently

$$\begin{aligned} \int _{S(2I)}\int _I\int _I \frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \ \mathrm{d}\mu (z)\le 2C|I|^{1-\gamma }\int _{S(2I)}\mathrm{d}\mu (z)\le C|I|^{1+s-\gamma }. \end{aligned}$$

(4)

Since $P_z(\theta )\le 4$ for $|z|\le \frac{1}{2}$, we get

$$\begin{aligned}&\int _{|z|\le \frac{1}{2} }\int _I\int _I\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \ \mathrm{d}\mu (z) \le 8\mu ({\mathbb {D}})\int _I\int _I\frac{\mathrm{d}\theta \mathrm{d}\varphi }{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma } \\&\qquad \le C|I|^{2-\gamma }\le C |I|^{1+s-\gamma }. \end{aligned}$$

Now, we assume that $\frac{1}{2}\le |z|<1$ and $z=|z|\mathrm{e}^{i\omega }\in S(2^{n+1}I){\setminus } S(2^nI)$. We consider two cases: (i) $\mathrm{e}^{i\omega } \in 2^nI$ and (ii) $\mathrm{e}^{i\omega } \in 2^{n+1}I{\setminus } 2^nI$.

In case (i), we have

$$\begin{aligned} \frac{2^n|I|}{2\pi }<1-|z|\le \frac{2^{n+1}|I|}{2\pi }. \end{aligned}$$

Thus

$$\begin{aligned} |P_z(\theta )-P_z(\varphi )|= & {} \frac{(1-|z|^2)2|z||\cos (\theta -\omega )-\cos (\varphi -\omega )|}{\left( (1-|z|)^2+4|z|\sin ^2\frac{\theta -\omega }{2}\right) \left( (1-|z|)^2+4|z|\sin ^2\frac{\varphi -\omega }{2}\right) } \\\le & {} \frac{8|\sin \frac{(\theta -\omega )+(\varphi -\omega )}{2}||\sin \frac{(\theta -\varphi )}{2}|}{(1-|z|)^3}\\\le & {} 2\frac{\left( |\theta -\omega |+|\varphi -\omega |\right) |\theta -\varphi |}{(1-|z|)^3}. \end{aligned}$$

Therefore, if $\mathrm{e}^{i\theta },\mathrm{e}^{i\varphi }\in I$, then

$$\begin{aligned} \frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\le & {} C\frac{\left( |\theta -\omega |+|\varphi -\omega |\right) |\theta -\varphi |^{1-\gamma }}{(1-|z|)^3}\nonumber \\\le & {} C\frac{2^n|I||I|^{1-\gamma }}{(2^n|I|)^3}=C\frac{|I|^{-1-\gamma }}{2^{2n}}. \end{aligned}$$

(5)

Now, we turn to case (ii). Then, for $\mathrm{e}^{i\psi }\in I$,

$$\begin{aligned} 2^{n-2}|I|\le |\psi -\omega |\le 2^n|I|. \end{aligned}$$

Consequently, for $\mathrm{e}^{i\theta },\mathrm{e}^{i\varphi }\in I$, we get

$$\begin{aligned} \frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\le & {} 2\frac{\left| |(1-z\mathrm{e}^{-i\theta }|^2-|1-z\mathrm{e}^{-i\varphi }|^2\right| }{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma |1-z\mathrm{e}^{-i\theta }||1-z\mathrm{e}^{-i\varphi }|^2}\nonumber \\\le & {} C \frac{\left( |\theta -\omega |+|\varphi -\omega |\right) |\theta -\varphi |^{1-\gamma }}{|\theta -\omega ||\varphi -\omega |^2}\nonumber \\\le & {} C\frac{|I|^{-1-\gamma }}{2^{2n}}. \end{aligned}$$

(6)

Now, we put $Q_n=S(2^nI),\ n=1,2,\ldots$ Then, by (5) and (6),

$$\begin{aligned}&\int _{\begin{array}{c} {Q_{n+1}{\setminus } Q_n}\\ |z|\ge \frac{1}{2} \end{array}}\int _I\int _I \frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}\mu (z) \le C \frac{|I|^{1-\gamma }}{2^{2n}}\int _{\begin{array}{c} Q_{n+1} \end{array}}\mathrm{d}\mu (z)\le C \frac{|I|^{1+s-\gamma }}{2^{n(2-s)}}. \end{aligned}$$

The above inequality and (4) imply

$$\begin{aligned}&\int _{\mathbb D}\int _I\int _I\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\,\mathrm{d}\theta \,\mathrm{d}\varphi \ \mathrm{d}\mu (z)\le \int _{Q_{1}}\int _I\int _I\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}\mu (z)\\&\qquad + \sum _{n=1}^\infty \int _{Q_{n+1}{\setminus } Q_n}\int _I\int _I\frac{|P_z(\theta )-P_z(\varphi )|}{|\mathrm{e}^{i\theta }-\mathrm{e}^{i\varphi }|^\gamma }\mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}\mu (z)\\&\quad \le C|I|^{s+1-\gamma }\sum _{n=1}^\infty \frac{1}{2^{n(2-s)}} =C|I|^{1+s-\gamma }. \end{aligned}$$

$\square$

The next theorem shows that if $\mu$ is an s-Carleson measure, $0<s\le 1$, then $S_\mu$ is in the Campanato space $\mathcal {L}^{1,s}$.

Theorem 3

If $\mu$ is an s-Carleson measure on ${\mathbb {D}}$, $0<s\le 1$ and $S_\mu (t)=S_\mu (\mathrm{e}^{it})$ is the balayage operator of $\mu$ given by (1), then there exists a positive constant C, such that for any $I\subset {\mathbb {T}}$

$$\begin{aligned} \frac{1}{|I|^s}\int _I|S_\mu (t)-(S_\mu )_I|{\text {d}}t\le C. \end{aligned}$$

Proof

It is enough to observe that

$$\begin{aligned}&\frac{1}{|I|^s}\int _I|S_\mu (t)-(S_\mu )_I|{\text {d}}t \le \frac{1}{|I|^{s+1}}\int _I\int _I|S_\mu (t)-S_\mu (u)|{\text {d}}t{\text {d}}u \end{aligned}$$

and the inequality follows from Theorem 2 with $\gamma = 0$. $\square$

3 B-Balayage for Weighted Bergman Spaces $A^p_\alpha$

Recall that, for $0<p<\infty$, $-1<\alpha <\infty$, the weighted Bergman space $A_\alpha ^p$ is the space of all holomorphic functions in $L^p({\mathbb {D}},\mathrm{d}A_\alpha )$, where

$$\begin{aligned} \mathrm{d}A_\alpha (z)=(\alpha +1)(1-|z|^2)^\alpha \mathrm{d}A(z) \end{aligned}$$

and $\mathrm{d}A$ is the normalized Lebesgue measure on ${\mathbb {D}}$; that is, $\int _{\mathbb {D}} \mathrm{d}A=1$. If f is in $L^p({\mathbb {D}}, \mathrm{d}A_\alpha )$, we write

$$\begin{aligned} \Vert f\Vert _{p,\alpha }^p=\int _{\mathbb {D}} |f(z)|^p \mathrm{d}A_\alpha (z). \end{aligned}$$

It is well known that, for $1<p<\infty$, the Bergman projection $P_\alpha$ given by

$$\begin{aligned} P_\alpha f(z)=\int _{\mathbb {D}} \frac{f(w)}{(1-z{{\bar{w}}})^{2+\alpha }}\mathrm{d}A_\alpha (w) \end{aligned}$$

is a bounded operator from $L^p({\mathbb {D}}, \mathrm{d}A_\alpha )$ onto $A_\alpha ^p$.

Let for $z,w\in {\mathbb {D}}$, the function

$$\begin{aligned} \varphi _z(w)=\frac{z-w}{1-{{\bar{z}}} w} \end{aligned}$$

denote the automorphism of the unit disk ${\mathbb {D}}$. The hyperbolic metric on ${\mathbb {D}}$ is given by

$$\begin{aligned} \beta (z,w)=\frac{1}{2} \log \frac{1+|\varphi _z(w)|}{1-|\varphi _z(w)|}. \end{aligned}$$

For $z\in {\mathbb {D}}$ and $r>0$, the hyperbolic disk with center z and radius r is

$$\begin{aligned} D(z,r)=\{w\in {\mathbb {D}}:\beta (z,w)<r\}. \end{aligned}$$

For $s>1$, the condition for an s-Carleson measure given in Introduction is equivalent to the condition where Carleson squares are replaced by hyperbolic disks. More exactly, the following result is known.

Proposition

[2, 10] Let $\mu$ be a positive Borel measure on ${\mathbb {D}}$ and $1<s<\infty$. Then, the following statements are equivalent

• (i) $\mu$ is an s-Carleson measure,

• (ii) $\mu (D(z,r))\le C(1-|z|^2)^s$ for some constant C depending only on r for all hyperbolic disk D(z, r), $z\in {\mathbb {D}}$.

A positive Borel measure $\mu$ on ${\mathbb {D}}$ is called an $A^p_\alpha$-Carleson measure if there exists a positive constant C, such that

$$\begin{aligned} \int _{\mathbb {D}} |f(z)|^p \mathrm{d}\mu (z)\le C\int _{\mathbb {D}} |f(z)|^p \mathrm{d}A_\alpha (z) \end{aligned}$$

for all $f\in A_\alpha ^p$.

It is well known that $\mu$ is an $A^p_\alpha$-Carleson measure if and only if $\mu$ is $(2+\alpha )$-Carleson measure (see [10, p. 133]). This means that $A^p_\alpha$-Carleson measures are independent of p.

The next corollary is an immediate consequence of the last proposition.

Corollary

[6] For $\alpha>-1,\ \sigma >0$, let $\mu ,\nu$ be positive Borel measures on ${\mathbb {D}}$, such that

$$\begin{aligned} \mathrm{d}\nu (z)=(1-|z|)^\sigma \mathrm{d}\mu (z). \end{aligned}$$

Then, $\mu$ is an $A^p_\alpha$-Carleson measure if and only if $\nu$ is an $A^p_{\alpha +\sigma }$-Carleson measure.

Recall that, for $1<p<\infty$, the Besov space $B_p$ is the space of all functions f analytic on ${\mathbb {D}}$, such that

$$\begin{aligned} \Vert f\Vert _{B_p}^p=\int _{\mathbb {D}} |f'(z)|^p(1-|z|^2)^p \mathrm{d}\tau (z)<\infty , \end{aligned}$$

where

$$\begin{aligned} \mathrm{d}\tau (z)=\frac{\mathrm{d}A(z)}{(1-|z|^2)^2} \end{aligned}$$

is the Möbius invariant measure on ${\mathbb {D}}$.

We will use the fact that the Besov space $B_p=P_\alpha (L^p,\mathrm{d}\tau ).$ The proof of this equality is given in [9, p. 119]. Moreover, if $f=P_\alpha g$, where $g\in L^p(\mathrm{d}\tau )$, then

$$\begin{aligned} (1-|z|^2)f'(z)=(\alpha +2)(1-|z|^2)\int _{\mathbb {D}} \frac{g(w){{\bar{w}}}}{(1-z\bar{w})^{3+\alpha }}\mathrm{d}A_\alpha (w). \end{aligned}$$

It then follows from [4, Thm. 1.9] that

$$\begin{aligned} \Vert f\Vert _{B_p}\le C_{p,\alpha }\Vert g\Vert _{L^p(\mathrm{d}\tau )}. \end{aligned}$$

(7)

The next theorem gives a Lipschitz type estimate for functions in the analytic Besov space.

Theorem 4

[8] For any $1<p<\infty$, there exists a constant $C_p>0$, such that

$$\begin{aligned} |f(z)-f(w)|\le C_p\Vert f\Vert _{B_p}(\beta (z,w))^\frac{1}{q} \end{aligned}$$

for all $f\in B_p$ and $z,w\in {\mathbb {D}}$, where $\frac{1}{p}+\frac{1}{q} =1$.

In [6], the authors also consider a version of the balayage of a measure $\mu$ on ${\mathbb {D}}$ defined by

$$\begin{aligned} G_{\mu ,\alpha }(z)= \int _{\mathbb D}\frac{(1-|z|^2)^{2+\alpha }}{|1-{{\bar{z}}} w|^{4+2\alpha }}\mathrm{d}\mu (w). \end{aligned}$$

They have proved the following generalization of inequality (2).

If $\mu$ is an $A^p_\alpha$-Carleson measure, then the generalized balayage $G_{\mu ,\alpha }$ satisfies the Lipschitz condition:

$$\begin{aligned} |G_{\mu ,\alpha }(z)-G_{\mu ,\alpha }(w)|\le C \beta (z,w),\quad z,w\in {\mathbb {D}}, \end{aligned}$$

where C is independent of z and w.

It is worth noting here that the balayage given by (1) is in a certain sense a limit case of $G_{\mu ,\alpha }$ as $\alpha \rightarrow -1$. Since an $A_\alpha ^p$-Carleson measure is actually a $(2+\alpha )$-Carleson measure, the last inequality gives a necessary condition for a measure $\mu$ to be an s-Carleson measure, as $1<s<\infty$.

Theorem 1 is a special case of the following more general theorem.

Theorem 5

Assume that $1< p<\infty$, $-1<\alpha <\infty$, and $\mu$ is a positive Borel measure on ${\mathbb {D}}$. If $\mu$ is a $p(2+\alpha )$-Carleson measure, then there exists a positive constant $C=C(p,\alpha )$, such that

$$\begin{aligned} |G_{\mu ,\alpha }(z)-G_{\mu ,\alpha }(w)|\le C\,\left( \beta (z,w)\right) ^{\frac{1}{p}} \end{aligned}$$

for all $z,w\in {\mathbb {D}}$.

Proof

For z, w, we have

$$\begin{aligned} |G_{\mu ,\alpha }(z)-G_{\mu ,\alpha }(w)|\le & {} \int _{\mathbb D}\left| \frac{(1-|a|^2)^{2+\alpha }}{|1-a\bar{z}|^{4+2\alpha }}-\frac{(1-|a|^2)^{2+\alpha }}{|1-a\bar{w}|^{4+2\alpha }}\right| \mathrm{d}\mu (a)\\\le & {} \int _{{\mathbb {D}}}\left| \frac{(1-|a|^2)^{2+\alpha }}{(1-a\bar{z})^{4+2\alpha }}-\frac{(1-|a|^2)^{2+\alpha }}{(1-a\bar{w})^{4+2\alpha }}\right| \mathrm{d}\mu (a). \end{aligned}$$

Since $\mu$ is a finite measure on ${\mathbb {D}}$, the Jensen’s inequality yields

$$\begin{aligned} |G_{\mu }(z)-G_{\mu }(w)|^p\le & {} C\int _{\mathbb D}\left| \frac{(1-|a|^2)^{2+\alpha }}{(1-a\bar{z})^{4+2\alpha }}-\frac{(1-|a|^2)^{2+\alpha }}{(1-a\bar{w})^{4+2\alpha }}\right| ^p\mathrm{d}\mu (a)\\= & {} C \int _{{\mathbb {D}}}\left| \frac{1}{(1-a\bar{z})^{4+2\alpha }}-\frac{1}{(1-a\bar{w})^{4+2\alpha }}\right| ^p(1-|a|^2)^{(2+\alpha )p}\mathrm{d}\mu (a).\\ \end{aligned}$$

By the Corollary, $(1-|a|^2)^{p(2+\alpha )}\mathrm{d}\mu (a)$ is an $A^p_{2p(2+\alpha )-2}$-Carleson measure, because $\mu$ is an $A^p_{p(2+\alpha )-2}$-Carleson measure. Consequently,

$$\begin{aligned}&\int _{{\mathbb {D}}}\left| \frac{1}{(1-a\bar{z})^{4+2\alpha }}-\frac{1}{(1-a\bar{w})^{4+2\alpha }}\right| ^p(1-|a|^2)^{p(2+\alpha )}\mathrm{d}\mu (a)\\&\qquad \le C \int _{{\mathbb {D}}}\left| \frac{1}{(1-a\bar{z})^4}-\frac{1}{(1-a\bar{w})^4}\right| ^p(1-|a|^2)^{2p(2+\alpha )-2}\mathrm{d}A(a)\\&\qquad = C \int _{{\mathbb {D}}}\left| \frac{(1-|a|^2)^{2+2\alpha }}{(1-\bar{a}z)^{4+2\alpha }}-\frac{(1-|a|^2)^{2+2\alpha }}{(1-{{\bar{a}}} w)^{4+2\alpha }}\right| ^p\mathrm{d}A_{\frac{2}{q-1}}(a), \end{aligned}$$

where q is the conjugate index for p, that is, $\frac{1}{p} +\frac{1}{q} =1$.

Now, set $\beta =\frac{2}{q-1}$ and note that

$$\begin{aligned}&\int _{{\mathbb {D}}}\left| \frac{(1-|a|^2)^{2+2\alpha }}{(1-\bar{a}z)^{4+2\alpha }}-\frac{(1-|a|^2)^{2+2\alpha }}{(1-{\bar{a}}w)^{4+2\alpha }}\right| ^p \mathrm{d}A_{\beta }(a)\\&\qquad = \left( \sup _{\Vert f\Vert _{q,\beta }\le 1}\left| \int _{\mathbb D}\left( \frac{(1-|a|^2)^{2+\alpha }}{(1-{{\bar{a}}} z)^{4+2\alpha }}-\frac{(1-|a|^2)^{2+2\alpha }}{(1-{{\bar{a}}} w)^{4+2\alpha }}\right) f(a)\mathrm{d}A_{\beta }(a)\right| \right) ^p. \end{aligned}$$

Put $g=(\alpha +1)^\frac{1}{q}(1-|a|^2)^{\frac{\beta +2}{q}}f$ and observe that $\Vert f\Vert _{q,\beta }\le 1$ if and only if $\Vert g\Vert _{L^q(\mathrm{d}\tau )}\le 1$. Moreover, since $\beta =\frac{2}{q-1}$ satisfies $\frac{\beta +2}{q}=\beta$, we get

$$\begin{aligned}&\sup _{\Vert f\Vert _{q,\beta }\le 1}\left| \int _{\mathbb D}\left( \frac{(1-|a|^2)^{2+2\alpha }}{(1-\bar{a}z)^{4+2\alpha }}-\frac{(1-|a|^2)^{2+2\alpha }}{(1-{{\bar{a}}} w)^{4+2\alpha }}\right) f(a)\mathrm{d}A_{\beta }(a)\right| \\&\qquad \quad = C\sup _{\Vert g\Vert _{L^q(\mathrm{d}\tau )}\le 1}\left| \int _{\mathbb D}\left( \frac{(1-|a|^2)^{2+2\alpha }}{(1-\bar{a}z)^{4+2\alpha }}-\frac{(1-|a|^2)^{2+2\alpha }}{(1-{{\bar{a}}} w)^{4+2\alpha }}\right) g(a)\mathrm{d}A(a)\right| \\&\qquad \quad = C\sup _{\Vert g\Vert _{L^q(\mathrm{d}\tau )}\le 1}\left| \int _{\mathbb D}\left( \frac{g(a)}{(1-{{\bar{a}}}z)^{4+2\alpha }}-\frac{g(a)}{(1-{{\bar{a}}} w)^{4+2\alpha }}\right) \mathrm{d}A_{2+\alpha }(a)\right| \\&\qquad \quad = C\sup _{\Vert g\Vert _{L^q(\mathrm{d}\tau )}\le 1}\left| P_{2+\alpha }g(z)-P_{2+\alpha }g(w)\right| \le C(\beta (z,w))^\frac{1}{p}, \end{aligned}$$

where the last inequality follows from Theorem 4 and inequality (7). $\square$