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# 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.

## 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 , 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 , 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 .

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 , 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  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.

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

### 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 $$\gamma = 0$$. $$\square$$

## 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

1. (i)

$$\mu$$ is an s-Carleson measure,

2. (ii)

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

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

 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

 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 , 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 zw, 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). $$\square$$

## Change history

• ### 15 February 2020

In the original publication, article title was incorrectly published as.

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## Acknowledgements

The authors are grateful to the referee for suggesting Theorem 5.

## Author information

Correspondence to Maria Nowak.