A Hankel matrix acting on spaces of analytic functions

TL;DR

This paper studies Hankel matrices defined by measures on [0,1) and their induced operators on analytic function spaces, extending classical Hilbert operator results to Hardy and Möbius invariant spaces.

Contribution

It generalizes the classical Hilbert operator by analyzing Hankel matrices from measures and explores their action on Hardy and Möbius invariant spaces, improving recent results.

Findings

Extended the understanding of Hankel operators on Hardy spaces.

Provided new bounds and conditions for boundedness of these operators.

Enhanced the theoretical framework for operators acting on analytic function spaces.

Abstract

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D $. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_\mu $ on Hardy spaces and on M\"obius invariant spaces.

Equations

$${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n$$

$${\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n,$$

$$\mu \left(S(I)\right)\le C|I{|}^s,for\; any\; intervalI\subset \partial \mathbb{D}.$$

$$\frac{\mu \left(S(I)\right){\left(\log \frac{2\pi }{|I|}\right)}^{\alpha }}{|I{|}^s}\le C,for\; any\; intervalI\subset \partial \mathbb{D}.$$

$$\tilde{\mu }(A)=\mu \left(A\cap [0,1)\right),\text{for any Borel subset A of D}.$$

$$\mu \left([t,1)\right)\le C(1-t{)}^s,0\le t<1.$$

$$I_{\mu }(f)(z)={\int }_{[0,1)}\frac{f(t)}{1-tz}d\mu (t),$$

$${\mathcal{H}}_{\mu }(f)=I_{\mu }(f),\text{for every f∈H1}.$$

$${\mathcal{H}}_{\mu }(f)=I_{\mu }(f),\text{for every f∈Hp}.$$

$$f\in \mathcal{B}\iff {\rho }_{\mathcal{B}}(f)\stackrel{\mathrm{def}}{=}\underset{z\in \mathbb{D}}{\sup }(1-|z{|}^2)|f^{\prime }(z)|<\infty .$$

$$\Vert f{\Vert }_{\star }\stackrel{\mathrm{def}}{=}\underset{a\in \mathbb{D}}{\sup }\Vert f\circ {\phi }_a-f(a){\Vert }_{H^2}<\infty .$$

$$H^{\infty }⊊BMOA⊊\bigcap _{0<p<\infty }{H}^p\text{and}BMOA⊊\mathcal{B}.$$

$${\rho }_{Q_s}(f)\stackrel{\mathrm{def}}{=}{\left(\underset{a\in \mathbb{D}}{\sup }{\int }_{\mathbb{D}}|f^{\prime }(z){|}^{2}g(z,a{)}^{s}dA(z)\right)}^{1/2}<\infty .$$

$$\mathcal{D}⊊Q_{s_1}⊊Q_{s_2}⊊BMOA,0<s_1<s_2<1.$$

$${\rho }_p(f)={\left({\int }_{\mathbb{D}}(1-|z{|}^2{)}^{p-2}|f^{\prime }(z){|}^{p}dA(z)\right)}^{1/p}<\infty .$$

$$B^p⊊B^q⊊BMOA,1<p<q<\infty .$$

$${\int }_{[0,1)}\log \frac{2}{1-t}d\mu (t)<\infty .$$

$$f_b(z)=\frac{1-b^2}{(1-bz{)}^2},z\in \mathbb{D}.$$

$$1\gtrsim \Vert {\mathcal{H}}_{\mu }(f_b){\Vert }_{H^1}.$$

$$f_b(z)=\sum _{k=0}^{\infty }{a}_{k,b}{z}^k,\text{with ak,b=(1−b2)(k+1)bk.}$$

$$1$$

$$=\sum _{n=1}^{\infty }\frac{1}{n}\left(\sum _{k=0}^{\infty }{a}_{k,b}{\int }_{[0,1)}{t}^{n+k}d\mu (t)\right)$$

$$\gtrsim (1-b^2)\sum _{n=1}^{\infty }\frac{1}{n}\left(\sum _{k=1}^{\infty }k{b}^k{\int }_{[b,1)}{t}^{n+k}d\mu (t)\right)$$

$$\gtrsim (1-b^2)\sum _{n=1}^{\infty }\frac{1}{n}\left(\sum _{k=1}^{\infty }k{b}^{n+2k}\mu \left([b,1)\right)\right)$$

$$=(1-b^2)\mu \left([b,1)\right)\sum _{n=1}^{\infty }\frac{b^n}{n}\left(\sum _{k=1}^{\infty }k{b}^{2k}\right)$$

$$=(1-b^2)\mu \left([b,1)\right)\left(\log \frac{1}{1-b}\right)\frac{b}{(1-b^2{)}^2}$$

$$\mu \left([b,1)\right)=O\left(\frac{1-b}{\log \frac{1}{1-b}}\right),\text{as b→1}.$$

$$f_b(z)={\left(\frac{1-b^2}{(1-bz{)}^2}\right)}^{1/p},z\in \mathbb{D}.$$

$$1\gtrsim \Vert {\mathcal{H}}_{\mu }(f_b){\Vert }_{H^p}.$$

$$f_b(z)=\sum _{k=0}^{\infty }{a}_{k,b}{z}^k,\text{with ak,b≈(1−b2)1/pkp2−1bk.}$$

$$1$$

$$=\sum _{n=1}^{\infty }{n}^{p-2}{\left(\sum _{k=0}^{\infty }{a}_{k,b}{\int }_{[0,1)}{t}^{n+k}d\mu (t)\right)}^p$$

$$\gtrsim (1-b^2)\sum _{n=1}^{\infty }{n}^{p-2}{\left(\sum _{k=1}^{\infty }{k}^{\frac{2}{p}-1}{b}^k{\int }_{[b,1)}{t}^{n+k}d\mu (t)\right)}^p$$

$$\gtrsim (1-b^2)\sum _{n=1}^{\infty }{n}^{p-2}{\left(\sum _{k=1}^{\infty }{k}^{\frac{2}{p}-1}{b}^{n+2k}\mu \left([b,1)\right)\right)}^p$$

Full text

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on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, in the unit disc ${\mathbb{D}}$. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_{\mu}$ on Hardy spaces and on Möbius invariant spaces.

This research is supported in part by a grant from “El Ministerio de Economía y Competitividad”, Spain (MTM2014-52865-P) and by a grant from la Junta de Andalucía FQM-210. The second author is also supported by a grant from “El Ministerio de de Educación, Cultura y Deporte”, Spain (FPU2013/01478).

1. Introduction and main results

We denote by $\mathbb{D}$ the unit disc in the complex plane $\mathbb{C}$, and by ${\mathcal{H}ol}(\mathbb{D})$ the space of all analytic functions in $\mathbb{D}$. We also let $H^{p}$ ($0<p\leq\infty$) be the classical Hardy spaces. We refer to [19] for the notation and results regarding Hardy spaces.

If  $\mu$ is a finite positive Borel measure on $[0,1)$ and $n\,=0,1,2,\dots$, we let $\mu_{n}$ denote the moment of order $n$ of $\mu$, that is, $\mu_{n}=\int_{[0,1)}t^{n}\,d\mu(t),$ and we define $\mathcal{H}_{\mu}$ to be the Hankel matrix $(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$. The matrix $\mathcal{H}_{\mu}$ can be viewed as an operator on spaces of analytic functions in the following way: if  $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\in{\mathcal{H}ol}({\mathbb{D}})$ we define

[TABLE]

whenever the right hand side makes sense and defines an analytic function in ${\mathbb{D}}$.

If $\mu$ is the Lebesgue measure on $[0,1)$ the matrix $\mathcal{H}_{\mu}$ reduces to the classical Hilbert matrix   $\mathcal{H}=\left({(n+k+1)^{-1}}\right)_{n,k\geq 0}$, which induces the classical Hilbert operator $\mathcal{H}$ which has extensively studied recently (see [1, 13, 14, 17, 27, 28]). Other related generalizations of the Hilbert operator have been considered in [20] and [32].

The question of describing the measures $\mu$ for which the operator $\mathcal{H}_{\mu}$ is well defined and bounded on distinct spaces of analytic functions has been studied in a good number of papers (see [8, 12, 21, 23, 30, 34, 38]). Carleson measures play a basic role in these works.

If $I\subset\partial{\mathbb{D}}$ is an interval, $|I|$ will denote the length of $I$. The Carleson square $S(I)$ is defined as $S(I)=\{re^{it}:\,e^{it}\in I,\quad 1-\frac{|I|}{2\pi}\leq r<1\}$.

If $\,s>0$ and $\mu$ is a positive Borel measure on ${\mathbb{D}}$, we shall say that $\mu$ is an $s$-Carleson measure if there exists a positive constant $C$ such that

[TABLE]

A $1$-Carleson measure will be simply called a Carleson measure.

We recall that Carleson [11] proved that $H^{p}\,\subset\,L^{p}(d\mu)$ ($0<p<\infty$), if and only if $\mu$ is a Carleson measure. This result was extended by Duren [18] (see also [19, Theorem 9. 4]) who proved that for $0<p\leq q<\infty$, $H^{p}\subset L^{q}(d\mu)$ if and only if $\mu$ is a $q/p$-Carleson measure.

If $X$ is a subspace of ${\mathcal{H}ol}({\mathbb{D}})$, $0<q<\infty$, and $\mu$ is a positive Borel measure in $\mathbb{D}$, $\mu$ is said to be a “${q}$-Carleson measure for the space ${X}$”  or an “${{(X,q)}}$-Carleson measure”  if $X\subset L^{q}(d\mu)$. The $q$-Carleson measures for the spaces $H^{p}$, $0<p,q<\infty$ are completely characterized. The mentioned results of Carleson and Duren can be stated saying that if $\,0<p\leq q<\infty$  then a positive Borel measure $\mu$ in ${\mathbb{D}}$ is a $q$-Carleson measure for $H^{p}$ if and only if $\mu$ is a $q/p$-Carleson measure. Luecking [29] and Videnskii [37] solved the remaining case $0<q<p$. We mention [9] for a complete information on Carleson measures for Hardy spaces.

Following [40], if $\mu$ is a positive Borel measure on ${\mathbb{D}}$, $0\leq\alpha<\infty$, and $0<s<\infty$ we say that $\mu$ is an $\alpha$-logarithmic $s$-Carleson measure if there exists a positive constant $C$ such that

[TABLE]

A positive Borel measure $\mu$ on $[0,1)$ can be seen as a Borel measure on $\mathbb{D}$ by identifying it with the measure $\tilde{\mu}$ defined by

[TABLE]

In this way a positive Borel measure $\mu$ on $[0,1)$ is an $s$-Carleson measure if and only if there exists a positive constant $C$ such that

[TABLE]

We have a similar statement for $\alpha$-logarithmic $s$-Carleson measures.

Widom [38, Theorem 3. 1] (see also [34, Theorem 3] and [33, p. 42, Theorem 7. 2]) proved that $\mathcal{H}_{\mu}$ is a bounded operator from $H^{2}$ into itself if and only $\mu$ is a Carleson measure. Galanopoulos and Peláez [21] studied the operators $\mathcal{H}_{\mu}$ acting on $H^{1}$ and Chatzifountas, Girela and Peláez [12] studied the action of $\mathcal{H}_{\mu}$ on $H^{p}$, $0<p<\infty$.

A key ingredient in [21] and [12] is obtaining an integral representation of $\mathcal{H}_{\mu}$. If $\mu$ is as above, we shall write throughout the paper

[TABLE]

whenever the right hand side makes sense and defines an analytic function in ${\mathbb{D}}$. It turns out that the operators $H_{\mu}$ and $I_{\mu}$ are closely related. Indeed, some of the results obtained in [21] and [12] are the following ones:

Theorem A** ([21]).**

Let $\mu$ be a positive Borel measure on $[0,1)$. Then:

• (i)

The operator $I_{\mu}$ is well defined on $H^{1}$ if and only if $\mu$ is a Carleson measure.

• (ii)

If $\mu$ is a Carleson measure, then the operator $\mathcal{H}_{\mu}$ is also well defined on $H^{1}$ and, furthermore,

[TABLE]

• (iii)

The operator $I_{\mu}$ is a bounded operator from $H^{1}$ into itself if and only if $\mu$ is a $1$-logarithmic $1$-Carleson measure.

Theorem B** ([12]).**

Suppose that $1<p<\infty$ and let $\mu$ be a positive Borel measure on $[0,1)$. Then:

• (i)

The operator $I_{\mu}$ is well defined on $H^{p}$ if and only if $\mu$ is a $1$-Carleson measure for $H^{p}$.

• (ii)

If $\mu$ is a $1$-Carleson measure for $H^{p}$, then the operator $\mathcal{H}_{\mu}$ is also well defined on $H^{p}$ and, furthermore,

[TABLE]

• (iii)

The operator $I_{\mu}$ is a bounded operator from $H^{p}$ into itself if and only if $\mu$ is a Carleson measure.

Theorem A and Theorem B immediately yield the following.

Theorem C**.**

Let $\mu$ be a positive Borel measure on $[0,1)$.

• (i)

If $\mu$ is a Carleson measure, then the operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{1}$ into itself if and only if $\mu$ is a $1$-logarithmic $1$-Carleson measure.

• (ii)

If $1<p<\infty$ and $\mu$ is a $1$-Carleson measure for $H^{p}$, then the operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{p}$ into itself if and only if $\mu$ is a Carleson measure.

Theorem C does not close completely the question of characterizing the measures $\mu$ for which $\mathcal{H}_{\mu}$ is a bounded operator from $H^{p}$ into itself. Indeed, in Theorem C we only consider $1$-Carleson measures for $H^{p}$. In principle, there could exist a measure $\mu$ which is not a $1$-Carleson measures for $H^{p}$ but so that the operator $\mathcal{H}_{\mu}$ is well defined and bounded on $H^{p}$. Our first result in this paper asserts that this is not the case.

Theorem 1.1**.**

Let $\mu$ be a positive Borel measure on $[0,1)$.

• (i)

The operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{1}$ into itself if and only if $\mu$ is a $1$-logarithmic $1$-Carleson measure.

• (ii)

If $1<p<\infty$ then the operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{p}$ into itself if and only if $\mu$ is a Carleson measure.

We have the following result for $p=\infty$, a case which was not considered in [12].

Theorem 1.2**.**

Let $\mu$ be a positive Borel measure on $[0,1)$. Then the following conditions are equivalent.

• (i)

$\int_{[0,1)}\frac{d\mu(t)}{1-t}\,<\,\infty$.

• (ii)

$\sum_{n=0}^{\infty}\mu_{n}\,<\,\infty$.

• (iii)

The operator $I_{\mu}$ is a bounded operator from $H^{\infty}$ into itself.

• (iv)

The operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{\infty}$ into itself.

In the paper [23] the authors have studied the operators $\mathcal{H}_{\mu}$ acting on certain conformally invariant spaces such as the Bloch space, BMOA, the analytic Besov spaces $B^{p}$ ($1<p<\infty$), and the $Q_{s}$ spaces. Let us introduce quickly these spaces.

It is well known that the set of all disc automorphisms (i.e., of all one-to-one analytic maps $f$ of $\mathbb{D}$ onto itself), denoted ${\rm Aut}(\mathbb{D})$, coincides with the set of all Möbius transformations of $\mathbb{D}$ onto itself: ${\rm Aut}(\mathbb{D})=\{\lambda\varphi_{a}:|a|<1,|\lambda|=1\}\,,$ where $\varphi_{a}(z)=(a-z)/(1-\overline{a}z)$.

A space $X$ of analytic functions in ${\mathbb{D}}$, defined via a semi-norm $\rho$, is said to be conformally invariant or Möbius invariant if whenever $f\in X$, then also $f\circ\varphi\in X$ for any $\varphi\in\operatorname{Aut}({\mathbb{D}})$ and, moreover, $\rho(f\circ\varphi)\leq C\rho(f)$ for some positive constant $C$ and all $f\in X$. We mention [3, 15, 42] as references for Möbius invariant spaces.

The Bloch space $\mathcal{B}$ consists of all analytic functions $f$ in $\mathbb{D}$ with bounded invariant derivative:

[TABLE]

A classical reference for the Bloch space is [2]; see also [42]. Rubel and Timoney [35] proved that $\mathcal{B}$ is the biggest “natural”  conformally invariant space.

The space $BMOA$ consists of those functions $f$ in $H^{1}$ whose boundary values have bounded mean oscillation on the unit circle. Alternatively, $BMOA$ can be characterized in the following way:

If $f$ is an analytic function in $\mathbb{D}$, then $f\in BMOA$ if and only if

[TABLE]

The seminorm $\|\cdot\|_{\star}$ is conformally invariant. We mention [22] as a general reference for the space $BMOA$. Let us recall that

[TABLE]

If $0\leq s<\infty$, we say that $f\in Q_{s}$ if $f$ is analytic in ${\mathbb{D}}$ and

[TABLE]

Here, $g(z,a)$ is the Green’s function in ${\mathbb{D}}$, given by $g(z,a)=\log\left|\frac{1-\overline{a}z}{z-a}\right|$, while $dA(z)=\frac{dx\,dy}{\pi}$ is the normalized area measure on $\mathbb{D}$. All $Q_{s}$ spaces ($0\leq s<\infty$) are conformally invariant with respect to the semi-norm $\rho_{Q_{s}}$ (see e.g., [39, p. 1] or [15, p. 47]).

These spaces were introduced by Aulaskari and Lappan in [5] while looking for new characterizations of Bloch functions. They proved that for $s>1$, $Q_{s}$ is the Bloch space. Using one of the many characterizations of the space $BMOA$ (see [22, Theorem 6. 2]) we have that $Q_{1}=BMOA$. In the limit case $s=0$, $Q_{s}$ is the classical Dirichlet space $\mathcal{D}$ of those analytic functions $f$ in ${\mathbb{D}}$ satisfying $\,\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\,dA(z)<\infty$.

Aulaskari, Xiao and Zhao proved in [7] that

[TABLE]

We mention [39] as an excellent reference for the theory of $Q_{s}$-spaces.

For $1<p<\infty$, the analytic Besov space $B^{p}$ is defined as the set of all functions $f$ analytic in $\mathbb{D}$ such that

[TABLE]

All $B^{p}$ spaces ($1<p<\infty$) are conformally invariant with respect to the semi-norm $\rho_{p}$ (see [3, p. 112] or [15, p. 46]). We have that $\mathcal{D}\,=\,B^{2}$. A lot of information on Besov spaces can be found in [3, 15, 16, 25, 41, 42]. Let us recall that

[TABLE]

Among others, the following results have been proved in [23].

Theorem D**.**

Let $\mu$ be a positive Borel measure on $[0,1)$.

• (i)

For any given $s>0$, the operator $I_{\mu}$ is well defined in $Q_{s}$ if and only if

[TABLE]

• (ii)

For any given $s>0$, the operator $I_{\mu}$ is a bounded operator from $Q_{s}$ into $BMOA$ if and only if $\mu$ is a $1$-logarithmic $1$-Carleson measure.

• (iii)

If $\mu$ is a $1$-logarithmic $1$-Carleson measure then $\mathcal{H}_{\mu}(f)=I_{\mu}(f)$, for all $f\in\mathcal{B}$.

• (iv)

If $\mu$ is a $1$-logarithmic $1$-Carleson measure then $\mathcal{H}_{\mu}$ is a bounded operator from $Q_{s}$ into $BMOA$ for any $s>0$.

It is natural to look for a characterization of those $\mu$ for which $I_{\mu}$ and/or $\mathcal{H}_{\mu}$ is a bounded operator from $\mathcal{B}$ into itself or, more generally, from $Q_{s}$ into itself for any $s>0$. We have the following result.

Theorem 1.3**.**

Let $\mu$ be a positive Borel measure on $[0,1)$. Then the following conditions are equivalent.

• (i)

The operator $I_{\mu}$ is bounded from $Q_{s}$ into itself for some $s>0$.

• (ii)

The operator $I_{\mu}$ is bounded from $Q_{s}$ into itself for all $s>0$.

• (iii)

The operator $\mathcal{H}_{\mu}$ is bounded from $Q_{s}$ into itself for some $s>0$.

• (iv)

The operator $\mathcal{H}_{\mu}$ is bounded from $Q_{s}$ into itself for all $s>0$.

• (v)

The measure $\mu$ is a $1$-logarithmic $1$-Carleson measure.

In [23] we also studied the operators $\mathcal{H}_{\mu}$ acting on Besov spaces. Theorem 3. 8 of [23] asserts that $\mu$ being a $\gamma$-logarithmic $1$-Carleson measure for some $\gamma>1$ is a sufficient condition for the boundedness of $\mathcal{H}_{\mu}$ from $B^{p}$ into itself, for any $p>1$. On the other hand, Theorem 3. 7 of [23] asserts that if $1<p<\infty$ and the operator $\mathcal{H}_{\mu}$ is bounded from $B^{p}$ to itself then $\mu$ is a $\gamma$-logarithmic $1$-Carleson measure for any $\gamma<\,1-\frac{1}{p}$. We can improve this result as follows.

Theorem 1.4**.**

Suppose that $1<p<\infty$ and let $\mu$ be a positive Borel measure on $[0,1)$ such that the operator $\mathcal{H}_{\mu}$ is bounded from $B^{p}$ into itself. Then $\mu$ is a $\left(1-\frac{1}{p}\right)$-logarithmic $1$-Carleson measure.

The paper is organized as follows. The results concerning Hardy spaces will be proved in Section 2; Section 3 will be devoted to prove Theorem 1.3 and Theorem 1.4. We close this section noticing that, as usual, we shall be using the convention that $C=C(p,\alpha,q,\beta,\dots)$ will denote a positive constant which depends only upon the displayed parameters $p,\alpha,q,\beta\dots$ (which sometimes will be omitted) but not necessarily the same at different occurrences. Moreover, for two real-valued functions $E_{1},E_{2}$ we write $E_{1}\lesssim E_{2}$, or $E_{1}\gtrsim E_{2}$, if there exists a positive constant $C$ independent of the arguments such that $E_{1}\leq CE_{2}$, respectively $E_{1}\geq CE_{2}$. If we have $E_{1}\lesssim E_{2}$ and $E_{1}\gtrsim E_{2}$ simultaneously then we say that $E_{1}$ and $E_{2}$ are equivalent and we write $E_{1}\asymp E_{2}$.

2. The operator $\mathcal{H}_{\mu}$ acting on Hardy

spaces

This section is devoted to prove Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1 (i). Suppose that $\mathcal{H}_{\mu}$ is a bounded operator from $H^{1}$ into itself. For $0<b<1$, set

[TABLE]

We have that $f_{b}\in H^{1}$ and $\|f_{b}\|_{H^{1}}=1$. Since $\mathcal{H}_{\mu}$ is bounded on $H^{1}$, this implies that

[TABLE]

We also have,

[TABLE]

Since the $a_{k,b}$’s are positive, it is clear that the sequence $\{\sum_{k=0}^{\infty}\mu_{n+k}a_{k,b}\}_{n=0}^{\infty}$ of the Taylor coefficients of $\mathcal{H}_{\mu}(f_{b})$ is a decreasing sequence of non-negative real numbers. Using this, Theorem 1. 1 of [31], (2.1), and the definition of the $a_{k,b}$’s, we obtain

[TABLE]

Then it follows that

[TABLE]

Hence, $\mu$ is a $1$-logarithmic $1$-Carleson measure.

The converse follows from Theorem C (i). $\Box$

Proof of Theorem 1.1 (ii). Suppose that $1<p<\infty$ and that $\mu$ is a positive Borel measure on $[0,1)$ such that the operator $\mathcal{H}_{\mu}$ is a bounded operator from $H^{p}$ into itself.

For $0<b<1$, set

[TABLE]

We have that $f_{b}\in H^{p}$ and $\|f_{b}\|_{H^{p}}=1$. Since $\mathcal{H}_{\mu}$ is bounded on $H^{p}$, this implies that

[TABLE]

We also have,

[TABLE]

Since the $a_{k,b}$’s are positive, it is clear that the sequence $\{\sum_{k=0}^{\infty}\mu_{n+k}a_{k,b}\}_{n=0}^{\infty}$ of the Taylor coefficients of $\mathcal{H}_{\mu}(f_{b})$ is a decreasing sequence of non-negative real numbers. Using this, Theorem A of [31], (2.1), and the definition of the $a_{k,b}$’s, we obtain

[TABLE]

Then it follows that

[TABLE]

and, hence, $\mu$ is a Carleson measure.

The other implication follows from Theorem C (ii). $\Box$

Proof of Theorem 1.2. The equivalence (i) $\Leftrightarrow$ (ii) is clear because

[TABLE]

The implication (i) $\Rightarrow$ (iii) is obvious.

(iii) $\Rightarrow$ (i): Suppose (iii). Let $f$ be the constant function $f(z)=1$, for all $z$. Then (iii) implies that there exists a positive constant $C$ such that

[TABLE]

Taking $z=0$ in this inequality, (i) follows.

(iii) $\Rightarrow$ (iv): Suppose (iii). We have seen that then (i) holds, and it is easy to see that (i) implies that $\mu$ is a Carleson measure. Using part (ii) of Theorem A, it follows that $\mathcal{H}_{\mu}$ is well defined in $H^{\infty}$ and that $\mathcal{H}_{\mu}(f)=I_{\mu}(f)$ for all $f$ in $H^{\infty}$. Then (iii) gives that $\mathcal{H}_{\mu}$ is bounded from $H^{\infty}$ into itself.

(iv) $\Rightarrow$ (iii): Suppose that (iv) is true and, as above, let $f$ be the constant function $f(z)=1$, for all $z$. Then $\mathcal{H}_{\mu}(f)\in H^{\infty}$. But $\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}\mu_{n}z^{n}$ and then it is clear that

[TABLE]

Thus we have seen that (iv) $\Rightarrow$ (ii). Since (ii) $\Leftrightarrow$ (iii), this finishes the proof. $\Box$

3. The operator $\mathcal{H}_{\mu}$ acting on Möbius invariant spaces

A basic ingredient in the proof of Theorem 1.3 will be to have a characterization of the functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ whose sequence of Taylor coefficients $\{a_{n}\}_{n=0}^{\infty}$ is a decreasing sequence of nonnegative numbers which lie in the $Q_{s}$-spaces. This is quite simple for $s>1$ (recall that $Q_{s}=\mathcal{B}$ if $s>1$):

Hwang and Lappan proved in [26, Theorem 1] that if $\{a_{n}\}$ is a decreasing sequence of nonnegative numbers then $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ is a Bloch function if and only if $a_{n}\,=\,\operatorname{O}\left(\frac{1}{n}\right)$.

Fefferman gave a characterization of the analytic functions having nonnegative Taylor coefficients which belong to $BMOA$, proofs of this criterium can be found in [10, 22, 24, 36]. Characterizations of the analytic functions having nonnegative Taylor coefficients which belong to $Q_{s}$ ($0<s<1$) were obtained in [6, Theorem 1. 2] and [4, Theorem 2. 3]. Using the mentioned result in [6, Theorem 1. 2], Xiao proved in [39, Corollary 3. 3. 1, p. 29] the following result.

Theorem E**.**

Let $s\in(0,\infty)$ and let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ with $\{a_{n}\}$ being a decreasing sequence of nonnegative numbers. Then $f\in Q_{s}$ if and only if $a_{n}\,=\,\operatorname{O}\left(\frac{1}{n}\right)$.

Being based on Theorem 1. 2 of [6], Xiao’s proof of this result is complicated. We shall give next an alternative simpler proof. It will simply use the validity of the result for the Bloch space and the simple fact that the mean Lipschitz space $\Lambda^{2}_{1/2}$ is contained in all the $Q_{s}$ spaces ($0<s<\infty$) (see [4, Remark 4, p. 427] or [39, Theorem 4. 2. 1.]).

We recall [19, Chapter 5] that a function $f\in{\mathcal{H}ol}(\mathbb{D})$ belongs to the mean Lipschitz space $\Lambda^{2}_{1/2}$ if and only if

[TABLE]

We have the following simple result for the space $\Lambda^{2}_{1/2}$.

Lemma 3.1**.**

If $\{a_{n}\}_{n=0}^{\infty}$ is a decreasing sequence of nonnegative numbers and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ ($z\in\mathbb{D}$), then $f\in\Lambda^{2}_{1/2}$ if and only if  $a_{n}\,=\,\operatorname{O}\left(\frac{1}{n}\right).$

Proof. If  $a_{n}\,=\,\operatorname{O}\left(\frac{1}{n}\right)$, then

[TABLE]

and, hence, $f\in\Lambda^{2}_{1/2}$.

Suppose now that $\{a_{n}\}_{n=0}^{\infty}$ is a decreasing sequence of nonnegative numbers and $f\in\Lambda^{2}_{1/2}$. Then, for all $n$

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Taking $r=1-\frac{1}{n}$ in (3.1), we obtain

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Since $\{a_{n}\}$ is decreasing, using (3.2) we have

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and then it follows that $a_{n}\,=\,\operatorname{O}\left(\frac{1}{n}\right).$ $\Box$

Now Theorem E follows using the result of Hwang and Lappan for the Bloch space, Lemma 3.1, and the fact that

[TABLE]

Using (3.3), it is clear that Theorem 1.3 follows from the following result.

Theorem 3.1**.**

Let $\mu$ be a positive Borel measure on $[0,1)$ and let $X$ be a Banach space of analytic functions in $\mathbb{D}$ with $\Lambda^{2}_{1/2}\subset X\subset\mathcal{B}$. Then the following conditions are equivalent.

• (i)

The operator $I_{\mu}$ is well defined in $X$ and, furthermore, it is a bounded operator from $X$ into $\Lambda^{2}_{1/2}$.

• (ii)

The operator $\mathcal{H}_{\mu}$ is well defined in $X$ and, furthermore, it is a bounded operator from $X$ into $\Lambda^{2}_{1/2}$.

• (iii)

The measure $\mu$ is a $1$-logarithmic $1$-Carleson measure.

• (iv)

$\int_{[0,1)}t^{n}\log\frac{1}{1-t}d\mu(t)\,=\,\operatorname{O}\left(\frac{1}{n}\right)$.

Proof. According to Proposition 2. 5 of [23], $\mu$ is a $1$-logarithmic $1$-Carleson measure if and only if the measure $\nu$ defined by $d\nu(t)\,=\,\log\frac{1}{1-t}d\mu(t)$ is a Carleson measure and, using Proposition 1 of [12], this is equivalent to (iv). Hence, we have shown that (iii)  $\Leftrightarrow$  (iv).

Set $F(z)\,=\,\log\frac{1}{1-z}$ ($z\in\mathbb{D}$). We have that $F\in X$.

(i) $\Rightarrow$ (iv): Suppose (i). Then

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is well defined for all $z\in\mathbb{D}$. Taking $z=0$, we see that $\int_{[0,1)}\log\frac{1}{1-t}d\mu(t)<\infty$. Since $F\in X$ we have also that $I_{\mu}(F)\in\Lambda^{2}_{1/2}$, but

[TABLE]

Since the sequence $\left\{\int_{[0,1)}t^{n}\log\frac{1}{1-t}d\mu(t)\right\}_{n=0}^{\infty}$ is a decreasing sequence of nonnegative numbers, using Lemma 3.1 we see that (iv) holds.

(iv) $\Rightarrow$ (i): Suppose (iv) and take $f\in X$. Since $X\subset\mathcal{B}$, it is well known that $|f(z)|\lesssim\log\frac{2}{1-|z|}$, see [2, p. 13]. This and (iv) give

[TABLE]

Then it follows easily that $I_{\mu}(f)$ is well defined and that

[TABLE]

Now (3.4) implies that $\int_{[0,1)}t^{n}f(t)d\mu(t)\,=\,\operatorname{O}\left(\frac{1}{n}\right)$ and then it follows that $I_{\mu}(f)\in\Lambda^{2}_{1/2}$.

The implication (iv) $\Rightarrow$ (ii) follows using Theorem 2. 3 of [23] and the already proved equivalences (i) $\Leftrightarrow$  (iii)  $\Leftrightarrow$  (iv).

It remains to prove that (ii) $\Rightarrow$ (iv). Suppose (ii) then $\mathcal{H}_{\mu}(F)\in\Lambda^{2}_{1/2}$. Now

[TABLE]

Notice that the sequence $\{\sum_{k=1}^{\infty}\frac{\mu_{n+k}}{k}\}_{n=0}^{\infty}$ is a decreasing sequence of nonnegative numbers. Then, using Lemma 3.1 and the fact that $\mathcal{H}_{\mu}(F)\in\Lambda^{2}_{1/2}$, we deduce that

[TABLE]

Now

[TABLE]

Then (iv) follows using (3.5). $\Box$

Remark 3.1**.**

It is clear that Theorem 3.1 actually implies the following result.

Theorem 3.2**.**

Let $\mu$ be a positive Borel measure on $[0,1)$ and let $0<s_{1},s_{2}<\infty$. Then following conditions are equivalent.

• (i)

The operator $I_{\mu}$ is well defined in $Q_{s_{1}}$ and, furthermore, it is a bounded operator from $Q_{s_{1}}$ into $Q_{s_{2}}$.

• (ii)

The operator $\mathcal{H}_{\mu}$ is well defined in $Q_{s_{1}}$ and, furthermore, it is a bounded operator from $Q_{s_{1}}$ into $Q_{s_{2}}$.

• (iii)

The measure $\mu$ is a $1$-logarithmic $1$-Carleson measure.

Proof of Theorem 1.4. Suppose that $1<p<\infty$ and let $\mu$ be a positive Borel measure on $[0,1)$ such that the operator $\mathcal{H}_{\mu}$ is bounded from $B^{p}$ into itself. For $\frac{1}{2}<b<1$, set

[TABLE]

We have,

[TABLE]

and then, using Lemma 3. 10 of [42] with $t=p-2$ and $c=0$, we have

[TABLE]

In other words, we have that

[TABLE]

Since $\mathcal{H}_{\mu}$ is a bounded operator from $B^{p}$ into itself, this implies that

[TABLE]

We have

[TABLE]

Since the $a_{k,b}$’s are positive it follows that the sequence $\{\sum_{k=0}^{\infty}\mu_{n+k}a_{k,b}\}_{n=0}^{\infty}$ of the Taylor coefficients of $\mathcal{H}_{\mu}(g_{b})$ is a decreasing sequence of non-negative real numbers. Using this, [23, Theorem 3. 10], and (3.6) we see that

[TABLE]

Then it follows that $\mu\left([b,1)\right)\,\lesssim\,\frac{1-b}{\left(\log\frac{1}{1-b}\right)^{1-\frac{1}{p}}}.$ This finishes the proof. $\Box$