Abstract
Let \({\mathbb {D}}\) be the unit disc in the complex plane. Given a positive finite Borel measure \(\mu \) on the radius [0, 1), we let \(\mu _n\) denote the n-th moment of \(\mu \) and we deal with the action on spaces of analytic functions in \({\mathbb {D}}\) of the operator of Hibert-type \({\mathcal {H}}_\mu \) and the operator of Cesàro-type \({\mathcal {C}}_\mu \) which are defined as follows: If f is holomorphic in \({\mathbb {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathbb {D}})\), then \({\mathcal {H}}_\mu (f)\) is formally defined by \({\mathcal {H}}_\mu (f)(z) = \sum _{n=0}^\infty \left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) z^n\) (\(z\in {\mathbb {D}}\)) and \({\mathcal {C}}_\mu (f)\) is defined by \(\mathcal C_\mu (f)(z) = \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n\) (\(z\in {\mathbb {D}}\)). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in \({\mathbb {D}}\). In this paper we study the action of the operators \({\mathcal {H}}_\mu \) and \({\mathcal {C}}_\mu \) on the Dirichlet space \({\mathcal {D}}\) and, more generally, on the analytic Besov spaces \(B^p\) (\(1\le p<\infty \)).
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1 Introduction
The open unit disc in the complex plane \({\mathbb {C}}\) will be denoted by \({\mathbb {D}}\) and \(\textrm{Hol}({\mathbb {D}})\) will stand for the space of all analytic functions in \({\mathbb {D}}\). Also, dA will denote the area measure on \({\mathbb {D}}\), normalized so that the area of \(\mathbb D\) is 1. Thus \(dA(z) = \frac{1}{\pi } dx dy = \frac{1}{\pi } r dr d\theta \).
For \(0\,\le \,r\,<\,1\), \(0<p\le \infty \), and f analytic in \({\mathbb {D}} \), the integral means \(M_p(r, f)\) of f are defined by
For \(0<p\le \infty \) the Hardy space \(H^p\) consists of those functions f, analytic in \({\mathbb {D}} \), for which
We refer to [20] for the theory of Hardy spaces.
For \(0<p<\infty \) and \(\alpha >-1\) the weighted Bergman space \(A^p_\alpha \) consists of those \(f\in \textrm{Hol}({\mathbb {D}})\) such that
The unweighted Bergman space \(A^p_0\) is simply denoted by \(A^p\). We refer to [21, 31, 48] for the notation and results about Bergman spaces.
The space of Dirichlet type \({\mathcal {D}}^p_\alpha \) (\(0<p<\infty \), \(\alpha >-1\)) is the space of those \(f\in \textrm{Hol}({\mathbb {D}})\) such that \(f^\prime \in A^p_{\alpha }\). Thus, a function \(f\in \textrm{Hol}({\mathbb {D}})\) belongs to \({\mathcal {D}}^p_\alpha \) if and only if
In this paper we shall be mainly concerned with the Dirichlet space \({\mathcal {D}}={\mathcal {D}}^2_0\) which consists of those \(f\in \textrm{Hol}({\mathbb {D}})\) whose image Riemann surface has a finite area. We recall that if \(f\in {\mathcal {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathcal {D}}\)), then
Throughout the paper \(\mu \) will be a positive finite Borel measure on the radius [0, 1) and, for \(n=0, 1, 2, \dots \), we shall let \(\mu _n\) denote the moment of order n of \(\mu \), that is, \(\mu _n=\int _{[0, 1)}t^n\,d\mu (t)\). The matrices \({\mathcal {H}}_\mu \) and \({\mathcal {C}}_\mu \) are defined as follows
As we shall see in Sects. 2 and 3, these matrices induce operators acting on spaces of analytic functions which are natural generalizations of the classical Hilbert and Cesàro operators. Recently a good amount of work has been devoted to study the action of these operators of Hilbert type and of Cesàro type on distinct subspaces of \(\textrm{Hol}({\mathbb {D}})\). Carleson-type measures play a basic role in this work.
Let us recall that if \(\mu \) is a positive finite Borel measure on [0, 1) then:
-
If \(s>0\), then \(\mu \) is said to be an s-Carleson measure if there exists a positive constant C such that
$$\begin{aligned} \mu \left( [t, 1)\right) \,\le C(1-t)^{s},\quad 0\le t<1. \end{aligned}$$ -
If \(0\le \alpha <\infty \), and \(0<s<\infty \) we say that \(\mu \) is an \(\alpha \)-logarithmic s-Carleson measure if there exists a positive constant C such that
$$\begin{aligned} \mu \left( [t, 1)\right) \le C(1-t)^s\left( \log \frac{2}{1-t}\right) ^{-\alpha }, \quad 0\le t<1. \end{aligned}$$
Let us close this section by saying 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. Furthermore, for two real-valued functions \(K_1, K_2\) we write \(K_1\lesssim K_2\), or \(K_1\gtrsim K_2\), if there exists a positive constant C independent of the arguments such that \(K_1\le C K_2\), respectively \(K_1\ge C K_2\). If we have \(K_1\lesssim K_2\) and \(K_1\gtrsim K_2\) simultaneously, then we say that \(K_1\) and \(K_2\) are equivalent and we write \(K_1\asymp K_2\).
2 Hilbert-Type Operators
The matrix \({\mathcal {H}}_\mu \) induces formally an operator, which will be also called \({\mathcal {H}}_\mu \), on spaces of analytic functions by its action on the Taylor coefficients:
To be precise, if \(f(z)=\sum _{k=0}^\infty a_kz^k\in \textrm{Hol}({\mathbb {D}})\) we define
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 \({\mathcal {H}}= \left( {(n+k+1)^{-1}}\right) _{n,k\ge 0}\), which induces the classical Hilbert operator \({\mathcal {H}}\) which has extensively studied recently (see [1, 16, 17, 19, 32,33,34]).
The finite positive Borel measures \(\mu \) for which \({\mathcal {H}}_\mu \) is a bounded operator on distinct spaces of analytic functions in \({\mathbb {D}}\) have been characterized in a number of papers such as [9, 14, 25, 27,28,29, 35, 37, 38, 45]. Obtaining an integral representation of \({\mathcal {H}}_\mu \) plays a basic role in these works. If \(\mu \) is as above, we shall write throughout the paper
whenever the right hand side makes sense and defines an analytic function in \({\mathbb {D}}\). It turns out that the operators \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) are very closely related.
Let us mention the following results.
Theorem A
Let \(\mu \) be a positive Borel measure on [0, 1). Then
-
(i)
The operator \({\mathcal {H}}_\mu \) is bounded from \(H^1\) into itself if and only if \(\mu \) is a 1-logarithmic 1-Carleson measure. In such a case \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) coincide on \(H^1\).
-
(ii)
If \(1<p<\infty \), then \({\mathcal {H}}_\mu \) is a bounded operator from \(H^p\) into itself if and only if \(\mu \) is a 1-Carleson measure. In such a case \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) coincide on \(H^p\).
-
(iii)
If \(p>1\) and \(-1<\alpha <\,p-2\) then the operator \(\mathcal H_\mu \) is well defined on \(A^p_\alpha \) and it is bounded from \(A^p_\alpha \) into itself if and only if \(\mu \) is a 1-Carleson measure. In such a case \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) coincide on \(A^p_\alpha \).
-
(iv)
If \(p>1\) and \(p-2<\alpha \le p-1\), then \(\mathcal H_\mu \) is well defined on \({\mathcal {D}}^p_\alpha \) and it is bounded from \({\mathcal {D}}^p_\alpha \) into itself if and only if \(\mu \) is a 1-Carleson measure. In such a case \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) coincide on \({\mathcal {D}}^p_\alpha \).
-
(v)
If \(0<\alpha <2\), \({\mathcal {H}}_\mu \) is a bounded operator from \({\mathcal {D}}^2_\alpha \) into itself if and only if \(\mu \) is a 1-Carleson measure. In such a case \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) coincide on \(\mathcal D^2_\alpha \).
The questions of characterizing those \(\mu \) for which \(\mathcal H_\mu \) is bounded on either the Dirichlet space \({\mathcal {D}}\) or on the Bergman space \(A^2\) are more delicate and remain open. Regarding the Dirichlet space, the following results are proved in [28].
Theorem B
-
(i)
Let \(\mu \) be a positive and finite Borel measure on [0, 1). If \(\gamma >1\) and \(\mu \) is a \(\gamma \)-logarithmic 1-Carleson measure, then \({\mathcal {H}}_\mu \) is bounded from \({\mathcal {D}}\) into itself.
-
(ii)
If \(0<\beta \le \frac{1}{2}\), then there exists a positive and finite Borel measure \(\mu \) on [0, 1) which is a \(\beta \)-logarithmic 1-Carleson measure but such that \({\mathcal {H}}_\mu (\mathcal D)\not \subset {\mathcal {D}}\).
We improve this result showing that being a 1-logarithmic 1-Carleson measure is enough to insure that \({\mathcal {H}}_\mu \) is bounded from \({\mathcal {D}}\) into itself and closing the gap between (i) and (ii). Indeed, we shall prove the following result.
Theorem 1
-
(i)
Let \(\mu \) be a positive and finite Borel measure on [0, 1). If \(\mu \) is a 1-logarithmic 1-Carleson measure, then \({\mathcal {H}}_\mu \) is bounded from \({\mathcal {D}}\) into itself.
-
(ii)
If \(0<\beta <1\), then there exists a positive and finite Borel measure \(\mu \) on [0, 1) which is a \(\beta \)-logarithmic 1-Carleson measure but such that \({\mathcal {H}}_\mu (\mathcal D)\not \subset {\mathcal {D}}\).
As a corollary of part (i) we obtain the following.
Corollary 2
-
(a)
Let \(\mu \) be a positive and finite Borel measure on [0, 1) and suppose that \(\mu \) is a 1-logarithmic 1-Carleson measure. Then there exists a positive constant C such that
$$\begin{aligned} \int _{[0, 1)}\vert tf(t)f^\prime (t)\vert \,d\mu (t)\,\le C\Vert f\Vert _{{\mathcal {D}}}^2,\quad f\in \mathcal D.\end{aligned}$$(2.3) -
(b)
There exists a positive constant C such that
$$\begin{aligned} \int _0^1\vert tf(t)f^\prime (t)\vert \,\log \frac{2}{1-t}\,dt\,\le C\Vert f\Vert _{{\mathcal {D}}}^2,\quad f\in {\mathcal {D}}. \end{aligned}$$(2.4)
Regarding the Bergman space \(A^2\), Theorem 1.5 of [25] asserts the following.
Theorem C
Let \(\mu \) be a positive and finite Borel measure on [0, 1) and let \(h_\mu \) be defined by \(h_\mu (z)=\sum _{n=0}^\infty \mu _nz^n\) (\(z\in {\mathbb {D}}.\)) If \(\mu \) satisfies the condition
then \({\mathcal {H}}_\mu \) is bounded from \(A^2\) into itself if and only if the measure \(\vert h_\mu ^\prime (z)\vert ^2dA(z)\) is a Dirichlet-Carleson measure.
We recall that a finite positive Borel measure \(\nu \) on \({\mathbb {D}}\) is said to be a Dirichlet-Carleson messure if \({\mathcal {D}}\) is continuously embedded in \(L^2(d\nu )\). Stegenga [43] gave a characterization of these measures involving the logarithmic capacity of a finite union of intervals of \(\partial {\mathbb {D}}\). Shields [39] obtained a simpler characterization when dealing with measures supported on [0, 1). This result of Shields will be used below.
Using Theorem 1 we shall prove the following result.
Theorem 3
-
(i)
Let \(\mu \) be a positive and finite Borel measure on [0, 1). If \(\mu \) is a 1-logarithmic 1-Carleson measure, then \({\mathcal {H}}_\mu \) is bounded from \(A^2\) into itself.
-
(ii)
If \(0<\beta <1\), then there exists a positive and finite Borel measure \(\mu \) on [0, 1) which is a \(\beta \)-logarithmic 1-Carleson measure but such that \({\mathcal {H}}_\mu (A^2)\not \subset {\mathcal {A}}^2\).
In order to prove our results we start using the above mentioned result of Shields [39] to find a weak condition which insures that \({\mathcal {H}}_\mu \) and \({\mathcal {I}}_\mu \) are well defined in \({\mathcal {D}}\) and that \({\mathcal {H}}_\mu (f)=\mathcal I_\mu (f)\) for all \(f\in {\mathcal {D}}\).
Proposition 4
Let \(\mu \) be a positive and finite Borel measure on [0, 1). If there exists a positive constant C such that
then \(\mathcal H_\mu \) and \({\mathcal {I}}_\mu \) are well defined in \({\mathcal {D}}\) and, furthermore, \({\mathcal {H}}_\mu (f)={\mathcal {I}}_\mu (f)\) for all \(f\in {\mathcal {D}}\).
Proof
Suppose that \(\mu \) satisfies (2.6). Shields proved in [39, Theorem 2] that this is equivalent to saying that there exists a positive constant A such that
We can express (2.7) simply by saying that \(\mu \) is a radial Carleson-Dirichlet measure. Also, it is easy to see that (2.6) implies that there exists \(B>0\) such that
Take \(f\in {\mathcal {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)).
Let us prove that \({\mathcal {I}}_\mu (f)\) is well defined.
Using (2.7) and (2.8), we see that
for all n. Then we have
This implies that, for all \(z\in {\mathbb {D}}\), the integral
converges and that
So \(\mathcal I_\mu (f) \) is a well defined analytic function in \({\mathbb {D}}\) and
Let us see now that \({\mathcal {H}}_\mu (f)\) is also well defined and that \({\mathcal {H}}_\mu (f)={\mathcal {I}}_\mu (f)\). Using (2.8), for all n, we have
Clearly, this implies that \({\mathcal {H}}_\mu \) is a well defined analytic function in \({\mathbb {D}}\). Also,
for all k. Then (2.9) yields that \({\mathcal {H}}_\mu (f)={\mathcal {I}}_\mu (f)\). \(\square \)
Let us turn now to prove Theorem 1
Proof of Theorem 1 (i)
Suppose that \(\mu \) is a 1-logarithmic 1-Carleson measure. Take \(f\in {\mathcal {D}}\), \(f(z)=\sum _{k=0}^\infty a_kz^k\) (\(z\in {\mathbb {D}}\)). Proposition 4 implies that \({\mathcal {H}}_\mu (f)\) and \({\mathcal {I}}_\mu (f)\) are well defined and that \({\mathcal {H}}_\mu (f)={\mathcal {I}}_\mu (f)\). The above mentioned result of Shields yields that
Since \(\mu \) is a 1-logarithmic 1-Carleson measure,
(see e. g. [28, pp. 380-381]). Using (2.10) and (2.11), we obtain
where
Now, using a result of Holland and Walsh [30, Theorem 7] and simple estimates we deduce that
Also, since, for every n,
it follows that
Putting everything together, we obtain \(\Vert {\mathcal {H}}_\mu (f)\Vert _{{\mathcal {D}}}^2\lesssim \Vert f\Vert _{{\mathcal {D}}}^2\). \(\square \)
Proof of Theorem 1 (ii)
Suppose that \(0<\beta <1\). Take \(\alpha \in {\mathbb {R}}\) with
Let \(\mu \) be the Borel measure on [0, 1) defined by \(d\mu (t)=\left( \log \frac{2}{1-t}\right) ^{-\beta }\,dt\). Then (see [28, p. 392]) \(\mu \) is a \(\beta \)-logarithmic 1-Carleson measure and
Set \(a_n=\frac{1}{(n+1)\left[ \log (n+1)\right] ^\alpha }\) (\(n=1, 2, \dots \)) and \(g(z)=\sum _{n=1}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)).
The condition \(\alpha >\frac{1}{2}\) implies that \(g\in \mathcal D\). We are going to see that \({\mathcal {H}}_\mu (g)\notin \mathcal D\), this will finish the proof.
We have
Since \(2\alpha +2\beta -2<1\), \(\sum _{n=2}^\infty \frac{1}{n\left[ \log n\right] ^{2\beta +2\alpha -2}}=\infty \) and, hence, \(\mathcal H_\mu (g)\notin {\mathcal {D}}\) as desired. \(\square \)
Proof of Corollary 2
The Dirichlet space is a Hilbert space with the inner product
Hence, \({\mathcal {D}}\) is identifiable with its dual with this pairing.
Assume that \(\mu \) is a finite Borel measure on [0, 1) which is a 1-logarithmic 1-Carleson measure. If \(f\in {\mathcal {D}}\), using Theorem 1, we see that \({\mathcal {H}}_\mu (f)\in {\mathcal {D}}\) and \(\Vert {\mathcal {H}}_\mu (f)\Vert _{{\mathcal {D}}}\lesssim \Vert f\Vert _{{\mathcal {D}}}\). Then \({\mathcal {H}}_\mu (f)\) induces a bounded linear functional on \({\mathcal {D}}\) with norm controlled by \(\Vert f\Vert _{\mathcal D}\). Thus
Now, using the definitions, Fubini’s theorem, and the reproducing formula for the Bergman space \(A^2\), we have
Using (2.12), we obtain
Take \(f, g\in {\mathcal {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\), \(g(z)=\sum _{n=0}^\infty b_nz^n\) (\(z\in {\mathbb {D}}\)). Set
Then \(f_1, g_1\in {\mathcal {D}}\), \(\Vert f_1\Vert _{{\mathcal {D}}}=\Vert f\Vert _{{\mathcal {D}}}\), and \(\Vert g_1\Vert _{{\mathcal {D}}}=\Vert g\Vert _{{\mathcal {D}}}\). Using (2.13) with \(f_1\) and \(g_1\) in the places of f and g, we obtain
Taking \(f=g\), (2.3) follows.
Part (b) follows taking \(d\mu (t)=\log \frac{2}{1-t}\,dt\) in part (a). \(\square \)
Proof of Theorem 3
Our proof of Theorem 3 is based on the fact that the pairing
is a “duality paring” between the Dirichlet space \({\mathcal {D}}\) and the Bergman space \(A^2\). Notice that if \(f(z)=\sum _{n=0}^\infty a_nz^n\) and \(g(z)=\sum _{n=0}^\infty b_nz^n\) (\(z\in {\mathbb {D}}\)), then
It is a simple exercise to show that \(<{\mathcal {H}}_\mu (P), Q>=<P, {\mathcal {H}}_\mu (Q)>\) if P and Q are polynomials. Then it follows that if \({\mathcal {H}}_\mu \) is a bounded operator from \({\mathcal {D}}\) into itself then its adjoint (via this pairing) is \({\mathcal {H}}_\mu \), and then we see that \({\mathcal {H}}_\mu \) is a bounded operator from \(A^2\) into itself. Using this and Theorem 1 (i) we obtain part (a) of Theorem 3.
Similarly, if \({\mathcal {H}}_\mu \) is a bounded operator from \(A^2\) into itself, then \({\mathcal {H}}_\mu \) is also a bounded operator from \({\mathcal {D}}\) into itself and then part (b) of Theorem 3 follows using Theorem 1 (ii). \(\square \)
3 Cesàro-Type Operators
For \(\mu \) a finite positive Borel measure on [0, 1) as above, the matrix \({\mathcal {C}}_\mu \) induces a linear operator, also called \({\mathcal {C}}_\mu \), from \(\textrm{Hol}({\mathbb {D}})\) into itself as follows: If \(f\in \textrm{Hol}({\mathbb {D}})\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)),
Let us remark that the operator \({\mathcal {C}}_\mu \) has the following integral representation: If \(f\in \textrm{Hol}({\mathbb {D}})\) then
When \(\mu \) is the Lebesgue measure on [0, 1), the operator \({\mathcal {C}}_\mu \) reduces to the classical Cesàro operator \({\mathcal {C}}\).
The Cesàro operator \({\mathcal {C}}\) acting on distinct subspaces of \(\textrm{Hol}({\mathbb {D}})\) has been extensively studied in a good number of articles such as [2, 10, 12, 15, 23, 36, 40,41,42, 44]. Let us recall that it is bounded on \(H^p\) (\(0<p<\infty \)) and on \(A^p_\alpha \) (\(0<p<\infty \), \(\alpha >\,-1\)).
The operators \({\mathcal {C}}_\mu \) were introduced in [23] where, among other results, it was proved that the following conditions are equivalent:
-
(i)
\(\mu \) is a Carleson measure, that is, \(\mu ([t, 1))\le C(1-t)\) (\(0<t<1\)).
-
(ii)
\(\mu _n={{\,\textrm{O}\,}}\left( \frac{1}{n}\right) \).
-
(iii)
\(1\le p<\infty \) and \({\mathcal {C}}_\mu \) is bounded from \(H^p\) into itself.
-
(iv)
\(1<p<\infty \), \(\alpha >-1\), and \({\mathcal {C}}_\mu \) is bounded from \(A^p_\alpha \) into itself.
Blasco [12] has generalized the definition of the operators \({\mathcal {C}}_\mu \) by dealing with complex Borel measures on [0, 1) and he has extended results of [23] to this more general setting.
A further generalization has been given in [24] by working with the operators \({\mathcal {C}}_\mu \) associated to arbitrary complex Borel measures on \({\mathbb {D}}\), not necessarily supported on a radius. The complex Borel measures on \({\mathbb {D}}\) for which the operator \({\mathcal {C}}_\mu \) is bounded or Hilbert-Schmidt on \(H^2\) or on \(A^2_\alpha \) (\(\alpha >-1\)) are characterized in the mentioned paper [24].
We devote this section to study the operators \({\mathcal {C}}_\mu \) on the Dirichlet space, a question which has not been considered in the just mentioned papers. Our main results are contained in the following two theorems.
Theorem 5
Let \(\mu \) be a finite positive Borel measure on [0, 1).
-
(i)
If \(\mu \) is a 1-logarithmic 1-Carleson measure, then \({\mathcal {C}}_\mu \) is a bounded operator from the Dirichlet space \({\mathcal {D}}\) into itself.
-
(ii)
If \({\mathcal {C}}_\mu \) is a bounded operator from \({\mathcal {D}}\) into itself then \(\mu \) is a 1/2-logarithmic 1-Carleson measure.
Theorem 6
Suppose that \(\frac{1}{2}<\beta <1\). Then there exists a finite positive Borel measure \(\mu \) on [0, 1) which is \(\beta \)-logarithmic 1-Carleson measure for which \({\mathcal {C}}_\mu (\mathcal D)\not \subset {\mathcal {D}}\).
Proof of Theorem 5 (i)
Since \(\mu \) is a 1-logarithmic 1-Carleson measure, we have that
Take \(f\in {\mathcal {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)). Using (3.2) and Theorem 7 of [30], we obtain
\(\square \)
Proof of Theorem 5 (ii)
Suppose that \(\mathcal C_\mu \) is a bounded operator from \({\mathcal {D}}\) into itself. For \(N\in {\mathbb {N}}\), set
Then,
Since \({\mathcal {C}}_\mu \) is bounded on \({\mathcal {D}}\), bearing in mind that the sequence of moments \(\{ \mu _n\} \) is decreasing, we have
Then it follows that \(\mu _N={{\,\textrm{O}\,}}\left( \frac{1}{N[\log (N+1)]^{1/2}}\right) \). This implies that \(\mu \) is a 1/2-logarithmic 1-Carleson measure. \(\square \)
Proof of Theorem 6
Assume that \(1/2<\beta <1\). Let \(\mu \) be the Borel measure on [0, 1) defined by \(d\mu (t)=\left( \log \frac{2}{1-t}\right) ^{-\beta }\,dt\). Then, as mentioned before, \(\mu \) is a \(\beta \)-logarithmic 1-Carleson measure and \(\mu _n\asymp \frac{1}{n[\log (n+1)]^\beta }\).
Set \(\alpha =\beta -\frac{1}{2}\). Then \(0<\alpha <\frac{1}{2}\). Define
We have that
Since \(\alpha <\frac{1}{2}\), we have that \(g\in {\mathcal {D}}\). Also
\(\square \)
Danikas and Siskakis [15] proved that \({\mathcal {C}}(H^\infty )\not \subset H^\infty \) and that \(\mathcal C(H^\infty )\subset BMOA\). This was improved by Essén and Xiao who proved in [22] that \({\mathcal {C}}(H^\infty )\subset Q^p\) for \(0<p<\infty \). This result has been sharpened in [10].
We recall that BMOA is the space of those functions \(f\in H^1\) whose boundary values have bounded mean oscillation. Alternatively, a function \(f\in \textrm{Hol}({\mathbb {D}})\) belongs to BMOA if and only if
where \(\textrm{Aut}({\mathbb {D}})\) denotes the set of all Möbius transformations from \({\mathbb {D}}\) onto itself. We refer to [26] for the theory of BMOA-functions.
For \(0<s<\infty \) the space \(Q_s\) consists of those \(f\in \textrm{Hol}({\mathbb {D}})\) such that
The spaces \(Q_s\) were introduced in [6] and [7]. We refer to [46] for the theory of \(Q_s\) spaces. Let us recall that
For \(s>1\) the space \(Q_s\) coincides with the Bloch space \({\mathcal {B}}\) of those functions \(f\in \textrm{Hol}({\mathbb {D}})\) for which
The paper [3] is an excellent reference for the theory of Bloch functions. Let us recall that \(BMOA\subsetneq {\mathcal {B}}\).
Blasco [12] has proved that
Here, for \(p\ge 1\), \(\Lambda ^p_{1/p}\) is the space of those functions \(f\in \textrm{Hol}({\mathbb {D}})\) having a non-tangential limit at almost every point of \(\partial {\mathbb {D}}\) and so that \(\omega _ p(\cdot , f)\), the integral modulus of continuity of order p of the boundary values \(f(e^{i\theta })\) of f, satisfies \(\omega _ p(\delta , f)={{\,\textrm{O}\,}}(\delta ^{1/p})\), as \(\delta \rightarrow 0\). Classical results of Hardy and Littlewood (see [13] and [20, Chapter 5]) show that \(\Lambda ^p_{1/p}\subset H^p\) and that
In particular, \(\Lambda ^1_1\) is the space of those \(f\in \textrm{Hol}({\mathbb {D}})\) such that \(f^\prime \in H^1\). The spaces \(\Lambda ^p_{1/p}\) increase with p and they are all contained in BMOA [13]. Since \(\Lambda ^2_{1/2}\subset Q_s\) for all \(s>0\) (see [5, p. 427]), (3.3) improves the mentioned result in [22].
Bao, Sun and Wulan [8, Theorem 3.1] have proved that for any given \(s>0\), \({\mathcal {C}}_\mu (H^\infty )\subset Q_s\) if and only if \(\mu \) is a Carleson measure.
It is natural to look for a result like (3.3) with \({\mathcal {D}}\) in the place of \(H^\infty \). It is easy to see that
Indeed, set \(a_n=\frac{1}{(n+1)\log (n+1)}\) (\(n\ge 1\)) and \(f(z)=\sum _{n=1}^\infty a_nz^n\) (\(z\in \mathbb D\)). Then \(f\in {\mathcal {D}}\) and, setting \(A_n=\sum _{k=1}^na_k\), we have, for \(0<r<1\),
Hence, \(C(f)\not \in {\mathcal {B}}\).
The next natural step is trying to characterize the measures \(\mu \) such that \({\mathcal {C}}_\mu ({\mathcal {D}})\subset \mathcal B\). We have the following result.
Theorem 7
Let X be a Banach space of analytic functions in \({\mathbb {D}}\) with \(\Lambda ^2_{1/2}\subset X\subset {\mathcal {B}}\) and let \(\mu \) be a positive finite Borel measure on [0, 1).
-
(i)
If \(\mu \) is a \(\frac{1}{2}\)-logarithmic 1-Carleson measure, then \({\mathcal {C}}_\mu \) is a bounded operator from \({\mathcal {D}}\) into X.
-
(ii)
If \(\mathcal C_\mu \) is a bounded operator from \({\mathcal {D}}\) into X and \(0<\beta <\frac{1}{2}\), then \(\mu \) is a \(\beta \)-logarithmic 1-Carleson measure.
Proof
Suppose that \(\mu \) is a \(\frac{1}{2}\)-logarithmic 1-Carleson measure. Then
Take \(f\in {\mathcal {D}}\), \(f(z)=\sum _{n=0}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)). We have
where \(A_n=\mu _n\left( \sum _{k=0}^na_k\right) \). We have,
This and (3.5) imply that \(\vert A_n\vert \lesssim \frac{\Vert f\Vert _{{\mathcal {D}}}}{n}\) a fact which easily yields that \({\mathcal {C}}_\mu (f)\in \Lambda ^2_{1/2}\). This finishes the proof of (i).
Let us turn to prove (ii). Assume that \(0<\beta <\frac{1}{2}\) and that \({\mathcal {C}}_\mu \) is a bounded operator from \({\mathcal {D}}\) into X.
Since \(X\subset {\mathcal {B}}\), \({\mathcal {C}}_\mu \) is a bounded operator from \({\mathcal {D}}\) into \({\mathcal {B}}\).
Set \(\alpha =1-\beta \), and \(f(z)=\sum _{n=0}^\infty \frac{z^n}{(n+1)[\log (n+2)]^\alpha }\) (\(z\in {\mathbb {D}}\)).
Notice that \(\frac{1}{2}<\alpha <1\). This implies that \(f\in \mathcal D\) and, hence, \({\mathcal {C}}_\mu (f)\in {\mathcal {B}}\). Then, bearing in mind that the sequence \(\{ \mu _n\} \) is decreasing, we see that, for \(0<r<1\) and \(N\in {\mathbb {N}}\),
Taking \(r=1-\frac{1}{N}\), we obtain
and, hence, \(\mu _N\lesssim \frac{1}{N[\log (N+2)]^\beta }\). This implies that \(\mu \) is a \(\beta \)-logarithmic 1-Carleson measure. \(\square \)
4 Extensions to Besov Spaces
The Dirichlet space is one among the analytic Besov spaces \(B^p\). For \(1<p<\infty \), the analytic Besov space \(B^p\) is the space \({\mathcal {D}}^p_{p-2}\). Thus \(B^2\,=\,{\mathcal {D}}\).
The minimal Besov space \(B^1\) requires a special definition. It is the space of all \(f\in \textrm{Hol}({\mathbb {D}})\) such that \(f^{\prime \prime }\in A^1\). It is a Banach space with the norm \(\Vert \cdot \Vert _{B^1}\) defined by \(\Vert f\Vert _{B^1}\,=\,\vert f(0)\vert \,+\,\vert f^\prime (0)\vert \,+\,\Vert f^{\prime \prime }\Vert _{A^1}\).
The Besov spaces \(B^p\) form a nested scale of conformally invariant spaces and they are all contained in BMOA:
Also \(B^p\,\subsetneq \,\Lambda ^p_{1/p}\) for all \(p\in [1,\infty )\). We mention [4, 11, 18, 30, 47, 48] for information on Besov spaces. Let us remark that, letting \(d\lambda \) be the Möbius invariant measure on \({\mathbb {D}}\) defined by \(d\lambda (z)=\frac{dA(z)}{(1-\vert z\vert ^2)^2} \), we have:
-
(a)
The Bergman projection P is a continuous linear operator from \(L^\infty ({\mathbb {D}})\) onto the Bloch space \({\mathcal {B}}\),
-
(b)
For \(1<p<\infty \), the Bergman projection P is a continuous linear operator from \(L^p(d\lambda )\) onto \(B^p\)
(see [48, Chapter 5]).
Our aim in this section is trying to extend to the spaces \(B^p\) some of the results obtained in the preceding ones for the Dirichlet space.
For the space \(B^1\) we have the following result.
Theorem 8
Let \(\mu \) be positive finite 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 \({\mathcal {H}}_\mu \) is a bounded operator from \(B^1\) into itself.
-
(iv)
The operator \({\mathcal {C}}_\mu \) is a bounded operator from \(B^1\) into itself.
Proof
The equivalence (i) \(\Leftrightarrow \) (ii) is clear.
Suppose that (iii) holds. Let f be the constant function \(f(z)=1\), for all \(z\in {\mathbb {D}}\). Then \({\mathcal {H}}_\mu (f)={\mathcal {I}}_\mu (f)\in B^1\subset H^\infty \) and then
Thus (i) holds.
Conversely, suppose that (i) holds. Take \(f\in B^1\). We have
Then using Fubini’s theorem, [48, Lemma 3.10], and the fact that \(B^1\subset H^\infty \), we obtain
Thus, (iii) follows.
Let us prove next the equivalence (i) \(\Leftrightarrow \) (iv).
Suppose (i). Take \(f\in B^1\). Bearing in mind (3.1) and using Fubini’s theorem, we see that
We now estimate each of the three terms in the last formula separately. For the first one we have
For the second one, we use the fact that \(B^1\subset \Lambda ^1_1\) to obtain
For the last integral, we use that \(B^1\subset H^\infty \) and Lemma 3.10 of [48] to see that
Putting everything together we obtain (iv).
Suppose now that (iv) holds. Let f be the constant function given by \(f(z)=1\), for all \(z\in {\mathbb {D}}\). Then \({\mathcal {C}}_\mu (f)\in B^1\subset H^\infty \). Using the integral representation of \({\mathcal {C}}_\mu \) we see that
Thus, \(\int _{[0, 1)}\frac{d\mu (t)}{1-t}<\infty \). This is (i). \(\square \)
Let us turn now to deal with the possible extensions in the range \(1<p<\infty \). The following result comes from [28, Theorem 2.4] and [23, Theorem 7].
Theorem D
Let \(\mu \) be a positive finite Borel measure on [0, 1). If \(\mu \) is a 1-logarithmic 1-Carleson measure then the operators \({\mathcal {H}}_\mu \) and \({\mathcal {C}}_\mu \) are bounded from the Bloch space \({\mathcal {B}}\) into itself.
Using this result and those obtained in Sects. 2 and 3 we will prove the following.
Theorem 9
Suppose that \(2<p<\infty \) and let \(\mu \) be a positive finite Borel measure on [0, 1). If \(\mu \) is a 1-logarithmic 1-Carleson measure then the operators \({\mathcal {H}}_\mu \) and \({\mathcal {C}}_\mu \) are bounded from the Besov space \(B^p\) into itself.
Proof
We shall use complex interpolation in the proof. Let us refer to [48, Chapter 2] for the terminology and basic results concerning complex interpolation.
If \(X_0\) and \(X_1\) are two compatible Banach spaces then, for \(0<\theta <1\), \([X_0, X_1]_{\theta }\) stands for the space obtained by the complex method of interpolation of Calderón. As a consequence of the above mentioned results characterizing the spaces \(B^p\) as the image of \(L^p(d\lambda )\) under the Bergman projection and the Bloch space as the image of \(L^\infty (d\lambda )\) under the Bergman projection, Zhu proves in [48, Theorem 5.25] that if \(1<p_0<\infty \), \(0<\theta <1\), and \(1/p=(1-\theta )/p_0\), then
In particular,
Theorem 9 follows using (4.2), Theorem 1 (i), Theorem 5 (i), and the interpolation theorem of operators [48, Theorem 2.4]. \(\square \)
Regarding the sharpness of Theorem 9, we have the following result.
Theorem 10
Suppose that \(0<\beta <1\).
-
(i)
If \(1<p<\infty \) then there exists a positive Borel measure \(\mu \) on [0, 1) which is a \(\beta \)-logarithmic 1-Carleson measure with the property that \({\mathcal {H}}_\mu (B^p)\not \subset B^p\).
-
(ii)
If \(1<p\le 2\) then there exists a positive Borel measure \(\mu \) on [0, 1) which is a \(\beta \)-logarithmic 1-Carleson measure with the property that \({\mathcal {C}}_\mu (B^p)\not \subset B^p\).
Proof
Assume that \(1<p<\infty \) and \(0<\beta <1\). Take \(\alpha \in {\mathbb {R}}\) with
Let \(\mu \) be the Borel measure on [0, 1) defined by \(d\mu (t)=\left( \log \frac{2}{1-t}\right) ^{-\beta } \,dt\). We know that \(\mu \) is a \(\beta \)-logarithmic 1-Carleson measure and that \(\mu _n\asymp \frac{1}{(n+1)[\log (n+2)]^\beta }.\)
For \(n\ge 1\), set \(a_n=\frac{1}{n[\log (n+1)]^\alpha }\) and \(g(z)=\sum _{n=1}^\infty a_nz^n\) (\(z\in {\mathbb {D}}\)).
Since the sequence \(\{ a_n\} \) is decreasing and \(\sum _{n=1}^\infty n^{p-1}\vert a_n\vert ^p<\infty \), using [28, Theorem 3.10] we see that \(g\in B^p\).
We have that \({\mathcal {H}}_\mu (g)(z)=\sum _{n=0}^\infty \left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) z^n\) (\(z\in {\mathbb {D}}\)). Since the \(a_k\)’s are positive and the sequence of moments \(\{ \mu _n\} \) is decreasing, it follows that the sequence \(\left\{ \sum _{k=0}^\infty \mu _{n+k}a_k\right\} \) is also decreasing. Then using again [28, Theorem 3.10] we see that
Now,
Since \(p(\beta +\alpha -1)<1\), it follows that \(\sum _{n=1}^\infty n^{p-1}\left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) ^p=\infty \) and then (4.3) gives that \(H_\mu (g)\not \in B^p\).
Assume now that \(1<p\le 2\). We have
Using the fact that \(1<p\le 2\) and [20, Theorem 6.s2] it readily follows that
But,
Using (4.4) we obtain that \({\mathcal {C}}_\mu (g)\not \in B^p\). \(\square \)
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Funding for open access publishing: Universidad Málaga/CBUA This research is supported in part by a grant from “El Ministerio de Economía y Competitividad”, Spain (PGC2018-096166-B-I00 and PID2019-106870GB-I00) and by grants from la Junta de Andalucía (FQM-210 and UMA18-FEDERJA-002).
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Galanopoulos, P., Girela, D., Mas, A. et al. Operators Induced by Radial Measures Acting on the Dirichlet Space. Results Math 78, 106 (2023). https://doi.org/10.1007/s00025-023-01887-6
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DOI: https://doi.org/10.1007/s00025-023-01887-6
Keywords
- The Dirichlet space
- Hardy spaces
- weighted Bergman spaces
- analytic Besov spaces
- Hilbert-type operators
- Cesàro-type operators