1 Introduction
Let
$\mathbb C$
be the complex plane and let
$H(\mathbb C)$
be the family of all holomorphic functions on
$\mathbb C$
. Given any
$\alpha>0$
, set
$$ \begin{align*} d\lambda_{\alpha}(z)=\frac{\alpha}{\pi}e^{-\alpha|z|^{2}}\, dA(z), \end{align*} $$
where
$dA$
is the Euclidean area measure on
$ \mathbb C$
. For
$1\le p<\infty $
, the Fock space
$F^{p}_{\alpha }$
consists of those functions
$f\in H(\mathbb C)$
for which
$$ \begin{align*} \|f\|_{p, \alpha}^{p}=\frac{p\alpha}{2\pi}\int_{\mathbb C}|f(z)e^ {-{\alpha}|z|^{2}/2}|^{p}\,dA(z)<\infty. \end{align*} $$
Similarly, for
$p=\infty $
, we use the notation
$F^{\infty }_{\alpha }$
to denote the space of holomorphic functions f on
$ \mathbb C$
such that
$$ \begin{align*} \|f\|_{\infty, \alpha}={\textrm{ess}}\sup \{|f(z)|e^ {-{\alpha}|z|^{2}/2}: z\in \mathbb C\}<\infty. \end{align*} $$
The Fock space
$F^{2}_{\alpha }$
is a Hilbert space with the inner product
$$ \begin{align*} \langle f,g\rangle_\alpha=\int_{\mathbb C}f(z)\overline{g(z)}\,d\lambda_{\alpha}(z). \end{align*} $$
For any fixed
$w\in \mathbb {C}$
, the mapping
$f\mapsto f(w)$
is a bounded linear functional on
$F^2_\alpha $
. By the Riesz representation theorem in functional analysis, there exists a unique function
$K_w\in F^2_\alpha $
such that
$f(w)=\langle f, K_w \rangle _\alpha $
for all
$f\in F^{2}_\alpha $
. The function
$K_\alpha (z,w)=K_w(z)$
is called the reproducing kernel of
$F^{2}_{\alpha }$
. It is well known that
$K_{\alpha }(z,w)=e^{\alpha z\overline {w}}$
(see, for example, [Reference Zhu8]). Let
$k_{w}(z)=e^{\alpha z\overline {w}-{\alpha }|w|^2/2}$
be the normalised reproducing kernel at w. The orthogonal projection
$P_\alpha : L^2(\mathbb {C}, d\lambda _\alpha )\rightarrow F^{2}_{\alpha }$
can be represented as
$$ \begin{align*} P_\alpha f(z)=\int_{\mathbb C}f(w)K_{\alpha}(z,w)\,d\lambda_{\alpha}(w). \end{align*} $$
With this expression,
$P_\alpha $
can be extended to a bounded linear operator from
$L^p(\mathbb {C}, d\lambda _\alpha )$
to
$F^{p}_{\alpha }$
for
$1\le p\le \infty $
, and
$P_\alpha f=f$
for
$f\in F^{p}_\alpha $
. (See [Reference Janson, Peetre and Rochberg4, Reference Zhu8] for more details of the theory of Fock spaces.)
A positive Borel measure
$\mu $
on
$\mathbb C$
is called a Carleson measure for
$F^{p}_{\alpha }$
if there exists a positive constant C such that
$$ \begin{align*} \int_{\mathbb C}|f(z)e^{-\alpha|z|^{2}}|^{p}\,d\mu(z)\le C\|f\|^p_{p,\alpha} \end{align*} $$
for all
$f\in F^{p}_{\alpha }$
. Carleson measure plays a fundamental role in the theory of Toeplitz operators and interpolation (see, for example, [Reference Hu and Lv3, Reference Zhu8]).
The concept of Hankel measures was first proposed by Xiao in [Reference Xiao6, Reference Xiao7]. Motivated by the study of Hankel operators on the weighted Bergman space and Hardy space, as well as the parallel to the notion of Carleson measures, Xiao introduced the notion of Hankel measures. More precisely, a complex Borel measure
$\mu $
on the open unit disk
$\mathbb D$
is called a Hankel measure (or pseudo-Carleson measure) for the weighted Bergman space
$A^{2}_{\alpha }$
if there exists a positive constant C such that
$$ \begin{align*} \bigg|\int_{\mathbb D}f(z)^{2}\,d\mu(z)\bigg|\le C\|f\|^2_{A^{2}_{\alpha}} \end{align*} $$
for all
$f\in A^{2}_{\alpha }$
. In [Reference Xiao6], Xiao characterised the properties of the measure
$\mu $
in order that
$\mu $
be a Hankel measure for
$A^{2}_{\alpha }$
. The similar problem for Hardy space was also considered in [Reference Xiao7]. Later, Hankel measure was studied in some other contexts (see [Reference Arcozzi, Rochberg, Sawyer and Wick1, Reference Bao, Ye and Zhu2]).
Inspired by Xiao’s work, we extend the theory of Hankel measures to Fock space. A complex Borel measure
$\mu $
on
$\mathbb C$
is called a Hankel measure for
$F^{p}_{\alpha }\ ( p\ge 1)$
if there exists a positive constant C such that
$$ \begin{align*} \bigg|\int_{\mathbb C}f(z)^{p}e^{-{p\alpha}|z|^{2}/2}\,d\mu(z)\bigg|\le C\|f\|^p_{p,\alpha} \end{align*} $$
for all
$f\in F^{p}_{\alpha }$
. It is clear that every Carleson measure for
$F^{p}_{\alpha }$
must be a Hankel measure for
$F^{p}_{\alpha }$
, but not conversely. The main result of this paper characterises the Hankel measure for
$F^{p}_{\alpha }$
.
Theorem 1.1 (Main Theorem).
Let
$\alpha>0, p\ge 1$
and let
$\mu $
be a complex Borel measure on
$\mathbb {C}$
. Then the following statements are equivalent:
-
(1)
$\mu $
is a Hankel measure for
$F^{p}_{\alpha }$
; -
(2) for any
$f\in F^{1}_{\alpha }$
,
$$ \begin{align*} \bigg|\int_{\mathbb C}f(z)e^{-{\alpha}|z|^{2}/2}\,d\mu(z)\bigg|\le C\|f\|_{1,\alpha}; \end{align*} $$
-
(3)
$\sup _{w\in \mathbb {C}}|\int _{\mathbb C}e^{p\alpha z\overline {w}-{p\alpha }(|z|^{2}+|w|^{2})/2} \,d\mu (z)|<\infty $
; -
(4)
$P\overline {\mu }(z)=\int _{\mathbb C}e^{\alpha z\overline {w}-{\alpha }|z|^{2}/2}\,d\kern1pt\overline {\mu }(z)$
defines a function in
$F^{\infty }_{\alpha }$
; -
(5)
$K_\mu : f\mapsto \int _{\mathbb C}e^{\alpha zw} f(w) \,d\mu (z)$
exists as a bounded operator on
$F^{p}_{\alpha }$
for each
$p\ge 1$
.
Throughout this paper, we use C to denote positive constants whose value may change from line to line, but do not depend on the functions being considered. We call two quantities A and B equivalent, denoted by
$A\simeq B$
, if there exists some C such that
$C^{-1} A \leq B \leq C A$
. Given some exponent
$s\ge 1$
, we always use
$s'$
to denote the conjugate of s, defined by
$s^{-1}+ {s'}^{-1} =1$
.
2 Proof of the main theorem
We begin by stating some known results which are used in the proof of the main theorem.
Given some
$a\in \mathbb C$
and
$r>0$
, write
$D(a,r)=\{z\in \mathbb C: |z-a|<r\}$
for the Euclidean ball centred at a with radius r. A sequence
$\{a_{k}\}$
in
$\mathbb C$
is called an r-lattice if the following conditions are satisfied:
-
(1)
$\bigcup _{k=1}^{\infty }D(a_{k},r)=\mathbb C$
; -
(2)
$\{D(a_{k},{r}/{4})\}_{k=1}^{\infty }$
are mutually disjoint.
Given
$r>0$
, it is easy to pick
$a_{k}\in \mathbb C$
such that
$\{a_{k}\}$
is an r-lattice.
The atomic decomposition for Fock spaces is a powerful result in the theory.
Theorem 2.1 [Reference Janson, Peetre and Rochberg4].
Let
$1\leq p\leq \infty $
. There exists a positive constant
$r_{0}$
such that for any r with
$0<r<r_{0}$
, the space
$F^{p}_{\alpha }$
consists exactly of the functions
$$ \begin{align} f(z)=\sum_{k=1}^{\infty}\lambda_{k}e^{\alpha z\overline{a}_{k}-{\alpha}|a_{k}|^{2}/2}, \end{align} $$
where
$\{\lambda _{k}\}\in l^{p}$
and
$\{a_{k}\}$
is an r-lattice. Moreover, there exists a positive constant C (independent of f) such that
$$ \begin{align*} C^{\kern1pt-1}\|f\|_{p, \alpha}\leq \inf \|\{\lambda_{k}\}\|_{l^{p}} \leq C\|f\|_{p, \alpha} \end{align*} $$
for all
$f \in F^{p}_{\alpha }$
, where the infimum is taken over all sequences
$\{\lambda _{k}\}$
that give rise to the decomposition in (2.1).
We also need the duality theorem.
Theorem 2.2 [Reference Janson, Peetre and Rochberg4].
Suppose
$\alpha>0, 1\leq p<\infty $
. Then the dual space of
$F^{p}_{\alpha }$
can be identified with
$F^{p'}_{\alpha }$
under the integral pairing
$$ \begin{align*} \langle f,g\rangle_{\alpha}=\lim_{R\rightarrow \infty}\int_{|z|<R}f(z)\overline{g(z)}e^ {-\alpha|z|^{2}}\,dA(z). \end{align*} $$
Proof of the Main Theorem.
We first show the equivalence of (1), (2) and (3).
(2)
$\Rightarrow $
(1). Suppose (2) is valid. Replacing
$f(z)e^{-{\alpha }|z|^2/2}$
by
$(f(z)e^{-{\alpha }|z|^{2}/2})^p$
gives
$$ \begin{align*} \bigg|\int_{\mathbb C}f(z)^{p}e^{-{p\alpha}|z|^{2}/2}\,d\mu(z)\bigg|\le C\int_{\mathbb C}|f(z)|^{p}e^{-{p\alpha}|z|^{2}/2}\,dA(z)= C\|f\|^p_{p,\alpha}. \end{align*} $$
Thus,
$\mu $
is a Hankel measure for
$F^p_\alpha $
.
(1)
$\Rightarrow $
(3). Suppose (1) is true. Let
$f(z)=k_w(z)$
. Since
$\|k_w\|_{p,\alpha }=1$
(see [Reference Zhu8, Lemma 2.33]), we obtain
$$ \begin{align*} \bigg|\int_{\mathbb C}k_w(z)^pe^{-{p\alpha}|z|^{2}/2}\,d\mu(z)\bigg|\le C\|k_w\|_{p,\alpha}^p=C. \end{align*} $$
From this, the statement (3) follows.
(3)
$\Rightarrow $
(2). For
$f\in F^{1}_{\alpha }$
, fix
$0<r<r_0$
where
$r_0$
is as in Theorem 2.1, and let
$\{a_{k}\}$
be an r-lattice. By Theorem 2.1, for any
$\{\lambda _{k}\}\in l^{1}$
, the function f admits the decomposition
$$ \begin{align*} f(z)=\sum_{k=1}^{\infty} \lambda_{k}e^{\alpha z\overline{a}_{k}-{\alpha}|a_{k}|^{2}/2} \end{align*} $$
with
$\|\{\lambda _{k}\}\|_{l^{1}}\leq C\|f\|_{1, \alpha }$
. Therefore,
$$ \begin{align*} \bigg|\int_{\mathbb C}f(z)e^{-{\alpha}|z|^{2}/2}\,d\mu(z)\bigg| &=\bigg|\int_{\mathbb C}\sum_{k=1}^{\infty}\lambda_{k}e^{\alpha z\overline{a}_{k}-{\alpha}(|a_{k}|^2+|z|^2)/2}\,d\mu(z)\bigg|\\ &\le \sum_{k=1}^{\infty}|\lambda_{k}|\bigg|\int_{\mathbb C}e^{\alpha z\overline{a}_{k}-{\alpha}(|a_{k}|^2+|z|^2)/2}\,d\mu(z)\bigg|\\ &\le \|\{\lambda_{k}\}\|_{l^{1}}\cdot\sup_j\bigg|\int_{\mathbb C}e^{\alpha z\overline{a}_{k}-{\alpha}(|a_{k}|^2+|z|^2)/2}\,d\mu(z)\bigg|\\ &\le C\|f\|_{1, \alpha}, \end{align*} $$
which gives (2).
Next we prove the equivalence of (2), (4) and (5).
(2)
$\Leftrightarrow $
(4). The reproducing formula of
$F^{1}_{\alpha }$
implies
$$ \begin{align*} \int_{\mathbb C}f(z)e^{-\frac{\alpha}{2}|z|^{2}}\,d\mu(z) &=\int_{\mathbb C}\bigg(\int_{\mathbb C}f(w)e^{\alpha z\overline{w}}\,d\lambda_\alpha(w)\bigg)e^{-{\alpha}|z|^{2}/2}\,d\mu(z)\\ &= \int_{\mathbb C}f(w)\bigg(\int_{\mathbb C}e^{\alpha z\overline{w}-{\alpha}|z|^{2}/2}\,d\mu(z)\bigg)\,d\lambda_\alpha(w)\\ &= \int_{\mathbb C}f(w)\overline{P\overline{\mu}(w)}\,d\lambda_\alpha(w)\\ &= \langle f, P\overline{\mu}\rangle_\alpha. \end{align*} $$
Since
$(F^{1}_{\alpha })^\ast \simeq F^{\infty }_{\alpha }$
under the pairing
$\langle \cdot , \cdot \rangle _\alpha $
, we obtain the equivalence between (2) and (4).
(4)
$\Leftrightarrow $
(5). Following [Reference Luecking5, Reference Xiao6], we call
$K_\mu $
a (small) Hankel operator associated with the symbol
$\mu $
. For
$p\ge 1$
, we claim that for
$f\in F^{p}_{\alpha }, g\in F^{p'}_{\alpha }$
,
$$ \begin{align*} \langle K_\mu, g\rangle_\alpha=\int_{\mathbb C}f(z)\overline{g(\overline{z})}\,d\mu(z). \end{align*} $$
In fact, by the reproducing formula of
$F^{1}_{\alpha }$
,
$$ \begin{align*} \langle K_\mu, g\rangle_\alpha &=\int_{\mathbb C}K_\mu f(w)\overline{g(w)}\,d\lambda_\alpha(w)\\&= \int_{\mathbb C}\bigg(\int_{\mathbb C}e^{\alpha zw} f(z)\, d\mu(z)\bigg)\overline{g(w)}\,d\lambda_\alpha(w)\\&= \int_{\mathbb C}f(z)\bigg(\overline{\int_{\mathbb C}e^{\alpha \overline{zw}}g(w)\,d\lambda_\alpha(w)}\bigg)\,d\mu(z)\\&= \int_{\mathbb C}f(z)\overline{g(\overline{z})}\,d\mu(z). \end{align*} $$
Notice that
$\overline {g(\overline {z})}\in F^{p'}_{\alpha }$
whenever
$g\in F^{p'}_{\alpha }$
, so
$\overline {g(\overline {z})}$
is analytic on
$\mathbb {C}$
. From the reproducing formula,
$$ \begin{align*} f(z)\overline{g(\overline{z})}=\int_{\mathbb C}f(u)\overline{g(\overline{u})}e^{\alpha z\overline{u}}\,d\lambda_\alpha(u). \end{align*} $$
Set
$h(u)=\int _{\mathbb C}e^{\alpha z\overline {u}}\,d\mu (z)$
. By Fubini’s theorem,
$$ \begin{align*} \langle K_\mu, g\rangle_\alpha=\int_{\mathbb C}\int_{\mathbb C}f(u)\overline{g(\overline{u})}e^{\alpha z\overline{u}}\,d\lambda_\alpha(u)\,d\mu(z) =\int_{\mathbb C}f(u)\overline{g(\overline{u})}h(u)\,d\lambda_\alpha(u). \end{align*} $$
This shows that
$K_\mu $
is a bounded operator on
$F^{p}_{\alpha }$
if and only if
$\overline {h}\in F^{\infty }_{\alpha }$
. Indeed, if
$\overline {h}\in F^{\infty }_{\alpha }$
, then for any
$f\in F^{p}_{\alpha }, g\in F^{p'}_{\alpha }$
, the dual relation
$(F^{1}_{\alpha })^\ast \simeq F^{\infty }_{\alpha }$
under the pairing
$\langle \cdot , \cdot \rangle _\alpha $
, together with Hölder’s inequality, gives
$$ \begin{align*} |\langle K_\mu, g\rangle_\alpha|\le C\|f\|_{p,\alpha}\|g\|_{p',\alpha}\|\overline{h}\|_{\infty,\alpha}. \end{align*} $$
Since
$(F^{p}_{\alpha })^\ast \simeq F^{p'}_{\alpha }$
relative to
$\langle \cdot , \cdot \rangle _\alpha $
, it follows that
$K_\mu $
is bounded on
$F^{p}_{\alpha }$
. Conversely, if
$K_\mu $
is bounded on
$F^{p}_{\alpha }$
, for a sequence
$\{a_k\}\subset \mathbb {C}$
, as in Theorem 2.1, let
$$ \begin{align*} f_k(z)=(e^{\alpha z\overline{a}_k-{\alpha}|a_k|^{2}/2})^{{1}/{p}},\quad g_k(z)=(e^{\alpha za_k-{\alpha}|a_k|^{2}/2})^{{1}/{p'}}. \end{align*} $$
When
$F\in F^{1}_{\alpha }$
, one can find a sequence
$\{\lambda _{k}\}\in l^{1}$
such that
$$ \begin{align*} F(z)=\sum_{k=1}^{\infty}\lambda_{k}f_k(z)\overline{g_k(\overline{z})} \end{align*} $$
with
$\|\{\lambda _{k}\}\|_{l^{1}}\leq C\|F\|_{1, \alpha }$
. Consequently,
$$ \begin{align*} |\langle F, \overline{h}\rangle_\alpha| &=\bigg|\int_{\mathbb C}\sum_{k=1}^{\infty}\lambda_{k}f_k(z)\overline{g_k(\overline{z})}h(z)\,d\lambda_\alpha(z)\bigg|\\ &\le \sum_{k=1}^{\infty}|\lambda_{k}|\,|\langle K_\mu f_k, g_k\rangle_\alpha|\\ &\le C\|\{\lambda_{k}\}\|_{l^{1}}\cdot\sup_j\|K_\mu f_k\|_{p,\alpha}\cdot\|g_k\|_{p',\alpha}\\ &\le C\|F\|_{1, \alpha}. \end{align*} $$
This shows
$\overline {h}\in F^{\infty }_{\alpha }$
and completes the proof of the main theorem.
Acknowledgements
I would like to thank the referees for their careful reading and valuable suggestions. I would also like to express my gratitude to Professor Zhangjian Hu (Huzhou University) for his support and encouragement along the way.
