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Bergman projections on weighted Fock spaces in several complex variables

Journal of Inequalities and Applications20172017:286

https://doi.org/10.1186/s13660-017-1560-3

  • Received: 28 March 2017
  • Accepted: 9 November 2017
  • Published:

Abstract

Let ϕ be a real-valued plurisubharmonic function on \({\mathbb {C}}^{n}\) whose complex Hessian has uniformly comparable eigenvalues, and let \(\mathcal{F}^{p}(\phi)\) be the Fock space induced by ϕ. In this paper, we conclude that the Bergman projection is bounded from the pth Lebesgue space \(L^{p}(\phi )\) to \(\mathcal{F}^{p}(\phi)\) for \(1\leq p \leq\infty\). As a remark, we claim that Bergman projections are also well defined and bounded on Fock spaces \(\mathcal{F}^{p}(\phi)\) with \(0< p<1\). We also obtain the estimates for the distance induced by ϕ and the \(L^{p}(\phi)\)-norm of Bergman kernel for \(\mathcal{F}^{2}(\phi)\).

Keywords

  • Bergman kernel
  • Bergman projection
  • reverse-Hölder inequality

1 Introduction

The symbol dv denotes the Lebesgue volume measure on \({\mathbb {C}}^{n}\), and
$$B(z,r)= \bigl\{ w\in{\mathbb {C}}^{n}: \vert w-z \vert < r \bigr\} \quad \text{for } z\in{\mathbb {C}}^{n} \text{ and } r>0. $$
Suppose \(\phi: {\mathbb {C}}^{n}\rightarrow{\mathbb {R}}\) is a \(C^{2}\) plurisubharmonic function. We say that ϕ belongs to the weight class W if ϕ satisfies the following statements:
(I) There exists \(c> 0\) such that for \(z\in{\mathbb {C}}^{n}\)
$$ \inf_{z\in{\mathbb {C}}^{n}}\sup_{w\in B(z,c)}\Delta \phi(w)>0; $$
(1)
(II) Δϕ satisfies the reverse-Hölder inequality
$$ \Vert \Delta\phi \Vert _{L^{\infty}(B(z,r) )}\leq Cr^{-2n} \int_{B(z,r)}\Delta\phi \,dv, \quad\forall z\in{\mathbb {C}}^{n}, r > 0 $$
(2)
for some \(0< C < +\infty\);
(III) The eigenvalues of \(H_{\phi}\) are comparable, i.e., there exists \(\delta_{0} > 0\) such that
$$ \bigl(H_{\phi}(z)u, u \bigr) \geq\delta_{0} \Delta\phi(z) \vert u \vert ^{2},\quad\forall z, u\in{\mathbb {C}}^{n}, $$
where
$$ H_{\phi}= \biggl(\frac{\partial^{2}\phi}{\partial z_{j}\,\partial\overline {z}_{k}} \biggr)_{j,k}. $$
Suppose \(0< p<\infty\), \(\phi\in\mathbf{W}\). The space \(L^{p}(\phi)\) consists of all Lebesgue measurable functions f on \({\mathbb {C}}^{n}\) for which
$$\Vert f \Vert _{p,\phi}= \biggl( \int_{{\mathbb {C}}^{n}} \bigl\vert f(z) \bigr\vert ^{p}e^{-p\phi(z)} \,dv(z) \biggr)^{\frac {1}{p}}< \infty. $$
\(L^{\infty}(\phi)\) is the set of all Lebesgue measurable functions f on \({\mathbb {C}}^{n}\) with
$$\Vert f \Vert _{\infty, \phi}=\sup_{z\in {\mathbb {C}}^{n}} \bigl\vert f(z) \bigr\vert e^{-\phi(z)} < \infty. $$
Let \(H({\mathbb {C}}^{n})\) be the family of all holomorphic functions on \({\mathbb {C}}^{n}\). The weighted Fock space is defined as
$$\mathcal{F}^{p}(\phi)=L^{p}(\phi)\cap H\bigl({\mathbb {C}}^{n}\bigr) $$
with the same norm \(\Vert \cdot \Vert _{p, \phi}\). It is easy to check that \(\mathcal{F}^{p}(\phi)\) is a Banach space under \(\Vert \cdot \Vert _{p, \phi}\) if \(1\leq p<\infty \), and \(\mathcal{F}^{p}(\phi)\) is a Fréchet space with the metric \(\varrho(f,g)= \Vert f-g \Vert _{p, \phi}^{p}\) whenever \(0< p<1\). Taking \(\phi(z)=\frac{1}{2} \vert z \vert ^{2}\), \(\mathcal {F}^{p}(\phi)\) is the classical Fock space which has been studied by many authors, see [13] and the references therein. Notice that the weight function φ on \({\mathbb {C}}^{n}\) with the restriction that \(d d^{c} \varphi\simeq d d^{c} \vert z \vert ^{2}\) in [4] and [5] belongs to W.

In the one-dimensional case, an important contribution to weighted Fock spaces was given by Christ [6] (but see also [7, 8]). They work under the assumption that ϕ is subharmonic and that \(\Delta\phi\, dA\) is a doubling measure, where dA is the area measure on \({\mathbb {C}}\). Notice that the hypotheses on \(\Delta\phi\, dA\) are a sort of finite-type assumption and are automatically verified when ϕ is a subharmonic non-harmonic polynomial.

The result of Christ was extended by Delin to several complex variables under the assumption of strict plurisubharmonicity of the weight in [9]. Dall’Ara [10] tried to extend Christ’s approach to \(n\geq2\). Given \(\phi\in\mathbf{W}\), let \(K(\cdot, \cdot)\) be the weighted Bergman kernel for \(\mathcal{F}^{2}(\phi)\). In particular, Theorem 20 of [10] proves that there is a constant \(C,\epsilon>0\) such that
$$ \bigl\vert K(z,w) \bigr\vert \leq C e^{\phi(z)+\phi(w)} \frac {e^{-\epsilon d(z,w)}}{\rho_{\phi}(z)^{n}\rho_{\phi}(w)^{n}} $$
(3)
for \(z,w\in{\mathbb {C}}^{n}\), where \(d(\cdot , \cdot)\), \(\rho_{\phi }(\cdot)\) described in Section 2.
In the setting of Bergman spaces, the Bergman projection is bounded on p-Bergman spaces for \(1< p<\infty\), it also maps \(L^{\infty}\) into Bloch spaces, see [11] for details. With the Bergman kernel \(K(\cdot, \cdot)\) for \(\mathcal{F}^{2}(\phi)\), the Bergman projection P can be represented as
$$Pf(z)= \int_{{\mathbb {C}}^{n}}K(z,w)f(w)e^{-2\phi(w)}\,dv(w),\quad z\in {\mathbb {C}}^{n}. $$
It is well known that \(P(f)=f\) for \(f\in\mathcal{F}^{2}(\phi)\). The purpose of this work is to discuss the boundedness of Bergman projection acting on \(\mathcal{F}^{p}(\phi)\) for general p. Section 2 is devoted to some basic estimates, including the distance \(d(\cdot , \cdot)\) and the \(L^{p}(\phi)\)-norm of the Bergman kernel. In Section 3, we will discuss the boundedness of Bergman projections from \(L^{p}(\phi)\) to \(\mathcal{F}^{p}(\phi)\) with \(1 \leq p \leq\infty\). We also show that the Bergman projection is well defined and bounded on \(\mathcal{F}^{p}(\phi)\) for \(p<1\).

In what follows, we always suppose \(\phi\in\mathbf{W}\) and use C to denote positive constants whose values may change from line to line but do not depend on the functions being considered. Two quantities A and B are called equivalent, denoted by ‘\(A\simeq B\)’, if there exists some C such that \(C^{-1 }A\le B \le C A\).

2 Some basic estimates

In this section, we are going to give some estimates, which will be useful in the following section. At the beginning, we will give some notations.

For \(z\in{\mathbb {C}}^{n}\), set
$$ \rho_{\phi}(z)=\sup \Bigl\{ r>0: \sup_{w\in B(z,r)} \Delta\phi (w)\leq r^{-2} \Bigr\} . $$
(4)
By (1), there exist \(c, s > 0\) such that for \(z\in{\mathbb {C}}^{n}\)
$$ \sup_{w\in B(z, c)}\Delta\phi(w)\geq s. $$
We then have some \(M>0\) such that
$$ \sup_{z\in{\mathbb {C}}^{n}}\rho_{\phi}(z)\leq M. $$
Moreover, there are some positive constants C, \(M_{1}\) and \(M_{2}\) such that for all \(z,w\in{\mathbb {C}}^{n}\), we have
$$ C^{-1}\theta^{-M_{1}}\rho_{\phi}(w)\leq \rho_{\phi}(z)\leq C\theta ^{M_{2}}\rho_{\phi}(w), $$
(5)
where \(\theta=\max (1,\frac{ \vert z-w \vert }{\rho _{\phi}(w)} )\). We can see this in Proposition 10 of [10].
Given \(r>0\), write
$$B^{r}(z)=B\bigl(z,r\rho_{\phi}(z)\bigr) \quad \text{and} \quad B(z)=B^{1}(z). $$
Then (5) implies that there is some C such that for \(z\in {\mathbb {C}}^{n}\)
$$ C^{-1}\rho_{\phi}(w)\leq\rho_{\phi}(z) \leq C\rho_{\phi}(w)\quad \text{for }w\in B(z). $$
(6)
By (6) and the triangle inequality, we have \(m_{1}, m_{2}>0\) such that
$$ B(z)\subseteq B^{m_{1}}(w), \qquad B(w)\subseteq B^{m_{2}}(z) \quad\text{whenever } w\in B(z). $$
(7)
Given a sequence \(\{a_{k}\}_{k=1}^{\infty}\) in \({\mathbb {C}}^{n}\), we say that \(\{a_{k}\}_{k=1}^{\infty}\) is a lattice if \(\{B(a_{k}) \}_{k=1}^{\infty}\) covers \({{\mathbb {C}}^{n}}\) and the balls of \(\{B^{\frac{1}{5}}(a_{k}) \}_{k=1}^{\infty}\) are pairwise disjoint. This lattice exists by a standard covering lemma, see Theorem 2.1 in [12], or Proposition 7 in [10] as well. Moreover, for the lattice \(\{a_{k}\}_{k}\) and any \(m> 0 \), there exists some integer N such that each \(z\in{{\mathbb {C}}^{n}}\) can be in at most N balls of \(\{B^{m}(a_{k}) \}_{k}\). Equivalently,
$$ \sum_{k=1}^{\infty}\chi_{B^{m}(a_{k})}(z)\le N \quad \text{for } z\in{{\mathbb {C}}^{n}}. $$
(8)
To the radius function \(\rho_{\phi}\) defined as (4), we associate the Riemannian metric \(\rho_{\phi}(z)^{-2}\,dz\otimes d\overline{z}\). In fact, we are interested only in the associated Riemannian distance, which we describe explicitly. If \(\gamma: [0, 1]\rightarrow{\mathbb {C}}^{n}\) is piecewise \(C^{1}\) curves, we define
$$ L_{\rho_{\phi}}(\gamma)= \int_{0}^{1}\frac{ \vert \gamma '(t) \vert }{\rho_{\phi}(\gamma(t))}\,dt. $$
Given \(z, w \in{\mathbb {C}}^{n}\), we put
$$ d(z, w)=\inf_{\gamma}L_{\rho_{\phi}}(\gamma), $$
where the inf is taken as γ varies over the collection of curves with \(\gamma(0)=z\) and \(\gamma(1)=w\). We then have the estimate for this distance as follows.

Lemma 1

There exist \(\alpha,\beta, C>0\) such that for \(z,w\in{\mathbb {C}}^{n}\)
$$\frac{1}{C} \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(z)} \biggr)^{\alpha}\leq d(z,w)\leq C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi}(z)} \biggr)^{\beta}. $$

Proof

First, we claim that there is some \(C>0\) such that
$$ d(z,w)\geq C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(z)} \biggr)^{\alpha}. $$
(9)
In fact, set μ to be
$$ \mu\bigl(B(z,r)\bigr)=r^{2} \Vert \Delta\phi \Vert _{L^{\infty}(B(z,r))},\quad z\in{\mathbb {C}}^{n}, r>0. $$
(10)
By (2), it is easy to check that there is some \(M>2\) such that
$$ \mu \bigl(B(z,2r) \bigr)\leq M\mu \bigl(B(z,r) \bigr). $$
(11)
Moreover,
$$ \mu \bigl(B\bigl(z,\rho_{\phi}(z)\bigr) \bigr)=1 $$
(12)
because of (4). Given any \(r\leq R\), it is easy to check that
$$ \mu \bigl(B(z,r) \bigr)\leq \biggl(\frac{r}{R} \biggr)^{2}\mu \bigl(B(z,R) \bigr)\leq\mu \bigl(B(z,R) \bigr) $$
(13)
for \(z\in{\mathbb {C}}^{n}\) because of (10). Also, there is a positive integer m such that \(2^{m-1}r< R\leq2^{m}r\). Hence, (11) and (12) tell us
$$\mu \bigl(B(z,R) \bigr)\leq\mu \bigl(B\bigl(z,2^{m}r\bigr) \bigr)\leq M\mu \bigl(B\bigl(z,2^{m-1}r\bigr) \bigr)\leq\cdots\leq M^{m} \mu \bigl(B(z,r) \bigr). $$
Since \(M^{m-1}=2^{(m-1)\log_{2}M}\leq (\frac{R}{r} )^{\log_{2}M}\), we get
$$ \mu \bigl(B(z,R) \bigr)\leq M \biggl(\frac{R}{r} \biggr)^{\log _{2}M}\mu \bigl(B(z,r) \bigr). $$
(14)
For \(z,w\in{\mathbb {C}}^{n}\), notice that \(B(w, \vert w-z \vert )\subset B(z,2 \vert w-z \vert )\). If \(\vert w-z \vert < \rho_{\phi}(z)\), take any piecewise \(C^{1}\) curve \(\gamma:[0, 1]\rightarrow{\mathbb {C}}^{n}\) connecting z and w, and let \(T_{0}\) be the minimum time such that \(\vert z -\gamma(T_{0}) \vert =\rho_{\phi}(z)\). By (6), \(\rho_{\phi}(\gamma(t))\simeq\rho_{\phi}(z)\) for \(t\in[0,T_{0})\). This implies
$$\int_{0}^{1}\frac{ \vert \gamma'(t) \vert }{\rho_{\phi }(\gamma(t))}\,dt\geq \frac{C}{\rho_{\phi}(z)} \int_{0}^{T_{0}} \bigl\vert \gamma'(t) \bigr\vert \,dt\geq C\frac{ \vert z-w \vert }{\rho_{\phi}(z)}. $$
If \(\vert z-w \vert \geq\rho_{\phi}(w)\), then (11), (10), (13) and (12) give
$$\begin{aligned} \mu \biggl(B \biggl(z,\frac{1}{4} \vert z-w \vert \biggr) \biggr)&\geq C\mu \bigl(B\bigl(z, 2 \vert z-w \vert \bigr) \bigr)\geq C\mu \bigl(B\bigl(w, \vert z-w \vert \bigr) \bigr) \\ &\geq C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(w)} \biggr)^{2}\mu \bigl(B\bigl(w, \rho_{\phi}(w)\bigr) \bigr) \\ &=C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(w)} \biggr)^{2}. \end{aligned}$$
On the other hand, for \(\zeta\in\overline{B (z, \frac {1}{4} \vert z-w \vert )}\), there are
$$B \biggl(\zeta, \frac{1}{4} \vert z-w \vert \biggr)\subset B \biggl(z, \frac{1}{2} \vert z-w \vert \biggr) $$
and
$$B \biggl(z, \frac{1}{4} \vert z-w \vert \biggr)\subset B \biggl( \zeta, \frac{1}{2} \vert z-w \vert \biggr). $$
Combining the above with (11), we know
$$\mu \biggl(B \biggl(z,\frac{1}{4} \vert z-w \vert \biggr) \biggr) \simeq\mu \biggl(B \biggl(\zeta,\frac{1}{2} \vert z-w \vert \biggr) \biggr). $$
By the fact \(\log_{2}M>0\), (13), (14) and (12), there exists \(t>0\) such that
$$\begin{aligned} \mu \biggl(B \biggl(z,\frac{1}{4} \vert z-w \vert \biggr) \biggr) &\simeq\mu \biggl(B \biggl(\zeta,\frac{1}{2} \vert z-w \vert \biggr) \biggr) \\ &\leq C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(\zeta)} \biggr)^{t}\mu \bigl(B \bigl( \zeta,\rho_{\phi}(\zeta ) \bigr) \bigr) \\ &\simeq \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(\zeta)} \biggr)^{t}. \end{aligned}$$
Hence, \((\frac{ \vert z-w \vert }{\rho_{\phi }(w)} )^{2}\leq C (\frac{ \vert z-w \vert }{\rho _{\phi}(\zeta)} )^{t}\). This implies
$$\rho_{\phi}(\zeta)\leq C \vert z-w \vert \biggl(\frac { \vert z-w \vert }{\rho_{\phi}(w)} \biggr)^{-\alpha },\qquad\zeta\in\overline{B \biggl(z, \frac{1}{4} \vert z-w \vert \biggr)}, $$
where \(\alpha=\frac{2}{t}>0\). For any piecewise \(C^{1}\) curves Γ, defined as \(\gamma: [0,1]\rightarrow{\mathbb {C}}^{n}\) with \(\gamma(0)=z\) and \(\gamma(1)=w\), we have
$$\begin{aligned} \int_{\Gamma}\frac{ \vert \gamma'(t) \vert }{\rho _{\phi}(\gamma(t))}\,dt&\geq \int_{\Gamma\cap\overline{B (z, \frac{1}{4} \vert z-w \vert )}}\frac{ \vert \gamma'(t) \vert }{\rho_{\phi}(\gamma(t))}\,dt \\ &\geq\frac{1}{ \vert z-w \vert (\frac{ \vert z-w \vert }{\rho_{\phi}(w)} )^{-\alpha}} \int _{\Gamma\cap\overline{B (z, \frac{1}{4} \vert z-w \vert )}} \bigl\vert \gamma'(t) \bigr\vert \,dt \\ &\geq C \biggl(\frac{ \vert z-w \vert }{\rho_{\phi }(w)} \biggr)^{\alpha}. \end{aligned} $$
This yields (9) is true. Now, we are going to prove the other direction. For \(z, w\in{\mathbb {C}}^{n}\), take \(\gamma (t)=z+t(w-z)\) and \(\gamma(t_{0})\in\partial B(z)\) (set \(t_{0}=1\) if \(w\in B(z)\)). Then (5) gives
$$\begin{aligned} d(z,w)&\leq \vert w-z \vert \int_{0}^{1}\frac{dt}{\rho _{\phi}(\gamma(t))} \\ &\leq C \vert w-z \vert \biggl( \int_{0}^{t_{0}}+ \int _{t_{0}}^{1} \biggr)\frac{dt}{\rho_{\phi}(\gamma(t))} \\ &\leq C\frac{ \vert w-z \vert }{\rho_{\phi}(z)} \int_{0}^{1}dt+ C \biggl(\frac{ \vert w-z \vert }{\rho_{\phi}(z)} \biggr)^{1+M_{1}} \int_{0}^{1} t^{M_{1}}\,dt \\ &\leq C \biggl(\frac{ \vert w-z \vert }{\rho_{\phi }(z)} \biggr)^{\beta}, \end{aligned}$$
where \(\beta>0\). The proof is completed. □

Now, we can estimate the following integral.

Lemma 2

Given \(p>0\) and \(k\in\mathbb {R}\), we have
$$\int_{ {\mathbb {C}}^{n}}\rho_{\phi}(\zeta)^{k}e^{ - pd(z,\zeta )}\,dv( \zeta)\leq C\rho_{\phi}(z)^{k+2n}, $$
where \(C>0\) is a constant depending only on n, p and k.

Proof

By (6), it is easy to check that
$$\begin{aligned} \int_{B(z)}\rho_{\phi}(\zeta)^{k}e^{ - pd(z,\zeta)}\,dv( \zeta) \leq \int_{B(z)} \rho_{\phi}(\zeta)^{k}\,dv(\zeta) \leq C\rho_{\phi }(z)^{k+2n}. \end{aligned}$$
Estimate (9) gives
$$\begin{aligned} \int_{{{\mathbb {C}}^{n}}\backslash B(z)}\rho_{\phi}(\zeta)^{k}e^{ - pd(z,\zeta)}\,dv( \zeta) &\leq \int_{{{\mathbb {C}}^{n}}\backslash B(z)}\rho_{\phi}(\zeta)^{k}e^{ - pC_{1} (\frac{ \vert z-\zeta \vert }{\rho_{\phi }(z)} )^{\alpha}}\,dv( \zeta) \\ &\leq \int_{{{\mathbb {C}}^{n}}\backslash B(z)}\rho_{\phi}(\zeta )^{k} \int_{pC_{1} (\frac{ \vert z-\zeta \vert }{\rho _{\phi}(z)} )^{\alpha}}^{\infty}e^{-s}\,ds \,dv(\zeta) \\ &\leq \int_{pC_{1}}^{\infty}e^{-s} \int_{ B^{ (\frac {s}{pC_{1}} )^{\frac{1}{\alpha}}}(z)}\rho_{\phi}(\zeta )^{k}\,dv( \zeta)\,ds. \end{aligned}$$
By (5), the inequality above is no more than
$$\begin{aligned}& \int_{pC_{1}}^{\infty} \sup_{\zeta\in B^{ (\frac{s}{pC_{1}} )^{\frac {1}{\alpha}}}(z)} \rho_{\phi}(\zeta)^{k}v \bigl(B^{ (\frac {s}{pC_{1}} )^{\frac{1}{\alpha}}}(z) \bigr)e^{-s}\,ds \\& \quad \le C\rho_{\phi}(z)^{k+2n} \int_{pC_{1}}^{\infty}s^{\frac{2n+\max \{kM_{2}, -kM_{1}\}}{\alpha}}e^{-s}\,ds =C \rho_{\phi}(z)^{k+2n}. \end{aligned}$$
Therefore,
$$\int_{ {\mathbb {C}}^{n}}\rho_{\phi}(w)^{k}e^{ - pd(z,w) }\,dv(w) \leq C\rho_{\phi}(z)^{k+2n}. $$
The proof is completed. □

Next, we will give the \(L^{p}(\phi)\)-norm of the Bergman kernel \(K(\cdot , \cdot)\) for \(\mathcal{F}^{2}(\phi)\).

Proposition 3

For \(0< p<\infty\), we have
$$\bigl\Vert K(\cdot, z) \bigr\Vert _{p,\phi}\leq C e^{\phi(z)}\rho _{\phi}(z)^{2n (\frac{1}{p}-1 )}, \quad z\in{\mathbb {C}}^{n}. $$

Proof

By (3) and Lemma 2, we obtain
$$\begin{aligned} \int_{{\mathbb {C}}^{n}} \bigl\vert K(w, z) \bigr\vert ^{p}e^{-p\phi(w)}\,dv(w) &\le C\frac{e^{p\phi(z)}}{\rho_{\phi}(z)^{pn}} \int_{{\mathbb {C}}^{n}}\rho_{\phi}(w)^{-pn}e^{-p\epsilon d(z,w)}\,dv(w) \\ &\leq Ce^{p\phi(z)}\rho_{\phi}(z)^{2n(1-p)}. \end{aligned}$$
The proof is completed. □

Lemma 4

For \(0< p<\infty\), there is a constant \(C>0\) such that for all \(r\in (0,1]\), \(f\in H({\mathbb {C}}^{n})\) and \(z\in{\mathbb {C}}^{n}\), we have
$$ \bigl\vert f(z) \bigr\vert e^{-\phi(z)}\leq \frac{C}{r^{\frac {2n}{p}}\rho_{\phi}(z)^{\frac{2n}{p}}} \biggl( \int_{ B^{r}(z)} \bigl\vert f(w)e^{-\phi(w)} \bigr\vert ^{p}\,dv(w) \biggr)^{\frac{1}{p}}. $$
(15)

Proof

If \(p=2\), (15) is just Lemma 13 in [10]. For \(p\neq2\), we borrow the idea in Lemma 19 of [7] and Lemma 13 in [10]. The details are omitted. □

3 Boundedness of Bergman projections

Recall that the Bergman projection P on \(L^{p}(\phi)\) is defined as
$$Pf(z)= \int_{{\mathbb {C}}^{n}}K(z,w)f(w)e^{-2\phi(w)}\,dv(w),\quad z\in {\mathbb {C}}^{n}. $$
In this section, we focus on the boundedness of Bergman projections P from \(L^{p}(\phi)\) to \(\mathcal{F}^{p}(\phi)\) for \(1\leq p\leq \infty\).

Theorem 5

Let \(1\leq p\leq\infty\). Then the Bergman projection P is bounded as a map from \(L^{p}(\phi)\) to \(\mathcal{F}^{p}(\phi)\).

Proof

By the definition of P, we can conclude Pf is holomorphic on \({\mathbb {C}}^{n}\). Fubini’s theorem and Proposition 3 yield
$$\begin{aligned} \Vert Pf \Vert _{1,\phi} &\leq \int_{{\mathbb {C}}^{n}}e^{-\phi(z)}\,dv(z) \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w)f(w) \bigr\vert e^{-2\phi(w)}\,dv(w) \\ &= \int_{{\mathbb {C}}^{n}} \bigl\vert f(w) \bigr\vert e^{-2\phi (w)}\,dv(w) \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w) \bigr\vert e^{-\phi(z)}\,dv(z) \\ &\leq C \Vert f \Vert _{1,\phi} \end{aligned}$$
for \(f\in L^{1}(\phi)\). Given \(f\in L^{\infty}(\phi)\), we obtain
$$\begin{aligned} \Vert Pf \Vert _{\infty,\phi} &\leq\sup_{z\in{\mathbb {C}}^{n}}e^{-\phi(z)} \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w)f(w) \bigr\vert e^{-2\phi(w)}\,dv(w) \\ &\leq \Vert f \Vert _{\infty,\phi}\sup_{z\in{\mathbb {C}}^{n}}e^{-\phi(z)} \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w) \bigr\vert e^{-\phi(w)}\,dv(w) \\ &\leq C \Vert f \Vert _{\infty,\phi}. \end{aligned}$$
If \(1< p<\infty\), Hölder’s inequality and Fubini’s theorem give
$$\begin{aligned}& \Vert Pf \Vert ^{p}_{p,\phi} \\& \quad \leq \int_{{\mathbb {C}}^{n}}e^{-p\phi(z)}\,dv(z) \biggl( \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w)f(w) \bigr\vert e^{-2\phi(w)}\,dv(w) \biggr)^{p} \\& \quad \leq \int_{{\mathbb {C}}^{n}} \int_{{\mathbb {C}}^{n}} \bigl\vert f(w) \bigr\vert ^{p}e^{-p\phi(w)} \bigl\vert K(z,w) \bigr\vert e^{-\phi(w)}\,dv(w) \bigl\Vert K(z,\cdot) \bigr\Vert _{1,\phi}^{p-1}e^{-p\phi(z)}\,dv(z) \\& \quad \leq C \int_{{\mathbb {C}}^{n}}e^{-\phi(z)}\,dv(z) \int_{{\mathbb {C}}^{n}} \bigl\vert f(w) \bigr\vert ^{p}e^{-p\phi(w)} \bigl\vert K(z,w) \bigr\vert e^{-\phi(w)}\,dv(w) \\& \quad \leq C \int_{{\mathbb {C}}^{n}} \bigl\vert f(w) \bigr\vert ^{p}e^{-p\phi (w)}e^{-\phi(w)}\,dv(w) \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w) \bigr\vert e^{-\phi(z)}\,dv(z) \\& \quad \leq C \Vert f \Vert ^{p}_{p,\phi} \end{aligned}$$
for \(f\in L^{p}(\phi)\). Thus, P is bounded from \(L^{p}(\phi)\) to \(\mathcal{F}^{p}(\phi)\) for \(1\leq p\leq\infty\). The proof is ended. □

In addition, we observe that the Bergman projection is also well defined and bounded on the weighted Fock space \(\mathcal{F}^{p}(\phi )\) with \(p<1\).

Remark 6

For \(p<1\), the Bergman projection P is bounded on \(\mathcal {F}^{p}(\phi)\).

Proof

First, we claim that P is well defined on \(\mathcal {F}^{p}(\phi)\). In fact, given any \(f\in\mathcal{F}^{p}(\phi)\), by (3), (15) and Lemma 2, we obtain
$$\begin{aligned}& \begin{gathered} \int_{{\mathbb {C}}^{n}} \bigl\vert K(z,w)f(w) \bigr\vert e^{-2\phi(w)}\,dv(w) \\ \quad \leq C \Vert f \Vert _{p,\phi} \int_{{\mathbb {C}}^{n}}\rho _{\phi}(w)^{-\frac{2n}{p}} \bigl\vert K(z,w) \bigr\vert e^{-\phi(w)}\,dv(w) \end{gathered} \\& \begin{gathered} \quad \leq Ce^{\phi(z)}\rho_{\phi}(z)^{-n} \int_{{\mathbb {C}}^{n}}\rho _{\phi}(w)^{-\frac{2n}{p}-n}e^{ - \epsilon d(z,w)}\,dv(w) \\ \quad \leq Ce^{\phi(z)}\rho_{\phi}(z)^{-\frac{2n}{p}}< \infty. \end{gathered} \end{aligned}$$
Now, we deal with the boundedness of P. In fact, let \(\{a_{k}\}_{k}\) be the lattice. For \(f\in\mathcal {F}^{p}(\phi)\), we get
$$\begin{aligned} \bigl\vert P f(z) \bigr\vert ^{p}&\leq \Biggl(\sum _{k=1}^{\infty} \int_{B(a_{k})} \bigl\vert f(w)K(w,z) \bigr\vert e^{-2\phi(w)}\,dv(w) \Biggr)^{p} \\ &\leq\sum_{k=1}^{\infty} \biggl( \int_{B(a_{k})} \bigl\vert f(w)K(w,z) \bigr\vert e^{-2\phi(w)}\,dv(w) \biggr)^{p} \\ &\leq\sum_{k=1}^{\infty}v \bigl(B(a_{k}) \bigr)^{p} \Bigl(\sup_{w\in B(a_{k})} \bigl\vert f(w)K(w,z) \bigr\vert e^{-2\phi(w)} \Bigr)^{p}. \end{aligned}$$
Notice that the associated function \(\rho_{2\phi}=\frac{\sqrt {2}}{2}\rho_{\phi}\), which follows from (4). Applying Lemma 4 with weight 2ϕ instead of ϕ, there then is some constant \(C>0\) such that \(\vert P f(z) \vert ^{p}\) is no more than C times
$$\begin{aligned} \sum_{k=1}^{\infty}\rho_{\phi }(a_{k})^{2np-2n} \sup_{w\in B(a_{k})} \int_{B(w)} \bigl\vert f(u) \bigr\vert ^{p} \bigl\vert K(u,z) \bigr\vert ^{p} e^{-2p\phi(u)}\,dv(u). \end{aligned}$$
Combining (7) with (8), we obtain
$$\begin{aligned} \bigl\vert P f(z) \bigr\vert ^{p} &\leq C\sum _{k=1}^{\infty} \int_{B^{m_{2}}(a_{k})}\rho_{\phi }(u)^{2np-2n} \bigl\vert f(u) \bigr\vert ^{p} \bigl\vert K(u,z) \bigr\vert ^{p} e^{-2p\phi(u)}\,dv(u) \\ &\leq C N \int_{{\mathbb {C}}^{n}}\rho_{\phi}(u)^{2np-2n} \bigl\vert f(u) \bigr\vert ^{p} \bigl\vert K(u,z) \bigr\vert ^{p} e^{-2p\phi(u)}\,dv(u). \end{aligned}$$
Therefore, applying Fubini’s theorem and Proposition 3, we get
$$\begin{aligned}& \int_{{\mathbb {C}}^{n}} \bigl\vert P f(z) \bigr\vert ^{p}e^{-p\phi (z)}\,dv(z) \\& \quad \leq C \int_{{\mathbb {C}}^{n}} \int_{{\mathbb {C}}^{n}} \bigl\vert K(u,z) \bigr\vert ^{p} e^{-p\phi(z)}\,dv(z)\rho_{\phi}(u)^{2np-2n} \bigl\vert f(u) \bigr\vert ^{p} e^{-2p\phi(u)}\,dv(u) \\& \quad \leq C \int_{{\mathbb {C}}^{n}} \bigl\vert f(u) \bigr\vert ^{p} e^{-p\phi (u)}\,dv(u). \end{aligned}$$
This means that P is bounded on \(\mathcal{F}^{p}(\phi)\). The proof is ended. □

4 Conclusion

In this paper, we show the boundedness of Bergman projection from the pth Lebesgue space \(L^{p}(\phi)\) to the weighted Fock space \(\mathcal {F}^{p}(\phi)\) for \(1\leq p\leq\infty\). We also remak that the Bergman projection is bounded on \(\mathcal{F}^{p}(\phi)\) with \(p<1\). Meanwhile, we get the estimates for the distance induced by ϕ and the \(L^{p}(\phi)\)-norm of Bergman kernel for \(\mathcal{F}^{2}(\phi )\).

Declarations

Acknowledgements

The author would like to thank the referees for their good suggestions. This work is funded by the National Natural Science Foundation of China (11601149, 11771139, 11571105) and the Natural Science Foundation of Zhejiang province (LY15A010014).

Authors’ contributions

The author wrote this paper by herself. She has read and approved the final manuscript.

Competing interests

The author declares that she has no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, China

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