Bergman projections on weighted Fock spaces in several complex variables

Let ϕ be a real-valued plurisubharmonic function on Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb {C}}^{n}$\end{document} whose complex Hessian has uniformly comparable eigenvalues, and let Fp(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}^{p}(\phi)$\end{document} be the Fock space induced by ϕ. In this paper, we conclude that the Bergman projection is bounded from the pth Lebesgue space Lp(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}(\phi )$\end{document} to Fp(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}^{p}(\phi)$\end{document} for 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq p \leq\infty$\end{document}. As a remark, we claim that Bergman projections are also well defined and bounded on Fock spaces Fp(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}^{p}(\phi)$\end{document} with 0<p<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p<1$\end{document}. We also obtain the estimates for the distance induced by ϕ and the Lp(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}(\phi)$\end{document}-norm of Bergman kernel for F2(ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}^{2}(\phi)$\end{document}.


Introduction
The symbol dv denotes the Lebesgue volume measure on C n , and B(z, r) = w ∈ C n : |w -z| < r for z ∈ C n and r > .
Suppose φ : C n → R is a C  plurisubharmonic function. We say that φ belongs to the weight class W if φ satisfies the following statements: (I) There exists c >  such that for z ∈ C n inf z∈C n sup w∈B (z,c) φ(w) > ; () (II) φ satisfies the reverse-Hölder inequality φ L ∞ (B(z,r)) ≤ Cr -n for some  < C < +∞; (III) The eigenvalues of H φ are comparable, i.e., there exists δ  >  such that where H φ = ∂  φ ∂z j ∂z k j,k .
Suppose  < p < ∞, φ ∈ W. The space L p (φ) consists of all Lebesgue measurable functions f on C n for which L ∞ (φ) is the set of all Lebesgue measurable functions f on C n with Let H(C n ) be the family of all holomorphic functions on C n . The weighted Fock space is defined as is the classical Fock space which has been studied by many authors, see [-] and the references therein. Notice that the weight function ϕ on C n with the restriction that dd c ϕ dd c |z|  in [] and [] belongs to W.
In the one-dimensional case, an important contribution to weighted Fock spaces was given by Christ [] (but see also [, ]). They work under the assumption that φ is subharmonic and that φ dA is a doubling measure, where dA is the area measure on C. Notice that the hypotheses on φ 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 []. Dall' Ara [] tried to extend Christ's approach to n ≥ . Given φ ∈ W, let K(·, ·) be the weighted Bergman kernel for F  (φ). In particular, Theorem  of [] proves that there is a constant C, >  such that for z, w ∈ C n , where d(·, ·), ρ φ (·) described in Section .
In the setting of Bergman spaces, the Bergman projection is bounded on p-Bergman spaces for  < p < ∞, it also maps L ∞ into Bloch spaces, see [] for details. With the Bergman kernel K(·, ·) for F  (φ), the Bergman projection P can be represented as It is well known that P(f ) = f for f ∈ F  (φ). The purpose of this work is to discuss the boundedness of Bergman projection acting on F p (φ) for general p. Section  is devoted to some basic estimates, including the distance d(·, ·) and the L p (φ)-norm of the Bergman kernel. In Section , we will discuss the boundedness of Bergman projections from L p (φ) to F p (φ) with  ≤ p ≤ ∞. We also show that the Bergman projection is well defined and bounded on F p (φ) for p < .
In what follows, we always suppose φ ∈ 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 B' , if there exists some C such that C - A ≤ B ≤ CA.

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 ∈ C n , set We then have some M >  such that Moreover, there are some positive constants C, M  and M  such that for all z, w ∈ C n , we have where θ = max(, |z-w| ρ φ (w) ). We can see this in Proposition  of []. Given r > , write By () and the triangle inequality, we have m  , m  >  such that k= are pairwise disjoint. This lattice exists by a standard covering lemma, see Theorem . in [], or Proposition  in [] as well. Moreover, for the lattice {a k } k and any m > , there exists some integer N such that each z ∈ C n can be in at most To the radius function ρ φ defined as (), we associate the Riemannian metric ρ φ (z) - dz ⊗ dz. In fact, we are interested only in the associated Riemannian distance, which we describe explicitly. If γ : [, ] → C n is piecewise C  curves, we define dt.
Given z, w ∈ C n , we put where the inf is taken as γ varies over the collection of curves with γ () = z and γ () = w. We then have the estimate for this distance as follows.
Proof First, we claim that there is some C >  such that In fact, set μ to be μ B(z, r) = r  φ L ∞ (B(z,r)) , z ∈ C n , r > .
By (), it is easy to check that there is some M >  such that because of (). Given any r ≤ R, it is easy to check that for z ∈ C n because of (). Also, there is a positive integer m such that  m- r < R ≤  m r. Hence, () and () tell us For z, w ∈ C n , notice that B(w, |w -z|) ⊂ B(z, |w -z|). If |w -z| < ρ φ (z), take any piecewise C  curve γ : [, ] → C n connecting z and w, and let T  be the minimum time such that |zγ (T  )| = ρ φ (z). By (), ρ φ (γ (t)) ρ φ (z) for t ∈ [, T  ). This implies If |z -w| ≥ ρ φ (w), then (), (), () and () give On the other hand, for ζ ∈ B(z, Combining the above with (), we know By the fact log  M > , (), () and (), there exists t >  such that Hence, ( |z-w| ρ φ (w) )  ≤ C( |z-w| ρ φ (ζ ) ) t . This implies where α =  t > . For any piecewise C  curves , defined as γ : [, ] → C n with γ () = z and γ () = w, we have This yields () is true. Now, we are going to prove the other direction. For z, w ∈ C n , take where β > . The proof is completed. Now, we can estimate the following integral.
Lemma  Given p >  and k ∈ R, we have where C >  is a constant depending only on n, p and k.
Proof By (), it is easy to check that By (), the inequality above is no more than Therefore, The proof is completed.
Next, we will give the L p (φ)-norm of the Bergman kernel K(·, ·) for F  (φ).
Proof By () and Lemma , we obtain The proof is completed.
Lemma  For  < p < ∞, there is a constant C >  such that for all r ∈ (, ], f ∈ H(C n ) and z ∈ C n , we have

Boundedness of Bergman projections
Recall that the Bergman projection P on L p (φ) is defined as In this section, we focus on the boundedness of Bergman projections P from L p (φ) to F p (φ) for  ≤ p ≤ ∞.
Theorem  Let  ≤ p ≤ ∞. Then the Bergman projection P is bounded as a map from L p (φ) to F p (φ).
Proof By the definition of P, we can conclude Pf is holomorphic on C n . Fubini's theorem and Proposition  yield If  < p < ∞, Hölder's inequality and Fubini's theorem give for f ∈ L p (φ). Thus, P is bounded from L p (φ) to F p (φ) for  ≤ p ≤ ∞. The proof is ended.
In addition, we observe that the Bergman projection is also well defined and bounded on the weighted Fock space F p (φ) with p < .
Remark  For p < , the Bergman projection P is bounded on F p (φ).
Proof First, we claim that P is well defined on F p (φ). In fact, given any f ∈ F p (φ), by (), () and Lemma , we obtain C n K(z, w)f (w) e -φ(w) dv(w) ≤ C f p,φ C n ρ φ (w) -n p K(z, w) e -φ(w) dv(w)