Characterizations of Fock-type spaces of eigenfunctions on $\mathbb{R}^n$

1.   Introduction

For $n\geq 2$, let $\mathbb{R}^n$ denote the $n$-dimensional real vector space. For two column vectors $x, y \in \mathbb{R}^n$, we use $\langle x, y\rangle$ to denote the inner product of $x$ and $y$. The ball in $\mathbb{R}^n$ with center $a$ and radius $r$ is denoted by $\mathbb{B}(a, r)$. In particular, we write $\mathbb{B} = \mathbb{B}(0, 1)$ and $\mathbb{B}_r = \mathbb{B}(0, r)$. Let $dv$ be the volume measure on $\mathbb{R}^n$ and $d\sigma$ the normalized surface measure on the unit sphere $\mathbb{S} = \partial \mathbb{B}$.

Given $\alpha > 0, m\in \mathbb{N}$ and $t\in \mathbb{R}$, the $t$-weighted $(\alpha, m)$-Gaussian measure $dG_{\alpha, m, t}$ on $\mathbb{R}^n$ is given by

where $C_{\alpha, m, t}$ is the positive constant to be the normalized volume measure. In particular, if $m = 2, t = 0$, $dG_{\alpha, 2, 0}$ is the classical Gaussian measure on $\mathbb{R}^n$ (cf. ^[1]).

For $\lambda\geq 0$, we denote by $H_\lambda(\mathbb{R}^n)$ the set of all eigenfunctions of the Laplacian with eigenvalue $\lambda$ on $\mathbb{R}^n$, i.e.,

where $\Delta$ is the ordinary Laplace operator on $\mathbb{R}^n$. Obviously, if $\lambda = 0$, $H_0(\mathbb{R}^n)$ is the set of all harmonic functions on $\mathbb{R}^n$.

Let $0 < p < \infty$, $s > -1$ and $f$ be a holomorphic function on the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. The famous Hardy-Littlewood theorem asserts that

where $dA$ is the normalized area measure on $\mathbb{C}$ so that $A(\mathbb{D}) = 1$ (cf. ^[2]).

It is known that the integral estimate (1.1) plays an important role in the theory of holomorphic functions. For the generalizations and applications of (1.1) to the spaces of holomorphic functions, harmonic functions, and solutions to certain PDEs, see ^{[3,4,5,6,7,8,9,10]} and the references therein.

Let ${\mathbb C}^n$ be the $n$-dimensional complex vector space. In recent years a special class of holomorphic function spaces, the so-called holomorphic Fock space $\mathbb{F}({\mathbb C}^n)$, has attracted much attention. See ^{[10,11,12,13,14,15,16]} for a summary of recent research on $\mathbb{F}({\mathbb C}^n)$. For $0 < p < \infty$ and $\alpha > 0$, recall that an entire function $f$ on ${\mathbb C}^n$ is said to belong to the Fock space $\mathbb{F}({\mathbb C}^n)$ if

In ^[12], Hu considered an analog of (1.1) in the setting of $\mathbb{F}({\mathbb C}^n)$ and proved that

As a consequence of (1.2), he obtained the boundedness and compactness of Cesàro operators from one Fock space to another. For the further generalizations of (1.2) to holomorphic Fock spaces with some general differential weights, see ^[11,14,15]. By applying these results, Cho et al. characterized Fock-type spaces in terms of Lipschitz type conditions and double integral conditions (cf. ^[13,14]).

Since the eigenfunctions can be viewed as extensions of holomorphic functions on the complex vector space, it is interesting to establish analogous of the equivalence of norms $(1.1)$ and $(1.2)$ in the setting of $H_\lambda({\mathbb R}^n)$. In ^[8], Stoll extended $(1.1)$ to the setting of $H_\lambda({\mathbb B})$ ([8, Theorem 5.1]). Furthermore, by using this result, he established some harmonic majorants criteria for eigenfunctions with finite Dirichlet integrals on a bounded domain $\Omega$ of ${\mathbb R}^n$ ([8, Theorem 5.2]). Motivated by the results in ^{[11,12,13,14]}, we consider a similar norm equivalence $(1.2)$ in the setting of $H_\lambda({\mathbb R}^n)$ in this note.

For $1 < p < \infty$ and $\alpha > 0$, the Fock-type space $F^p_{\alpha, m, t}({\mathbb R}^n)$ consists of all $f\in H_\lambda({\mathbb R}^n)$ such that

Especially, when $m = 2$, $t = \lambda = 0$, $F^p_{\alpha, 2, 0}({\mathbb R}^n)$ becomes the harmonic Fock space (cf. ^[17]).

Theorem 1.1. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t\in {\mathbb R}$. Then

for all $f\in H_\lambda({\mathbb R}^n)$.

As an application of Theorem 1.1, we obtain a Lipschitz type characterization for the Fock-type space $F^p_{\alpha, m, t}({\mathbb R}^n)$.

Theorem 1.2. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $q\geq 0$, $t\in {\mathbb R}$ and $f\in H_\lambda({\mathbb R}^n)$. Then the following two statements are equivalent on ${\mathbb R}^n$:

$(a)$ $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$;

$(b)$ There exists a positive continuous function $g\in L^p(dG_{\alpha p, m, t-pq(m-1)})$ such that

for all $x, y \in {\mathbb R}^n$ with $x\neq y$.

For $m\in \mathbb{N}$, $s\in {\mathbb R}$, $r > 0$ and $f\in H_0({\mathbb R}^n)$ (i.e. $f$ is harmonic), we define

and

where $\chi _{E_r(x)}$ denotes the characteristic function of Euclidean ball $E_r(x)$ (see its definition in Section 2).

In our final result, we discuss the double integral characterization for harmonic Fock-type spaces.

Theorem 1.3. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t, s\in {\mathbb R}$, $q\geq 0$ and $f\in H_0({\mathbb R}^n)$. Then the following statements are equivalent on ${\mathbb R}^n$:

$(a)$ $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$;

$(b)$ $Lf\in L^p(dG_{\alpha p, m, t}\times dG_{\alpha p, m, t})$;

$(c)$ $L_r^s f\in L^p(dG_{\beta p, m, \gamma}\times dG_{\beta p, m, \gamma})$, where $\beta = \frac{s+\alpha}{2}, \gamma = \frac{t-n(m-1)}{2}$.

Lipschitz type characterization for Bergman spaces with standard weights on the unit disc ${\mathbb D}$ in the complex plane ${\mathbb C}$ in terms of the Euclidean, hyperbolic, and pseudo-hyperbolic metrics was original established by Wulan and Zhu ([9, Theorem 1.1]). As an application, double integral characterizations for weighted Bergman spaces in the unit ball in ${\mathbb C}^n$ were proved in ^[19]. For the further generalizations of these results to harmonic Bergman space and holomorphic Fock space, we refer to ^{[3,4,6,13,14]}.

The rest of this paper is organized as follows. In Section 2, some necessary terminology and notation will be introduced. In Section 3, we shall prove Theorem 1.1. The proof of Theorem 1.2 will be presented in Section 4 by applying Theorem 1.1. The final Section 5 is devoted to the proof of Theorems 1.3. Throughout this paper, constants are denoted by $C$, they are positive and may differ from one occurrence to the other. For nonnegative quantities $X$ and $Y$, $X \lesssim Y$ means that $X$ is dominated by $Y$ times some inessential positive constant. We write $X\thickapprox Y$ if $Y\lesssim X\lesssim Y$.

2.   Preliminaries

In this section, we introduce notations and collect some preliminaries results that involve eigenfunctions on ${\mathbb R}^n$.

For $0 < p < \infty$, $\lambda\geq 0$ and $f\in H_\lambda({\mathbb R}^n)$, the $p$-th integral mean of $f$ on $r\mathbb{S}$ is defined as

Lemma 2.1. Let $1\leq p < \infty$, $\lambda\geq 0$ and $f\in H_\lambda({\mathbb B})$. Then both $M^p_p(f, r)$ and $M^p_p(\nabla f, r)$ are increasing with $0 < r < 1$.

Proof. We first prove the monotonicity of $M^p_p(f, r)$. Let $f\in H_\lambda({\mathbb B})$ and $\mathcal{Z}_f$ be the zero set of $f$ on ${\mathbb B}$. Then

which implies that $|f|^p$ is subharmonic on ${\mathbb B} \setminus \mathcal{Z}_f$. Note that at each point of $\mathcal{Z}_f$ the mean value inequality trivially holds, and thus $|f|^p$ is subharmonic on ${\mathbb B}$. It follows from Green's theorem, we know that $M^p_p(f, r)$ is increasing with $0 < r < 1$.

Now we come to prove the monotonicity of $M^p_p(\nabla f, r)$. In view of the definition of $H_\lambda({\mathbb B})$, it is easy to see that if $f\in H_\lambda({\mathbb B})$, then $f\in \mathcal{C}^{\infty}$. This gives

which implies that the partial derivative $\partial_i f$ also belongs to $H_\lambda({\mathbb B})$. By a discussion similar to the above, the monotonicity of $M^p_p(\nabla f, r)$ follows.

For $m\in \mathbb{N}$, $r > 0$ and $a\in {\mathbb R}^n$, the Euclidean ball $E_r(a)$ in ${\mathbb R}^n$ is defined as

Lemma 2.2. Let $m\in \mathbb{N}$, $a\in {\mathbb R}^n$ and $r > 0$. Then for any $x \in E_r(a)$,

Lemma 2.3. Let $1 < p < \infty$, $0 < \alpha < \infty$, $k\in {\mathbb R}, m\in \mathbb{N}$ and $f$ be a locally integrable function on $[0, \infty)$. Then there exists a constant $C$ such that

Proof. Let $\phi(r) = (1+r)^k e^{-\alpha r^m}$, $\varphi(r) = (1+r)^{k-(m-1)p}e^{-\alpha r^m}$ and $p'$ be the conjugate of $p$, i.e., $\frac{1}{p}+\frac{1}{p'} = 1$. By simple computations, we have

and

This gives that

Since $\int_0^\infty \phi(r)dr < \infty$ and $\varphi(r)^{1-p'} \in C[0, R]$ for $R > 0$, it concludes that

Applying Riemann-Liouville integral theorem in ^[20], the assertion of this lemma follows.

We end this section with some inequalities concerning eigenfunctions in $H_\lambda({\mathbb R}^n)$ which are useful for our investigations (cf. ^[8]).

Lemma 2.4. Let $1\leq p < \infty$, $r > 0$ and $f \in H_\lambda({\mathbb R}^n)$. Then there exists some positive constant $C$ such that

3.   Proof of Theorem 1.1

In this section, we divide the proof of Theorem 1.1 into the following two parts.

Proposition 3.1. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t\in {\mathbb R}$. Then

for all $f\in H_\lambda({\mathbb R}^n)$.

Proof. By the subharmonicity of $|f(x)|^p$ and Lemma 2.4, we have

where $\omega_n$ is the volume of the unit ball in ${\mathbb R}^n$. It follows Lemma 2.2, $(3.2)$ can be rewritten as

Combing this with Fubini's theorem, we obtain that

This proves the result.

Proposition 3.2. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t\in {\mathbb R}$. Then

for all $f\in H_\lambda({\mathbb R}^n)$.

Proof. To simplify our notation, set $\partial_\rho f(\rho \zeta) = \frac{\partial f(\rho \zeta)}{\partial \rho}$, where $\rho > 0$ and $\zeta \in \mathbb{S}$. By the fundamental theorem of calculus,

where the last inequality follows from Lemma 2.3.

Hence, by the monotonicity of $M_p^p(\nabla f, r)$, we have

as required. The proof of this proposition is finished.

Proof of Theorem 1.1. Gathering Propositions 3.1 and 3.2, (1.3) follows.

4.   Lipschitz type characterization

In this section, we discuss the Lipschitz type characterization for the space $F^p_{\alpha, m, t}({\mathbb R}^n)$ by applying Theorem 1.1.

For $x\in {\mathbb R}^n, r > 0$ and $m\in \mathbb{N}$, set

Obviously, we have $\Omega_r(x) \subset E_r(x)$.

Proof of Theorem 1.2. We first prove $(b)\Rightarrow (a)$. Assume that $(b)$ holds. Fixing $x$ and letting $y$ approach $x$ in the direction of each real coordinate axis, we get

for each $i\in \{1, 2, ..., n\}$. Thus, we have

and

It follows from the assumption $g\in L^p(dG_{\alpha p, m, t-pq(m-1)})$ that

Hence $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$ by Theorem 1.1.

For the converse, we assume $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$. Fix $r > 0$ and consider any two points $x, y\in {\mathbb R}^n$ with $y\in \Omega_r(x)$. Since $sy+(1-s)x\in E_r(x)$ for $0\leq s \leq 1$, it is given that

Note that for each $\xi \in E_r(x)$,

and thus

where

If $y\notin \Omega_r(x)$, that is,

then the triangle inequality implies

By letting $g(x) = h(x)+\frac{|f(x)|}{\; r(1+|x|^{m-1}~~)^q}$, we obtain

for all $x, y \in {\mathbb R}^n$. It is clear that $\frac{|f(x)|}{\; r(1+|x|^{m-1}~~)^q} \in L^p(dG_{\alpha p, m, t-pq(m-1)})$ from the assumption $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$ and thus $g$ is the desired function provided that $h\in L^p(dG_{\alpha p, m, t-pq(m-1)})$.

Now, we claim that $h\in L^p(dG_{\alpha p, m, t-pq(m-1)})$. From the definition of $E_r(x)$, it is easy for us to find $r_1 > r$ such that $E_r(\xi) \subset E_{r_1}(x)$ for each $\xi \in E_r(x)$. By Lemmas 2.2 and 2.4, we deduces that

Taking the supremum over all $\xi \in E_r(x)$ leads to

Integrating both sides of the above inequality against the measure $dG_{\alpha p, m, t-pq(m-1)}$ and applying Fubini's theorem, we have

It follows from Lemma 2.2 again that

which is what we need.

The proof of Theorem 1.2 is complete.

From the proof of Theorem 1.2, the following local version of Theorem 1.2 can be easily derived for arbitrary $q\in {\mathbb R}$.

Theorem 4.1. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t, q\in {\mathbb R}$ and $f\in H_\lambda({\mathbb R}^n)$. Then the following two statements are equivalent on ${\mathbb R}^n$:

$(a)$ $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$;

$(b)$ There exists a positive continuous function $g\in L^p(dG_{\alpha p, m, t-pq(m-1)})$ such that

for all $x, y \in {\mathbb R}^n$ with $y \in \Omega_r(x)$ and $x \neq y$.

5.   Double integral characterization

In this section, we shall prove Theorem 1.3.

Theorem 5.1. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t\in {\mathbb R}$ and $f\in H_0({\mathbb R}^n)$. Then the following two statements are equivalent on ${\mathbb R}^n$:

$(a)$ $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$;

$(b)$ $Lf\in L^p(dG_{\alpha p, m, t}\times dG_{\alpha p, m, t})$.

Proof. Let $f\in H_0({\mathbb R}^n)$. We first assume that $(a)$ holds. Then

and thus $(b)$ holds.

Conversely, assume $(b)$ holds. Fixing $x\in {\mathbb B}$ and replacing $f$ by $f-f(x)$, it follows from Lemma 2.4, we have

Integrating both sides of the above against the measure $dG_{\alpha p, m, t}(x)$ gives

from which we see that $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$. The proof of this theorem is finished.

Now, we come to characterize $F^p_{\alpha, m, t}({\mathbb R}^n)$ in terms of double integral of $L^s_rf$ as follows.

Theorem 5.2. Let $1 < p < \infty$, $\alpha > 0, m\in \mathbb{N}$, $t, s \in {\mathbb R}$ and $f\in H_0({\mathbb R}^n)$. Then the following two statements are equivalent on ${\mathbb R}^n$:

$(a)$ $f\in F^p_{\alpha, m, t}({\mathbb R}^n)$;

$(b)$ $L_r^s f\in L^p(dG_{\beta p, m, \gamma}\times dG_{\beta p, m, \gamma})$, where $\beta = \frac{s+\alpha}{2}, \gamma = \frac{t-n(m-1)}{2}$.

Proof. Let us first assume that $(a)$ holds. Then

By applying Lemma 2.2 and Fubini's theorem, we conclude that

and

Therefore

Conversely, we assume $(b)$ holds. Fixing $x\in {\mathbb B}$ and $f\in H_0({\mathbb R}^n)$, let

Then it is easy to check that $g_x(y)\in H_0({\mathbb R}^n)$ and $\nabla g_x(x) = \nabla f(x) e^{s|x|^m}$. Applying Lemmas 2.2 and 2.4, we obtain

By integrating both sides of the above against the measure $dG_{(s+\alpha) p, m, t}(x)$ and Lemma 2.2 again, we see that

Hence, by Theorem 1.1, we obtain

The proof of this theorem is complete.

6.   Conclusions

We obtain a norm equivalence for an exponential type weighted integral of an eigenfunction and its derivative on $\mathbb{R}^n$. By using this result, we characterize Fock-type spaces of eigenfunctions on $\mathbb{R}^n$ in terms of Lipschitz type conditions and double integral conditions. All of these results are extensions of the corresponding ones in classcial Fock space.

Acknowledgments

The authors heartily thank the referee for a careful reading of the paper as well as for many useful comments and suggestions.

Conflict of interest

The authors declare that there is no conflicts of interest regarding the publication of this article.