1 Introduction

For $n\geqslant 2$ , let $\mathbb {B}_n=\{x\in \mathbb {R}^n:|x|<1\}$ be the unit ball and $\partial \mathbb {B}_n=\{x\in \mathbb {R}^n:|x|=1\}$ be the unit sphere, where $|x|=(\sum _{i=1}^n|x_i|^2)^{1/2}$ is the standard Euclidean norm on $\mathbb {R}^n$ . The hyperbolic ball is the unit ball $\mathbb {B}_n$ equipped with the hyperbolic metric

$$\begin{align*}ds^2=\frac{4}{(1-|x|^2)^2}\sum\limits_{i=1}^n dx_i^2.\end{align*}$$

For $a\in \mathbb {B}_n$ , the Möbius transformation $\varphi _a$ exchanging a and $0$ in $\mathbb {B}_n$ is the following self-map on $\mathbb {B}_n$ :

$$\begin{align*}\varphi_a(x)=\frac{a|x-a|^2+(1-|a|^2)(a-x)}{|x-a|^2+(1-|a|^2)(1-|x|^2)}. \end{align*}$$

The Laplacian operator $\Delta _h$ with respect to the hyperbolic metric is the linear operator such that

(1.1) $$ \begin{align} (\Delta_hf)(a)=\Delta(f\circ\varphi_a)(0) \end{align} $$

for all $f\in C^2(\mathbb {B}_n)$ and for each $a\in \mathbb {B}_n$ . This operator is also known as the Laplace–Beltrami operator or the invariant Laplace operator. By calculation, it is easy to show that

$$\begin{align*}(\Delta_hf)(a)=(1-|a|^2)^2\Delta f(a)+2(n-2)(1-|a|^2)\langle a,\nabla f(a)\rangle, \end{align*}$$

where

$$\begin{align*}\Delta=\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2},\quad\nabla=\left(\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n}\right)\end{align*}$$

denote the standard Euclidean Laplacian and gradient operators, respectively.

We say that a complex-valued function $f\in C^2(\mathbb {B}_n)$ is $\mathcal {H}$ -harmonic if ${(\Delta _hf)(x)=0}$ for all $x\in \mathbb {B}_n$ . The linear space of $\mathcal {H}$ -harmonic functions on $\mathbb {B}_n$ is denoted by $\mathcal {H}(\mathbb {B}_n)$ . For further background and basic properties of $\mathcal {H}$ -harmonic functions, see [Reference Stoll28, Reference Üreyen29].

Let $\nu $ be the Lebesgue measure on $\mathbb {B}_n$ normalized so that $\nu (\mathbb {B}_n)=1.$ For $1\leqslant p<\infty $ , the $\mathcal {H}$ -harmonic Bergman space $\mathcal {B}^p $ is the Banach space defined by

$$\begin{align*}\mathcal{B}^p =\bigg\{f\in\mathcal{H}(\mathbb{B}_n):\int_{\mathbb{B}_n}|f(x)|^pd\nu (x)<\infty\bigg\}.\end{align*}$$

It is known that the point evaluation functionals $f\mapsto f(x)$ for every $x\in \mathbb {B}_n$ are bounded on $\mathcal {B}^2 $ [Reference Üreyen29]. Therefore, $\mathcal {B}^2 $ is a reproducing kernel Hilbert space. We denote the reproducing kernel by $\mathcal {R} (x,\cdot )\in \mathcal {B}^2 $ . Then, by the reproducing property, we have

(1.2) $$ \begin{align} f(x)=\int_{\mathbb{B}_n} f(y) \overline{\mathcal{R} (x,y)}d\nu (y),\quad\text{for all } f\in\mathcal{B}^2. \end{align} $$

Since the reproducing kernel $\mathcal {R} (\cdot ,\cdot )$ is real-valued [Reference Üreyen29], the conjugation in (1.2) can be removed. The $\mathcal {H}$ -Bergman projection $P : L^2(\mathbb {B}_n,d\nu )\to \mathcal {B}^2 $ is given by

$$\begin{align*}(P f)(x)=\int_{\mathbb{B}_n}f(y) \mathcal{R} (x,y)d\nu (y). \end{align*}$$

It is straightforward to verify that the $\mathcal {H}$ -Bergman projection P is well-defined as an integrable operator for each $x\in \mathbb {B}_n$ and any $f\in L^1(\mathbb {B}_n,d\nu )$ .

In fact, the invariant Laplace operator can also be defined as (1.1) on the Euclidean unit ball of $\mathbb {C}^n$ . The space of $\mathcal {M}$ -harmonic functions, which are annihilated by the invariant Laplace operator on the complex unit ball, has been studied extensively [Reference Ahern, Bruna and Cascante1, Reference Engliš and Youssfi12, Reference Rudin21, Reference Stoll25]. However, functional theory of $\mathcal {H}$ -harmonic function spaces on the real hyperbolic ball is less well-known. The concept of $\mathcal {H}$ -harmonic functions was first introduced by Jaming in 1999 [Reference Jaming18]. In the same paper, he investigated the Hardy spaces $H^p(\mathbb {B}_n),0<p<\infty ,$ whose elements are the hyperbolic harmonic extensions of distributions belonging to the Hardy spaces $H^p(\partial \mathbb {B}_n)$ of the sphere and obtained an atomic decomposition of these spaces. In 2003, Liu and Shi studied the invariant mean-value property of $\mathcal {H}$ -harmonic [Reference Liu and Shi20]. Furthermore, similar spaces defined for holomorphic functions can also be defined for $\mathcal {H}$ -harmonic functions, such as Hardy–Sobolev spaces, Lipschitz spaces [Reference Grellier and Jaming14], and weighted Dirichlet spaces [Reference Stoll26]. Additionally, Stoll obtained the reproducing kernel of the Hardy space and radial eigenfunctions of the invariant Laplacian $\Delta _h$ [Reference Stoll27].

This note focuses on the Bergman spaces of $\mathcal {H}$ -harmonic functions. The Bergman projection on reproducing kernel function spaces plays a crucial role in the study of function spaces and their operators. Among other things, one of the key issues regarding $P $ is the study of its Calderón–Zygmund theory, which includes the $L^p$ -boundedness for all $1\leqslant p\leqslant +\infty $ . In general, related orthogonal projections, such as classical Bergman projection on holomorphic and harmonic function space on $\mathbb {B}_n$ , are weak-type $(1,1)$ [Reference Deng, Huang, Zhao and Zheng9], hence, are bounded on $L^p(\mathbb {B}_n,d\nu )$ for $1<p<+\infty $ by interpolation. Indeed, Üreyen [Reference Üreyen29] proved that P is bounded on $L^p(\mathbb {B}_n,d\nu )$ if and only if $1<p<\infty $ and $P:L^\infty \to \mathcal {B}$ is bounded, where $\mathcal {B}$ is the $\mathcal {H}$ -harmonic Bloch space on $\mathbb {B}_n$ , but leaving the case $p=1$ open. Here, the author proves the weak-type $(1,1)$ estimate, closing the gap.

Theorem 1.1 The $\mathcal {H}$ -harmonic Bergman projection operator P is of weak-type (1,1), i.e., there exists a constant $C>0$ independent of f such that for any $t>0$

(1.3) $$ \begin{align} (P f)_*(t)\leqslant\frac{C}{t}||f||_{L^1(\mathbb{B}_n,d\nu)}, \qquad \text{holds for all }f\in L^1(\mathbb{B}_n,d\nu), \end{align} $$

where $f_*(t)=\nu (\{x\in \mathbb {B}_n:|f(x)|>t\})$ is the distribution function of f.

For a domain $\Omega \subset \mathbb {R}^n$ (or $\mathbb {C}^n$ ), one can see the harmonic (or holomorphic) Bergman spaces $\mathcal {B}^2(\Omega )$ (or $\mathcal {B}_a^2(\Omega )$ ) are reproducing kernel Hilbert spaces with kernel $K_\Omega (x,y)$ (or $k_\Omega (z,w)$ ). Denote the harmonic Bergman projection by ${P_\Omega : L^2(\Omega )\rightarrow \mathcal {B}^2(\Omega )}$ and

$$\begin{align*}P_\Omega f(x)=\int_\Omega f(y)K_\Omega(x,y) dy \end{align*}$$

for $f\in L^2(\Omega )$ (or the holomorphic Bergman projection $P_\Omega : L^2(\Omega )\rightarrow \mathcal {B}_a^2(\Omega )$ and

$$\begin{align*}P_\Omega f(z)=\int_\Omega f(z)k_\Omega(z,w) dw \end{align*}$$

for $f\in L^2(\Omega )$ .)

Since $P_\Omega $ is an orthogonal projection on $\mathcal {B}^2(\Omega )$ (or $\mathcal {B}_a^2(\Omega )$ ), it is a bounded operator on $L^2(\Omega )$ . Unlike the case of the unit ball, where the Bergman projection is $L^p$ -bounded for all $1<p<\infty $ , determining the range $1 < p_\Omega \leqslant q_\Omega < \infty $ for which $P_\Omega $ is $L^p$ -bounded on a general domain $\Omega $ is a nontrivial question. By interpolation and duality, one finds that $1/p_\Omega + 1/q_\Omega = 1$ . Therefore, the $L^p$ -boundedness problem of the Bergman projection for a given domain $\Omega $ reduces to determining the exponent $p_\Omega \in (1,2]$ such that $P_\Omega $ extends continuously to $L^p(\Omega )$ for all $p \in (p_\Omega , p_\Omega ')$ , where $p_\Omega '$ is the Hölder conjugate of $p_\Omega $ .

For a class of domains in complex plane $\mathbb {C}$ , Solovyov characterizes the boundedness of the Bergman projection on $L^p(\Omega )$ by geometric properties of the boundary of $\Omega $ [Reference Solov’ev22, Reference Solov’ev23], which are automatically fulfilled for $p\in (\frac {4}{3},4)$ . For a simply connected domain $\Omega $ in $\mathbb {C}$ , by conformal mapping, Hedenmalm gives a positive answer for some $p_0\in [\frac {4}{3},2)$ [Reference Hedenmalm15]. Here, $p_0$ is independent of $\Omega $ . Edholm and McNeal extend Hedenmalm’s result to some $p_\Omega $ for generalized Hartogs triangles, which are pseudoconvex but non-smooth domains in $\mathbb {C}^2$ [Reference Edholm and McNeal10, Reference Edholm and McNeal11]. Specifically, Chakrabarti and Zeytuncu show $p_\Omega =\frac {4}{3}$ when $\Omega $ is the classical Hartogs triangle [Reference Chakrabarti and Zeytuncu5]. For higher dimensions, such continuous extension of the Bergman projections also holds on monomial polyhedra in $\mathbb {C}^n$ [Reference Bender, Chakrabarti, Edholm and Mainkar2].

Recall that the classical holomorphic Bergman projection on $\mathbb {D}$ is weak-type $(1,1)$ . One may ask how the Bergman projection $P_\Omega $ behaves on $L^{p_\Omega }(\Omega )$ and $L^{p^{\prime }_\Omega }(\Omega )$ , where $p_\Omega $ is the endpoint of the bounded range of $P_\Omega $ . The works of Huo–Wick [Reference Huo and Wick16] and Christopherson–Koenig [Reference Christopherson and Koenig6–Reference Christopherson and Koenig8] show that $P_\Omega $ is not weak-type $(p,p)$ at the lower endpoint of boundedness, while it is weak-type $(p,p)$ at the upper endpoint for the classical and generalized Hartogs triangles, respectively. The mapping behavior of the holomorphic Bergman projection on monomial polyhedra at the lower endpoint is even worse: there is no way to extend the Bergman projection to $L^p(\Omega )$ using its integral representation [Reference Chakrabarti and Edholm3, Proposition 8.1]. In other words, there exists a function $f\in L^p(\Omega )$ such that the integral does not converge. However, it can be shown that on certain monomial polyhedra the Bergman projection is restricted weak-type $(p,p)$ , where p is the lower endpoint of $L^p$ boundedness [Reference Chakrabarti and Huo4].

A similar phenomenon for the harmonic Bergman projection can be observed when $\Omega $ is a punctured smooth domain in $\mathbb {R}^3$ . In such setting, Koenig and Wang show that $P_\Omega $ is bounded on $L^p(\Omega )$ if and only if $\frac {3}{2}<p<3$ [Reference Koenig and Wang19]. For endpoints, $P_\Omega $ is weak-type $(3,3)$ but is not weak-type $(\frac {3}{2},\frac {3}{2})$ . This limitation is missing when $n\neq 3$ , i.e., the harmonic Bergman projection $P_\Omega $ on such punctured domains in $\mathbb {R}^2$ or $\mathbb {R}^n$ , $n\geqslant 4$ , is always bounded on $L^p(\Omega )$ for $1<p<\infty $ . Besides, $P_\Omega $ is not weak-type $(1,1)$ for $n=2$ , while $P_\Omega $ is weak-type $(1,1)$ for $n\geqslant 4$ .