May 22, 2026
In this work, we introduce Orlicz-Hardy type spaces and Orlicz-Calderón Hardy type spaces on the Heisenberg group \(\mathbb{H}^{n}\) and study the relationship between them by means of the Heisenberg sub-Laplacian \(\mathcal{L}\). More precisely, we show, under suitable assumptions, that every distribution in the Orlicz-Hardy space \(H^{\Phi}(\mathbb{H}^{n})\) can be represented uniquely as the sub-Laplacian of a function in an appropriate Orlicz-Calderón Hardy space. In this way, for any \(f \in H^{\Phi}(\mathbb{H}^{n})\), we obtain a uniqueness and solvability result for the equation \(\mathcal{L}F=f\).
On the Heisenberg group \(\mathbb{H}^n\), we consider the following inhomogeneous equation \[\label{L32problem} \mathcal{L} F = f,\tag{1}\] where \(\mathcal{L}\) is the Heisenberg sub-Laplacian, \(f\) is a data distribution on \(\mathbb{H}^n\) and \(F\) is an unknown function. Recently, the author in [1] proved that if \(f \in H^p(\mathbb{H}^n)\) with \(Q \, (2 + \frac{Q}{q})^{-1} < p \leq 1\) (where \(Q = 2n+2\) and \(1 < q < \frac{n+1}{n}\)), then there exists a unique \(F \in \mathcal{H}^{p}_{q, 2}(\mathbb{H}^n)\) what solves (1 ). Here, \(\mathcal{H}^{p}_{q, 2}(\mathbb{H}^n)\) is the Calderón-Hardy space on \(\mathbb{H}^n\). This problem was also posed in the variable context. In [2], we solved the equation (1 ) for \(f \in H^{p(\cdot)}(\mathbb{H}^n)\), and certain variable exponents \(p(\cdot) : \mathbb{H}^n \to (0, \infty)\). In this work, for an Orlicz function \(\Phi\), we introduce the Orlicz-Hardy spaces on the Heisenberg group \(H^{\Phi}(\mathbb{H}^n)\) and posed the equation (1 ) for \(f \in H^{\Phi}(\mathbb{H}^n)\). Naturally, the classical Hardy spaces \(H^{p}(\mathbb{H}^n)\), \(0 < p < \infty\), are variable Hardy spaces and Orlicz-Hardy spaces. These last two spaces have different nature. That is, \(H^{p(\cdot)}(\mathbb{H}^n)\) mixes the value of \(p\) according to the position of \(z\) and \(H^{\Phi}(\mathbb{H}^n)\) mixes the value of \(p\) according to the value of \(f(z)\).
The counterpart of the equation (1 ) on \(\mathbb{R}^n\) was first considered by A. Gatto, J. Jiménez and C. Segovia in [3]. More precisely, they posed, for \(0< p \leq 1\), \(m \in \mathbb{N}\) and \(f \in H^p(\mathbb{R}^n)\) (see [4]), the equation \[\label{delta32problem} \Delta^m F = f,\tag{2}\] where \(\Delta\) is the Laplace operator on \(\mathbb{R}^n\). To address this problem, they introduced the Calderón-Hardy spaces \(\mathcal{H}^p_{q, \gamma}(\mathbb{R}^n)\), \(0 < p \leq 1 < q < \infty\) and \(\gamma > 0\), and proved for \(n(2m + n/q)^{-1} < p \leq 1\) that given \(f \in H^p(\mathbb{R}^n)\) there exists a unique \(F \in \mathcal{H}^p_{q, 2m}(\mathbb{R}^n)\) what solves (2 ).
In [5], R. Durán extended the definition of \(\mathcal{H}^p_{q, 2m}(\mathbb{R}^n)\) to the case of non-isotropic dilations on \(\mathbb{R}^n\), solving the problem (2 ) for more general elliptic operators with symbols of the form \(\xi_1^{2k_1} + \cdot \cdot \cdot + \xi_n^{2k_n}\), with \(k_1, ..., k_n \in \mathbb{N}\).
The equation (2 ), for \(f \in H^{p(\cdot)}(\mathbb{R}^{n})\) and \(f \in H^{p}(\mathbb{R}^{n}, w)\), was studied by the present author in [6] and [7] respectively, obtaining analogous results to those of Gatto, Jiménez and Segovia.
Recently, Z. Liu, Z. He and H. Mo in [8] extended the definition of Calderón-Hardy spaces to Orlicz setting on \(\mathbb{R}^n\) and solved the equation (2 ) when \(f \in H^{\Phi}(\mathbb{R}^{n})\), where \(H^{\Phi}(\mathbb{R}^{n})\) are the Orlicz-Hardy spaces defined in [9].
The purpose of this work is to solve the equation (1 ) for \(f \in H^{\Phi}(\mathbb{H}^n)\) and \(Q \, (2 + \frac{Q}{q})^{-1} < i(\Phi) \leq I(\Phi) < \infty\), where \(Q= 2n+2\), \(1 < q < \frac{n+1}{n}\) and the index \(i(\Phi)\) and \(I(\Phi)\) are given by (6 ) and (7 ) below. Once defined the spaces \(H^{\Phi}(\mathbb{H}^n)\) and \(\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\), we will prove that \(f\) has an atomic decomposition \(f = \sum_j \lambda_j a_j\) in \(\mathcal{S}'(\mathbb{H}^n)\). Then, as in [1], we consider the following potential of \(f\), \[f \ast c_n \rho^{-2n} = \sum_j \lambda_j (a_j \ast c_n \rho^{-2n}),\] where "\(\ast\)" is the convolution in \(\mathbb{H}^n\) and \(c_n \rho^{-2n}\) is the fundamental solution of \(\mathcal{L}\) obtained by G. Folland in [10], and show that a representative for the solution \(F \in \mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\) of (1 ) is \(\sum_j \lambda_j (a_j \ast_{\mathbb{H}^n} c_n \rho^{-2n})\). On the other hand, the case \(0 < I(\Phi) < Q \, (2 + \frac{Q}{q})^{-1}\) is trivial. Indeed, we have \(\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n}) = \{ 0\}\), when \(0 < I(\Phi) < Q \, (2 + \frac{Q}{q})^{-1}\). These results is contained in Theorems 25 and 26 of Section 6, respectively.
Although the fundamental solutions for the iterated Heisenberg sub-Laplacian \(\mathcal{L}^m\) are known for every integer \(m \geq 2\) (see [11]), the problem \(\mathcal{L}^m F = f\) on \(\mathbb{H}^n\) is much more complicated. For this reason we focus solely on the case \(m=1\).
The tools used in this work are the atomic series for elements in \(H^{\Phi}(\mathbb{H}^n)\) and the fundamental solution of \(\mathcal{L}\) already mentioned, together with the complementary function \(\Phi^{*}\), a Fefferman-Stein inequality for the Hardy-Littlewood maximal operator, and certain pointwise inequalities established in Section 5, among them, (?? ) is the most important.
Our results apply to the following Orlicz functions:
(i) \(\Phi(t) = t^p\), with \(t \geq 0\) and \(0 < p < \infty\). It this case, \(L^{\Phi}(\mathbb{H}^n) \equiv L^{p}(\mathbb{H}^n)\).
(ii) \(\Phi(t) = t^{p_1} + t^{p_2}\), with \(t \geq 0\) and \(0 < p_1 \leq p_2 < \infty\). In this case, \(L^{\Phi}(\mathbb{H}^n)\) is isomorphic to \(L^{p_1}(\mathbb{H}^n) \cap L^{p_2}(\mathbb{H}^n)\).
(iii) \(\Phi(t) = \min \{ t^{p_1}, t^{p_2} \}\), with \(t \geq 0\) and \(0 < p_1 \leq p_2 < \infty\). In this case, \(L^{\Phi}(\mathbb{H}^n)\) is isomorphic to \(L^{p_1}(\mathbb{H}^n) + L^{p_2}(\mathbb{H}^n)\).
(iv) \(\Phi(t) = t \log(e + t)\), with \(t \geq 0\). In this case, \(L^{\Phi}(\mathbb{H}^n)\) is isomorphic to \(L \log L (\mathbb{H}^n)\).
We observe that the functions (i)-(iv) are all of positive lower and upper type (see Definition 4 below).
This paper is organized as follows. In Section 2 we state the basics of the Heisenberg group. The properties of Orlicz functions and Orlicz spaces are presented in Section 3. The definition of Orlicz-Hardy spaces on the Heisenberg group as well as its atomic decomposition are presented in Section 4. We introduce the Orlicz-Calderón Hardy spaces on the Heisenberg group and investigate their properties in Section 5. Finally, our main results are proved in Section 6.
Notation: The symbol \(S \lesssim T\) stands for the inequality \(S \leq c T\) for some constant \(c\). The symbol \(S \approx T\) stands for \(T \lesssim S \lesssim T\). For a measurable subset \(E\subseteq \mathbb{H}^{n}\) we denote by \(\left\vert E\right\vert\) and \(\chi_{E}\) the Haar measure of \(E\) and the characteristic function of \(E\) respectively. Given a real number \(s \geq 0\), we write \(\lfloor s \rfloor\) for the integer part of \(s\).
Throughout this paper, \(C\) will denote a positive constant, not necessarily the same at each occurrence.
The Heisenberg group \(\mathbb{H}^{n}\) can be identified with \(\mathbb{R}^{2n} \times \mathbb{R}\) whose group law (noncommutative) is given by \[(x,t) \cdot (y,s) = \left( x+y, t+s + x^{t} J y \right),\] where \(J\) is the \(2n \times 2n\) skew-symmetric matrix given by \[J= 2 \left( \begin{array}{cc} 0 & -I_n \\ I_n & 0 \\ \end{array} \right)\] being \(I_n\) the \(n \times n\) identity matrix. One can easily check that \(e = (0,0)\) is the neutral element, and \((x, t)^{-1}=(-x, -t)\) is the inverse of \((x, t)\).
The Heisenberg group \(\mathbb{H}^n\) also admits the following homogeneous dilation: \[\lambda \cdot (x,t) = (\lambda x, \lambda^{2}t), \,\,\,\;\lambda > 0,\] which satisfies \(\lambda \cdot((x,t) \cdot (y,s)) = (\lambda \cdot(x,t)) \cdot (\lambda \cdot(y,s))\).
The Koranyi norm on \(\mathbb{H}^{n}\) is the function \(\rho : \mathbb{H}^{n} \to [0, \infty)\) defined by \[\label{Koranyi32norm} \rho(z) = \rho(x,t) = \left( |x|^{4} + \, t^{2} \right)^{1/4}, \,\,\,\, z=(x,t) \in \mathbb{H}^{n},\tag{3}\] where \(| \cdot |\) is the usual Euclidean norm on \(\mathbb{R}^{2n}\). It is easy to check that for any \(z, w \in \mathbb{H}^n\) and \(\lambda > 0\),
(i) \(\rho(\lambda \cdot z) = \lambda \rho(z)\),
(ii) \(\rho(z + w) \leq \rho(z) + \rho(w)\),
(iii) \(|x| \leq \rho(x,t)\) and \(|t| \leq \rho(x,t)^2\).
Moreover, \(\rho\) is continuous on \(\mathbb{H}^{n}\) and is smooth on \(\mathbb{H}^{n} \setminus \{ e \}\). The \(\rho\) -
ball centered at \(z_0 \in \mathbb{H}^{n}\) with radius \(\delta > 0\) is defined by \[B(z_0, \delta) := \{ w \in \mathbb{H}^{n} : \rho(z_0^{-1} \cdot w) <
\delta \}.\]
The topology in \(\mathbb{H}^{n}\) induced by the \(\rho\) - balls coincides with the Euclidean topology of \(\mathbb{R}^{2n} \times \mathbb{R} \equiv\mathbb{R}^{2n+1}\) (see [12]). So, the borelian sets of \(\mathbb{H}^{n}\) are identified with those of \(\mathbb{R}^{2n+1}\). The Haar measure in \(\mathbb{H}^{n}\) is the Lebesgue measure of \(\mathbb{R}^{2n+1}\), thus \(L^{p}(\mathbb{H}^{n}) \equiv L^{p}(\mathbb{R}^{2n+1})\), for every \(0 < p \leq \infty\). Moreover, for \(f \in L^{1}(\mathbb{H}^{n})\) and for \(r > 0\) fixed, we have \[\label{homog32dim} \int_{\mathbb{H}^{n}} f(r \cdot z) \, dz = r^{-Q} \int_{\mathbb{H}^{n}} f(z) \, dz,\tag{4}\] where \(Q= 2n+2\). The number \(2n+2\) is known as the homogeneous dimension of \(\mathbb{H}^{n}\) (we observe that the topological dimension of \(\mathbb{H}^{n}\) is \(2n+1\)).
Let \(|B(z_0, \delta)|\) be the Haar measure of the \(\rho\) - ball \(B(z_0, \delta) \subset \mathbb{H}^{n}\). Then, \[|B(z_0, \delta)| = c \delta^{Q},\] where \(c = |B(e,1)|\) and \(Q = 2n+2\). Given \(\lambda > 0\), we put \(\lambda B = \lambda B(z_0, \delta) = B(z_0, \lambda \delta)\). So \(|\lambda B| = \lambda^{Q} |B|\).
Remark 1. For any \(z, z_0 \in \mathbb{H}^{n}\) and \(\delta >0\), we have \[z_0 \cdot B(z, \delta) = B(z_0 \cdot z, \delta).\] In particular, \(B(z, \delta) = z \cdot B(e, \delta)\). It is also easy to check that \(B(e, \delta) = \delta \cdot B(e,1)\) for any \(\delta > 0\).
Remark 2. If \(f \in L^{1}(\mathbb{H}^{n})\), then for every \(\rho\) - ball \(B\) and every \(z_0 \in \mathbb{H}^{n}\), we have \[\int_{B} f(w) \, dw = \int_{z_{0}^{-1} \cdot B} f(z_0 \cdot u) \, du.\]
Definition 1. A function \(f : \mathbb{H}^n \to \mathbb{C}\) is said to be radial if there exists a function \(f_0 : [0, \infty) \to \mathbb{C}\) such that \[f(x,t) = f_0(\rho(x,t)), \,\,\,\, \text{for all} \,\, (x,t) \in \mathbb{H}^n.\]
Remark 3. We observe that the literature is not unanimous in the use of this terminology. Sometimes, a function \(f : \mathbb{H}^n \to \mathbb{C}\) is said to be radial if there exists \(f_0 : [0, \infty) \times \mathbb{R} \to \mathbb{C}\) such that \(f(x,t) = f_0 (|x|, t)\).
The Hardy-Littlewood maximal operator \(M\) is defined by \[Mf(z) = \sup_{B \ni z} |B|^{-1}\int_{B} |f(w)| \, dw,\] where \(f\) is a locally integrable function on \(\mathbb{H}^{n}\) and the supremum is taken over all the \(\rho\) - balls \(B\) containing \(z\).
If \(f\) and \(g\) are measurable functions on \(\mathbb{H}^{n}\), their convolution \(f * g\) is defined by \[(f * g)(z) := \int_{\mathbb{H}^{n}} f(w) g(w^{-1} \cdot z) \, dw,\] when the integral is finite.
For every \(i = 1,2, ..., 2n+1\), \(X_i\) denotes the left invariant vector field given by \[X_i = \frac{\partial}{\partial x_i} + 2 x_{i+n} \frac{\partial}{\partial t}, \,\,\,\, i=1, 2, ..., n;\] \[X_{i+n} = \frac{\partial}{\partial x_{i+n}} - 2 x_{i} \frac{\partial}{\partial t}, \,\,\, i=1, 2, ..., n;\] and \[X_{2n+1} = \frac{\partial}{\partial t}.\] The sublaplacian on \(\mathbb{H}^n\), denoted by \(\mathcal{L}\), is the counterpart of the Laplacain \(\Delta\) on \(\mathbb{R}^n\). The sublaplacian \(\mathcal{L}\) is defined by \[\mathcal{L} =- \sum_{i=1}^{2n} X_i^2,\] where \(X_i\), \(i= 1, ..., 2n\), are the left invariant vector fields defined above.
Given a multi-index \(I=(i_1,i_2, ..., i_{2n}, i_{2n+1}) \in (\mathbb{N} \cup \{ 0 \})^{2n+1}\), we set \[|I| = i_1 + i_2 + \cdot \cdot \cdot + i_{2n} + i_{2n+1},d(I) = i_1 + i_2 + \cdot \cdot \cdot + i_{2n} + 2 \, i_{2n+1}.\] The amount \(|I|\) is called the length of \(I\) and \(d(I)\) the homogeneous degree of \(I\). We adopt the following multi-index notation for higher order derivatives and for monomials on \(\mathbb{H}^{n}\). If \(I=(i_1, i_2, ..., i_{2n+1})\) is a multi-index, \(z = (x,t) = (x_1, ..., x_{2n}, t) \in \mathbb{H}^{n}\) and \(X = \{ X_i \}_{i=1}^{2n+1}\), we put \[z^{I} := x_{1}^{i_1} \cdot \cdot \cdot x_{2n}^{i_{2n}} \cdot t^{i_{2n+1}} \,\,\,\,\,\, \text{and} \,\,\,\,\,\, X^{I} := X_{1}^{i_1} X_{2}^{i_2} \cdot \cdot \cdot X_{2n+1}^{i_{2n+1}}.\] A computation give \[(r\cdot z)^{I} = r^{d(I)} z^{I} \,\,\,\,\,\, \text{and} \,\,\,\,\,\, X^{I}(f(r \cdot z)) = r^{d(I)} (X^{I}f)(r\cdot z).\] So, the operators \(X^{I}\) and the monomials \(z^{I}\) are homogeneous of degree \(d(I)\). In particular, the sublaplacian \(\mathcal{L}\) is an operator homogeneous of degree \(2\). The operators \(X^{I}\), and \(\mathcal{L}\) interact with the convolutions in the following way \[X^{I}(f \ast g) = f \ast (X^{I}g), \,\,\,\,\,\, \text{and} \,\,\,\,\,\, \mathcal{L} (f \ast g) = f \ast \mathcal{L}g.\]
We recall that every polynomial \(p\) on \(\mathbb{H}^n\) can be written as a unique finite linear combination of the monomials \(z^I\), that is \[\label{polynomial} p(z) = \sum_{I \in \mathbb{N}_{0}^n} c_I \, z^I,\tag{5}\] where all but finitely many of the coefficients \(c_I \in \mathbb{C}\) vanish. The homogeneous degree of a polynomial \(p\) written as (5 ) is \(\max \{ d(I) : I \in \mathbb{N}_{0}^n \,\, \text{with} \,\, c_I \neq 0 \}\). Let \(k \in \mathbb{N} \cup \{ 0 \}\), with \(\mathcal{P}_{k}\) we denote the subspace formed by all the polynomials of homogeneous degree at most \(k\). So, every \(p \in \mathcal{P}_{k}\) can be written as \(p(z) = \sum_{d(I) \leq k} c_I \, z^I\), with \(c_I \in \mathbb{C}\).
The Schwartz space \(\mathcal{S}(\mathbb{H}^{n})\) is defined by \[\mathcal{S}(\mathbb{H}^{n}) = \left\{ \phi \in C^{\infty}(\mathbb{H}^{n}) : \sup_{z \in \mathbb{H}^{n}} (1+\rho(z))^{N} |(X^{I} \phi)(z)| < \infty \,\,\, \forall \,\, N \in \mathbb{N}_{0}, \, I \in (\mathbb{N}_{0})^{2n+1} \right\}.\] We topologize the space \(\mathcal{S}(\mathbb{H}^{n})\) with the following family of seminorms \[\| \phi \|_{\mathcal{S}(\mathbb{H}^{n}), \, N} := \sum_{d(I) \leq N} \sup_{z \in \mathbb{H}^{n}} (1+\rho(z))^{N} |(X^{I} \phi)(z)| \,\,\,\,\,\,\, (N \in \mathbb{N}_{0}),\] with \(\mathcal{S}'(\mathbb{H}^{n})\) we denote the dual space of \(\mathcal{S}(\mathbb{H}^{n})\).
A fundamental solution for the sublaplacian on \(\mathbb{H}^n\) was obtained by G. Folland in [10]. More precisely, he proved the following result.
Theorem 4. \(c_n \, \rho^{-2n}\) is a fundamental solution for \(\mathcal{L}\) with source at \(0\), where \[\rho(x,t) = (|x|^4 + t^2)^{1/4},\] and \[c_n = \left[ n(n+2) \int_{\mathbb{H}^n} |x|^2 (\rho(x,t)^4 + 1)^{-(n+4)/2} dxdt \right]^{-1}.\] In others words, for any \(u \in \mathcal{S}(\mathbb{H}^{n})\), \(\left( \mathcal{L}u, c_n \rho^{-2n} \right) = u(0)\).
We start this section with some basic notions and results about Orlicz functions and Orlicz spaces on \(\mathbb{H}^n\) (see e.g. [13]).
Definition 2. A function \(\Phi : [0, \infty) \to [0, \infty)\) is called an Orlicz function if
(i) it is non-decreasing and satisfies \(\lim_{t \to 0^{+}} \Phi(t) = \Phi(0) = 0\), \(\Phi(t) > 0\) for all \(t>0\) and \(\lim_{t \to \infty} \Phi(t) = \infty\);
(ii) the function \(z \to \Phi(\vert f(z) \vert)\) is measurable for every measurable function \(f\) on \(\mathbb{H}^n\).
Definition 3. Let \(\Phi\) be an Orlicz function, the Orlicz space \(L^{\Phi}(\mathbb{H}^n)\) is defined to be the set of all measurable functions \(f\) on \(\mathbb{H}^n\) such that \[\Vert f \Vert_{\Phi} := \inf \left\{ \lambda > 0 : \int_{\mathbb{H}^n} \Phi(\vert f(z) \vert/ \lambda) dz \leq 1 \right\} < \infty.\] The amount \(\Vert f \Vert_{\Phi}\) is known as the Luxemburg norm of \(f\) with respect to \(\Phi\).
Given an Orlicz function \(\Phi\) and any \(s \in (0, \infty)\) fixed, we define \[\Phi_s(t) := \Phi(t^s), \,\,\,\, t \in (0, \infty).\]
Lemma 1. Let \(\Phi\) be an Orlicz function. Then, for any \(s \in (0, \infty)\) and any measurable function \(f\), one has \[\Vert f \Vert_{\Phi}^{s} = \Vert \vert f \vert^s \Vert_{\Phi_{1/s}}.\]
Proof. Fix \(s \in (0, \infty)\). Then, by definition \[\Vert \vert f \vert^s \Vert_{\Phi_{1/s}} = \inf \left\{ \lambda > 0 : \int_{\mathbb{H}^n} \Phi(\vert f(z) \vert/ \lambda^{1/s}) dz \leq 1 \right\}\] \[= \inf \left\{ \mu^s > 0 : \int_{\mathbb{H}^n} \Phi(\vert f(z) \vert/ \mu) dz \leq 1 \right\} = \Vert f \Vert_{\Phi}^{s}.\] Then, the lemma follows. ◻
Lemma 2. Let \(\Phi\) be a continuous Orlicz function. If \(0 \neq f \in L^{\Phi}(\mathbb{H}^n)\), then \[\int_{\mathbb{H}^n} \Phi \left( \| f \|^{-1}_{\Phi} |f(z)| \right) dz \leq 1.\]
Proof. From the definition of the amount \(\| f \|_{\Phi}\), there exists a positive decreasing sequence \(\{ \alpha_j \}\) such that \(\int_{\mathbb{H}^n} \Phi \left( \alpha^{-1}_j |f(z)| \right) dz \leq 1\) for all \(j\) and \(\alpha_j \to \| f \|_{\Phi} > 0\). Finally, by Fatou’s lemma and the continuity of the Orlicz function \(\Phi\), we get \[\int_{\mathbb{H}^n} \Phi \left( \| f \|^{-1}_{\Phi} |f(z)| \right) dz \leq \limsup_{j \to \infty} \int_{\mathbb{H}^n} \Phi \left( \alpha^{-1}_j |f(z)| \right) dz \leq 1.\] This finishes the proof. ◻
Definition 4. An Orlicz function \(\Phi : [0, \infty) \to [0, \infty)\) is said to be of positive lower (respectively, positive upper) type \(p\) with \(p \in (0, \infty)\) if there exists a positive constant \(C\), depending on \(p\), such that, for any \(t >0\) and \(r \in (0,1]\) (respectively, \(r \in [1, \infty)\)),
\[\label{lower32upper32type} \Phi(rt) \leq C r^p \Phi(t).\qquad{(1)}\]
Lemma 3. Let \(\Phi\) be an Orlicz function with positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\). Then,
(i) \(\Phi(t) \gtrsim \, t^{p_{\Phi}^{-}}\), if \(t \geq 1\),
(ii) \(\Phi(t) \gtrsim \, t^{p_{\Phi}^{+}}\), if \(0 < t \leq 1\).
Proof. Write \(\Phi(1) = \Phi(t^{-1} t)\) and apply (?? ) according to the case. ◻
Given an Orlicz function \(\Phi\), for any measurable function \(f\) on \(\mathbb{H}^n\), we set \[\kappa_{\Phi}(f) := \int_{\mathbb{H}^n} \Phi(|f(z)|) dz.\]
Lemma 4. Let \(\Phi\) be an Orlicz function of positive upper type \(p_{\Phi}^{+}\). Then, \(f \in L^{\Phi}(\mathbb{H}^n)\) if and only if \(\kappa_{\Phi}(f) < \infty\).
Proof. Assume that \(f \neq 0\). If \(\kappa_{\Phi}(f) < \infty\), by the monotone convergence theorem, we obtain \(f \in L^{\Phi}(\mathbb{H}^n)\). Now, if \(f \in L^{\Phi}(\mathbb{H}^n)\), there exists \(\lambda > 1\) such that \(\kappa_{\Phi}(f/\lambda) \leq 1\). Then, (?? ) leads to \[\kappa_{\Phi}(f) = \kappa_{\Phi}(\lambda f/ \lambda) \leq C \lambda^{p_{\Phi}^{+}} \kappa_{\Phi}(f/\lambda) \leq C \lambda^{p_{\Phi}^{+}} < \infty.\] This concludes the proof. ◻
Lemma 5. Let \(\Phi\) be an Orlicz function of positive upper type \(p_{\Phi}^{+}\). If \(\{ f_j \}\) is a sequence of measurable functions on \(\mathbb{H}^n\) such that \(\kappa_{\Phi}(f_j) \to 0\), then \(\| f_j \|_{\Phi} \to 0\).
Proof. Suppose that \(0< \kappa_{\Phi}(f_j) \to 0\). Given \(0 < \epsilon < 1\) and \(C > 0\) as in (?? ), for all sufficiently large \(j\), we have \(0 < C \kappa_{\Phi}(f_j) < \epsilon^{p_{\Phi}^{+}}\) and \[\kappa_{\Phi} \left( C^{-1/p_{\Phi}^{+}} \kappa_{\Phi}(f_j)^{-1/p_{\Phi}^{+}} f_j \right) \leq \kappa_{\Phi}(f_j)^{-1} \kappa_{\Phi}(f_j) = 1,\] so \(\| f_j \|_{\Phi} \leq C^{1/p_{\Phi}^{+}} \kappa_{\Phi}(f_j)^{1/p_{\Phi}^{+}} < \epsilon\). Then, \(\| f_j \|_{\Phi} \to 0\). ◻
Lemma 6. Assume that \(s \in (0, \infty)\). If \(\Phi\) is an Orlicz function of positive lower (resp., positive upper) type \(p_{\Phi}^{-}\) (resp., type \(p_{\Phi}^{+}\)), then \(\Phi_s\) is of positive lower (resp., positive upper) type \(s p_{\Phi}^{-}\) (resp., type \(s p_{\Phi}^{+}\)).
Proof. It follows immediately from the definitions. ◻
Remark 5. Writing \(\Phi(t) = \Phi(r r^{-1} t)\) with \(r \in (0,1)\), it is easy to check that if an Orlicz function \(\Phi\) is both of positive lower type \(p_{\Phi}^{-}\) and of positive upper type \(p_{\Phi}^{+}\), then \(p_{\Phi}^{-} \leq p_{\Phi}^{+}\). Moreover, if \(\Phi\) is of positive lower (resp., positive upper) type \(p_{\Phi}^{-}\) (resp., type \(p_{\Phi}^{+}\)), then it also is of positive lower (resp., positive upper) type \(p\) for any \(p \in (0, p_{\Phi}^{-})\) (resp., type \(p\) for any \(p \in (p_{\Phi}^{+}, \infty)\)).
This remark leads to the following definition. Given an Orlicz function \(\Phi\), as in [14], define \[\label{p32menos} i(\Phi) := \sup \{ p_{\Phi}^{-} : \Phi \,\, \text{is of positive lower type} \,\, p_{\Phi}^{-} \},\tag{6}\] and \[\label{p32mas} I(\Phi) := \inf \{ p_{\Phi}^{+} : \Phi \,\, \text{is of positive upper type} \,\, p_{\Phi}^{+} \}.\tag{7}\]
The following result states that the couple \(\left(L^{\Phi}(\mathbb{H}^n), \Vert \cdot \Vert_{\Phi} \right)\) is a quasi-normed space when \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and of positive upper type \(p_{\Phi}^{+}\). In particular, if \(\Phi(t) = t^p\), \(0 < p < \infty\), then \(L^{\Phi}(\mathbb{H}^n) = L^p(\mathbb{H}^n)\) with quasi-norm coincidence.
Proposition 6. Let \(\Phi\) be an Orlicz function with positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\), then for any \(f, g : \mathbb{H}^n \to \mathbb{C}\) belonging to \(L^{\Phi}(\mathbb{H}^n)\) and \(\alpha \in \mathbb{C}\),
(i) \(\Vert f \Vert_{\Phi} \geq 0\) and \(\Vert f \Vert_{\Phi} = 0\) if and only if \(f(z) = 0\) a.e. \(z\);
(ii) \(\Vert \alpha f \Vert_{\Phi} = \vert \alpha \vert \Vert f \Vert_{\Phi}\);
(iii) \(\Vert f + g \Vert_{\Phi} \leq K (\Vert f \Vert_{\Phi} + \Vert g \Vert_{\Phi})\), where \(K \geq 1\) and does not depend on \(f\) and \(g\);
(iv) If \(0 \leq f \leq g\) a.e., then \(\| f \|_{\Phi} \leq \| g \|_{\Phi}\);
(v) If \(\Phi\) is continuous and \(0 \leq f_n \uparrow f\) a.e., then \(\| f_n \|_{{\Phi}} \uparrow \| f \|_{{\Phi}}\);
(vi) If \(\Phi\) is bijective and \(E \subset \mathbb{H}^n\) is measurable with \(|E| < \infty\), then \(\| \chi_E \|_{\Phi} = \frac{1}{\Phi^{-1}(|E|^{-1})}\).
Proof. The first part of (i) is obvious, the second one it follows from that \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-}\). From the definition of \(\Vert \cdot \Vert_{\Phi}\) follows (ii) and (vi). Now, (iii) is consequence of [15]. As \(\Phi\) is non-decreasing, (iv) follows. Finally, (v) follows from monotone convergence Theorem. ◻
Definition 5. Two Orlicz functions \(\Phi\) and \(\Psi\) are called equivalent, denoted by \(\Phi \sim \Psi\), if there exist constants \(C_0 \geq 1\) and \(C \geq 1\) such that \[C_{0}^{-1}\Phi(t/C) \leq \Psi(t) \leq C_0 \Phi(Ct),\] for all \(t \geq 0\).
Remark 7. It is clear that if \(\Phi \sim \Psi\), then \(L^{\Phi} = L^{\Psi}\) with equivalent Luxemburg norms. We observe that all our results are invariant under the change of equivalent Orlicz functions. Moreover, equivalent Orlicz functions have the same positive lower and upper type numbers. By [15], without loss of generality, we may always assume that an Orlicz function \(\Phi\) of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\) is continuous and strictly increasing.
Corollary 1. Let \(\Phi\) be an Orlicz function with positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\), then there exists an Orlicz function \(\Psi\) equivalent to \(\Phi\) such that \(L^{\Psi}(\mathbb{H}^n)\) is a quasi-Banach function space.
Definition 6. A function \(\Phi : [0, \infty) \to [0, \infty)\) is called a Young function* if \(\Phi\) is convex, left-continuous, \(\lim_{t \to 0^{+}} \Phi(t) = \Phi(0) = 0\), and \(\lim_{t \to \infty} \Phi(t) = \infty\).*
Remark 8. From the convexity and \(\Phi(0) = 0\), it follows that any Young function is non-decreasing. Moreover, if \(\Phi\) is a young function, from [13], it follows that the couple \((L^{\Phi}, \| \cdot \|_{\Phi})\) is a Banach space.
Remark 9. If an Orlicz function \(\Phi\) is also a Young function, then \(\Phi\) is a bijective continuous function from \([0, \infty)\) onto \([0, \infty)\).
For a Young function \(\Phi\), we define \(\Phi^{-1}\) and its complementary function \(\Phi^{*}\) on \([0, \infty)\) by \[\Phi^{-1}(s):= \inf\{ t \geq 0 : \Phi(t) > s \}\] and \[\Phi^{*}(s) := \sup\{ ts - \Phi(t) : t \in [0, \infty) \},\] respectively. From the definition of \(\Phi^{*}\), it follows that \[\label{Young32ineq} ts \leq \Phi(t) + \Phi^{*}(s), \,\,\,\, \forall t,s \geq 0,\tag{8}\] Then, the Köthe dual \((L^{\Phi})' = L^{\Phi^{*}}\) with comparable norms (see [13]). Moreover, by [17], we have that \[\label{prop321466} s \leq \Phi^{-1}(s) (\Phi^{*})^{-1}(s) \leq 2s, \,\,\,\, s \geq 0.\tag{9}\] If \(\Phi\) is a Young-Orlicz function, then \(\Phi^{-1}\) is the usual inverse function of \(\Phi\).
Definition 7. An Orlicz function \(\Phi\) is called an \(N\)-function if it is a continuous and convex function such that \[\label{232limites} \lim_{t \to 0^{+}} \frac{\Phi(t)}{t} = 0, \,\,\,\,\,\, \text{and} \,\,\,\,\,\, \lim_{t \to \infty} \frac{\Phi(t)}{t} = \infty.\qquad{(2)}\]
Remark 10. From [15] we have that if \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-} \in (1, \infty)\) and positive upper type \(p_{\Phi}^{+}\), then there exists an Orlicz \(N\)-function \(\Psi\) equivalent to \(\Phi\) of positive lower type \(p_{\Psi}^{-} = p_{\Phi}^{-}\) and positive upper type \(p_{\Psi}^{+} = p_{\Phi}^{+}\). Thus, without loss of generality, we may always assume that an Orlicz function \(\Phi\) of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\), with \(1 < p_{\Phi}^{-} \leq p_{\Phi}^{+} < \infty\), is also an \(N\)-function. In particular, an Orlicz \(N\)-function is a Young-Orlicz function.
Next, we will prove that the Hardy-Littlewood maximal operator \(M\) on the Heisenberg group is bounded on Orlicz spaces \(L^{\Phi}(\mathbb{H}^n)\), when \(\Phi\) is an Orlicz function with positive lower type \(p_{\Phi}^{-} > 1\) and positive upper type \(p_{\Phi}^{+}\). As an application of this result, we will obtain a vector-valued inequality for \(M\).
Lemma 7. If \(\Phi\) is an Orlicz \(N\)-function of positive upper type \(p_{\Phi}^{+} > 1\), then its complementary function \(\Phi^{*}\) is also an Orlicz \(N\)-function.
Proof. It is clear that \(\Phi\) is a Young-Orlicz function. By [13], \(\Phi^{*}\) is a Young function. From (?? ), it follows that \(\Phi^{*}(s) > 0\) for all \(s >0\) and so \(\Phi^{*}\) is continuous (see Remark 9). Now, from (8 ), we have that \(\Phi^{*}(s)/s \to \infty\) as \(s \to \infty\). As \(\Phi\) is of positive upper type \(p_{\Phi}^{+} > 1\), by [18], we have that \(\Phi^{*}\) is of positive lower type \((p_{\Phi}^{+})' > 1\) and thus, for \(0 < s < 1\), results \[0 \leq \frac{\Phi^{*}(s)}{s} \leq C s^{(p_{\Phi}^{+})' - 1} \Phi^{*}(1) \to 0, \,\,\,\, \text{as} \,\, s \to 0.\] Therefore, \(\Phi^{*}\) is an Orlicz \(N\)-function. ◻
Proposition 11. If \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and upper type \(p_{\Phi}^{+}\) with \(1 < p_{\Phi}^{-} \leq p_{\Phi}^{+} < \infty\), then the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{\Phi}(\mathbb{H}^n)\).
Proof. By Remark 10, we can assume that \(\Phi\) is an Orlicz \(N\)-function with \(1 < p_{\Phi}^{-} \leq p_{\Phi}^{+} < \infty\). From Lemma 7, it follows that \(\Phi^{*}\) is an Orlicz \(N\)-function. Since the Hardy-Littlewood maximal operator on \(\mathbb{H}^n\) is of weak type \((1, 1)\) and type \((\infty, \infty)\) (see [19]), the proposition then follows from [20]. ◻
Corollary 2. If \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and upper type \(p_{\Phi}^{+}\) with \(1 < p_{\Phi}^{-} \leq p_{\Phi}^{+} < \infty\), then the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{\Phi^{*}}(\mathbb{H}^n)\), where \(\Phi^{*}\) is the complementary function of \(\Phi\).
Proof. Assuming that \(\Phi\) is a \(N\)-function, it follows, by Lemma 7 and [18], that \(\Phi^{*}\) is an Orlicz \(N\)-function of positive lower type \(p_{\Phi^{*}}^{-} = (p_{\Phi}^{+})' > 1\) and upper type \(p_{\Phi^{*}}^{+} = (p_{\Phi}^{-})' \geq (p_{\Phi}^{+})'\). Then, the corollary follows from Proposition 11 applied to \(\Phi^{*}\). ◻
Theorem 12. Let \(\Phi\) be an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and upper type \(p_{\Phi}^{+}\) with \(1 < p_{\Phi}^{-} \leq p_{\Phi}^{+} < \infty\). Then, for any \(1 < r < \infty\), \[\label{vector32ineq32M} \left\| \left( \sum_{j=1}^{\infty} (M f_j)^r \right)^{1/r} \right\|_{\Phi} \lesssim \left\| \left( \sum_{j=1}^{\infty} |f_j|^r \right)^{1/r} \right\|_{\Phi}\qquad{(3)}\] holds for all sequences of locally integrable functions \(\{ f_j \}_{j=1}^{\infty}\) on \(\mathbb{H}^n\).
Proof. Given \(1 < p < \infty\), by [21], we have, for any weight \(w \in \mathcal{A}_p(\mathbb{H}^n)\) (see [21]), that the Hardy-Littlewood maximal operator \(M\) is a bounded operator \(L^p_w (\mathbb{H}^n) \to L^p_w (\mathbb{H}^n)\). Then, combining Proposition 11, Corollary 2 and proceeding as in the proof of [18] (we point out that such argument works as well on \(\mathbb{H}^n\)), (?? ) follows. ◻
We finish this section with the following auxiliary result.
Proposition 13. Let \(\Phi\) be an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\). Suppose that \(s > 1\) and \(0 < \theta < \min\{ 1, p_{\Phi}^{-} \}\) are such that \(s \theta > p_{\Phi}^{+}\) and \(\{ b_j \}_{j=1}^{\infty}\) is a sequence of non-negative functions in \(L^{s}(\mathbb{H}^{n})\) such that each \(b_j\) is supported in a \(\rho\) - ball \(B_j \subset \mathbb{H}^{n}\) and \[\| b_j \|_{L^{s}(\mathbb{H}^{n})} \leq A_j |B_j|^{1/s},\] where \(A_j >0\) for all \(j \geq 1\). Then, for any sequence of non-negative numbers \(\{ k_j \}_{j=1}^{\infty}\) we have \[\left\| \sum_{j=1}^{\infty} k_j b_j \right\|_{\Phi_{1/\theta}} \leq C \left\| \sum_{j=1}^{\infty} A_j k_j \chi_{B_j} \right\|_{\Phi_{1/\theta}},\] where \(C\) is a positive constant which does not depend on \(\{ b_j \}_{j=1}^{\infty}\), \(\{ A_j \}_{j=1}^{\infty}\), and \(\{ k_j \}_{j=1}^{\infty}\).
Proof. The argument used to proof [22] also works in this setting, but now considering the space \(L^{\Phi}(\mathbb{H}^n)\), [13], Lemma 1 and Corollary 2 applied with \(\left( (\Phi^{1/\theta})^{*} \right)^{1/s'}\) and taking into account that \(1 < s' < (p_{\Phi}^{+}/\theta)' \leq (p_{\Phi}^{-}/\theta)'\). ◻
In this section, we define the Orlicz-Hardy spaces \(H^{\Phi}(\mathbb{H}^n)\) where \(\Phi\) is an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\). We also provide an atomic decomposition for elements in \(H^{\Phi}(\mathbb{H}^n)\).
Given \(N \in \mathbb{N}\), define \[\mathcal{F}_{N}=\left\{ \varphi \in \mathcal{S}(\mathbb{H}^{n}) : \| \phi \|_{\mathcal{S}(\mathbb{H}^{n}), \, N} \leq 1\right\}.\] For any \(f \in \mathcal{S}'(\mathbb{H}^{n})\), the grand maximal function of \(f\) is defined by \[\mathcal{M}_N f(z)=\sup\limits_{t>0}\sup\limits_{\phi \in \mathcal{F}_{N}}\left\vert \left( f \ast \phi_t \right)(z) \right\vert,\] where \(\phi_t(z) := t^{-2n-2} \phi(t^{-1} \cdot z)\) with \(t > 0\).
Now, we introduce two maximal functions. Given \(\phi \in \mathcal{S}(\mathbb{H}^n)\) and \(f \in \mathcal{S}'(\mathbb{H}^n)\), we define the discrete maximal function \(M_{\phi}^{dis}f\) by \[(M_{\phi}^{dis}f)(z) := \sup \left\{ |(f \ast \phi_{2^{-j}})(z)| : j \in \mathbb{Z} \right\}, \,\,\,\, z \in \mathbb{H}^n.\] Given an integer \(L > 1\), we define the maximal function \(M_{\phi, L}^{*}f\) by \[M_{\phi, L}^{*}f(z) := \sup_{j \in \mathbb{Z}} \sup_{w \in \mathbb{H}^n} \frac{|(f \ast \phi_{2^{-j}})(w)|}{(1+ 4^{j} \rho(z^{-1} \cdot w)^2)^L}, \,\,\,\, z \in \mathbb{H}^n.\]
Lemma 8. ([23]) Let \(f \in \mathcal{S}'(\mathbb{H}^n)\), \(0 < \theta < 1\) and let \(\phi\) be a radial function in \(\mathcal{S}(\mathbb{H}^n)\) with \(\int \phi \neq 0\). Then, there exists \(L_{\theta}\) such that for all \(L \geq L_{\theta}\) and \(z \in \mathbb{H}^n\), \[M_{\phi, L}^{*}f(z) \lesssim \left[ M \left( (M_{\phi}^{dis}f)^{\theta} \right)(z) \right]^{1/\theta}.\]
In the next result, we show that, for \(N\) and \(L\) large enough, the quantities \(\| \mathcal{M}_{N}f \|_{\Phi}\), \(\| M_{\phi}^{dis}f \|_{\Phi}\) and \(\| M_{\phi, L}^{*} f \|_{\Phi}\) are mutually comparable, with bounds independent of \(f\).
Theorem 14. Let \(\Phi\) be an Orlicz function such that \(0 < i(\Phi) \leq I(\Phi) < \infty\). For a radial function \(\phi \in \mathcal{S}(\mathbb{H}^n)\) with \(\int \phi \neq 0\), we have \[\| \mathcal{M}_{N}f \|_{\Phi} \approx \| M_{\phi, L}^{*} f \|_{\Phi} \approx \| M_{\phi}^{dis}f \|_{\Phi},\] for all \(f \in \mathcal{S}'(\mathbb{H}^n)\), where \(N\) and \(L\) are large enough.
Proof. It is clear that \(M_{\phi}^{dis}f(z) \leq M_{\phi, L}^{*} f(z)\) for all \(z \in \mathbb{H}^n\). Now, for \(0 < \theta < \min \{1, i(\Phi) \}\), from Lemma 8, Lemma 1 and Proposition 11, it follows that \[\| M_{\phi, L}^{*} f \|_{\Phi} \approx \| M_{\phi}^{dis}f \|_{\Phi}.\] On the other hand, we have that \(M_{\phi}^{dis}f(z) \leq \mathcal{M}_N f(z)\) for all \(z \in \mathbb{H}^n\). Thus, \[\| M_{\phi, L}^{*} f \|_{\Phi} \lesssim \| \mathcal{M}_N f \|_{\Phi}.\] In the proof of [23], the authors shown, for \(N\) and \(L\) large enough and \(\tau \in \mathcal{S}(\mathbb{H}^n)\) satisfying \(\| \tau \|_{\mathcal{S}(\mathbb{H}^{n}), \, N} \leq 1\), that \[|(f \ast \tau_{2^{-j}})(z)| \lesssim M_{\phi, L}^{*} f (z), \,\,\,\, \text{for all} \,\, z \in \mathbb{H}^n.\] Being \(\tau\) and \(j\) arbitrary, one obtains \(\mathcal{M}_N f(z) \lesssim M_{\phi, L}^{*} f (z)\) for all \(z \in \mathbb{H}^n\), and with them the theorem. ◻
Definition 8. Given an Orlicz funtion \(\Phi\) of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\), we define the Orlicz-Hardy space \(H^{\Phi}(\mathbb{H}^{n})\) as the set of all \(f \in S^{\prime}(\mathbb{H}^{n})\) for which \(\mathcal{M}_{N}f \in L^{\Phi}(\mathbb{H}^{n})\), where \(N\) is large enough in the sense of Theorem 14. In this case we set \(\| f \|_{H^{\Phi}(\mathbb{H}^{n})} = \| \mathcal{M}_{N}f \|_{\Phi}\).
Now, we introduce the definition of \(\Phi\)-atom in \(\mathbb{H}^n\).
Definition 9. Let \(\Phi\) be an Orlicz function of positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\)
with \(0 < i(\Phi) \leq I(\Phi) < \infty\), \(\max\{ 1, I(\Phi) \} < p_0 \leq \infty\) and \(m \in \mathbb{N}\). Fix an integer \(m \geq m_{\Phi} : = Q \left(\lfloor i(\Phi)^{-1} - 1 \rfloor + 1 \right)\). A measurable function \(a(\cdot)\) on \(\mathbb{H}^{n}\) is called a \((\Phi, p_{0}, m)\) - atom if there exists a \(\rho\) - ball \(B\) such that
\(a_{1})\) \(\textit{supp}\left( a\right) \subset B\),
\(a_{2})\) \(\left\Vert a \right\Vert_{L^{p_{0}}(\mathbb{H}^{n})} \leq
\left\vert B \right\vert^{\frac{1}{p_{0}}} \Vert \chi_B \Vert_{\Phi}^{-1}\),
\(a_{3})\) \(\int a(z) \, z^{I} \, dz = 0\) for all multiindex \(I\) such that \(d(I) \leq m\).
A such atom is also called an atom centered at the \(\rho\) - ball \(B\). Following the ideas in the proof of [23], and adapting them to our context, one obtains that every \((\Phi, p_{0}, m)\) - atom \(a(\cdot)\) belongs to \(H^{\Phi}(\mathbb{H}^{n})\). Moreover, there exists an universal constant \(C > 0\) such that \(\| a \|_{H^{\Phi}(\mathbb{H}^n)} \leq C\) for all \((\Phi, p_{0}, m)\) - atom \(a(\cdot)\).
Remark 15. We observe that every \((\Phi, \infty, m)\) - atom is an \((\Phi, p_0, m)\) - atom for any \(p_0 \in (1, \infty)\).
Now, we shall formulate an atomic decomposition theorem in terms of \((\Phi, \infty, m)\) - atoms. Before establishing this result, we introduce the following three constants (see [24]) \[T_1 = 9 \gamma \beta^{N}, \,\,\,\,\,\, T_2 = 2 \gamma^2 T_1, \,\,\,\, \text{and} \,\,\,\, T_3 = 3 \gamma T_2,\] where \(\gamma\) is the constant in the inequality [24] (when \(G = \mathbb{H}^n\) and \(|\cdot|=\rho(\cdot)\) is the Koranyi norm given by (3 ), \(\gamma = 1\)), \(\beta\) is the constant in [24] (we observe that \(\beta \geq 1\), see [24]), and \(N\) is as in Definition 8.
Theorem 16. Let \(\Phi\) be an Orlicz function such that \(0 < i(\Phi) \leq I(\Phi) < \infty\). Then every \(f \in H^{\Phi}(\mathbb{H}^n)\) can be written as \[f=\sum\limits_{j=1}^{\infty } \lambda_{j}a_{j} \label{serie32atomica}\qquad{(4)}\] in \(S^{\prime }(\mathbb{H}^{n}),\) where \(\left\{ \lambda_{j} \right\}_{j=1}^{\infty}\) is a sequence of non-negative numbers, the \(a_{j}\)’s are \((\Phi, \infty, m)\) - atoms supported on \(\rho\) - balls \(B_j\) and \[\label{norma32atomica} \left\Vert \left\{ \sum_{j} \left( \frac{\lambda_j \chi_{B_j}}{\Vert \chi_{B_j} \Vert_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\Vert_{\Phi} \leq C \Vert f \vert_{H^{\Phi}}, \,\,\,\,\,\,\,\, \forall \,\,\, 0 < \theta \leq 1,\qquad{(5)}\] where \(C\) is an universal positive constant which does not depend on \(\left\{ \lambda_{j}\right\}_{j=1}^{\infty}\), \(\left\{ B_{j}\right\}_{j=1}^{\infty}\) and \(f\).
Proof. Let \(f \in H^{\Phi}(\mathbb{H}^n) \cap L^{s}(\mathbb{H}^n)\) with \(s \in (\max \{I(\Phi), 1\},\infty)\) and \(m \geq m_{\Phi}\). For any \(j \in \mathbb{Z}\) and \(N\) large enough, we set \[O_j = \{ z \in \mathbb{H}^n : \mathcal{M}_N f(z) > 2^{j} \}.\] It is clear that \(O_j \supseteq O_{j+1}\) for all \(j \in \mathbb{Z}\). For every \(j \in \mathbb{Z}\), by [24] and taking into account the constants \(T_1\), \(T_2\) and \(T_3\) previously defined, there exist a sequence of points \(\{ z_{j,k} \}_{k \in \mathbb{N}} \subset O_j\) and a sequence of scalars \(\{ r_{j,k} \}_{k \in \mathbb{N}} \subset (0, \infty)\) such that
(w1) \(O_j = \cup_{k=1}^{\infty} B(z_{j,k}, r_{j,k})\),
(w2) the family of balls \(\{ B(z_{j,k}, r_{j,k}/4) \}_{k \in \mathbb{N}}\) are disjoint,
(w3) \(B(z_{j,k}, T_2 r_{j,k}) \cap O_j^c = \emptyset\), but \(B(z_{j,k}, T_3 r_{j,k}) \cap O_j^c \neq \emptyset\),
(w4) there exists \(L \in \mathbb{N}\) such that no point of \(O_j\) lies in more than \(L\) of the balls \(B(z_{j,k}, T_2 r_{j,k})\).
Now, by Calderón-Zygmund decomposition given in [24], we can decomposed to \(f\) as \[f = b_j + g_j, \,\,\,\,\,\,\,\, b_j = \sum_k b_{j,k}, \,\,\,\,\,\,\,\, b_{j,k} = (f - P_{j,k}) \zeta_{j,k},\] where \(\zeta_{j,k} \in C^{\infty}_{0}(B(z_{j,k}, 2 r_{j,k}))\), \(\int b_{j,k}(z) z^{I} dz = 0\) for all \(d(I) \leq m\) and \(|g_j| \lesssim 2^j\). We put \(\widetilde{B}_{j,k} := B(z_{j,k}, T_1 r_{j,k})\). By [24], we have that \[\mathcal{M}_N b_{j,k}(z) \lesssim \mathcal{M}_N f(z), \,\,\,\,\, \text{if} \,\, z \in \widetilde{B}_{j,k},\] and \[\mathcal{M}_N b_{j,k}(z) \lesssim 2^{j} \left( r_{j,k}/\rho(z_{j,k}^{-1} \cdot z) \right)^{Q+m}, \,\,\,\,\, \text{if} \,\, z \notin \widetilde{B}_{j,k}.\] So, \[\| \mathcal{M}_N b_j \|_{\Phi} \lesssim \left\| \sum_k \mathcal{M}_N b_{j, k} \right\|_{\Phi} \lesssim \left\| \sum_k \mathcal{M}_N f \cdot \chi_{\widetilde{B}_{j,k}} \right\|_{\Phi}\] \[+ \left\| \sum_k 2^{j} \left( r_{j,k}/\rho(z_{j,k}^{-1} \cdot (\cdot)) \right)^{Q+m} \cdot \chi_{\widetilde{B}_{j,k}^{c}} \right\|_{\Phi}\] \[\lesssim \left\| \mathcal{M}_N f \cdot \chi_{O_j} \right\|_{\Phi} + \left\| \sum_k 2^{j} \left( M\chi_{\widetilde{B}_{j,k}} \right)^{\frac{Q+m}{Q}} \right\|_{\Phi}\] Since \(m \geq m_{\Phi}\), from Theorem 12, it follows that \[\| \mathcal{M}_N b_j \|_{\Phi} \lesssim \left\| \mathcal{M}_N f \cdot \chi_{O_j} \right\|_{\Phi},\] and so, by Lemma 5, \[\| f - g_j \|_{H^{\Phi}(\mathbb{H}^n)} = \| b_j \|_{H^{\Phi}(\mathbb{H}^n)} = \| \mathcal{M}_N b_j \|_{\Phi} \lesssim \left\| \mathcal{M}_N f \cdot \chi_{O_j} \right\|_{\Phi} \to 0\] as \(j \to +\infty\). On the other hand, we have \(g_j \to 0\) uniformly as \(j \to -\infty\). Then, \[f = \sum_{j=-\infty}^{+\infty} (g_{j+1} - g_j) \,\,\,\,\, \text{in} \,\, \mathcal{S}'(\mathbb{H}^n).\] Now, by [24], we can write \(g_{j+1} - g_j = \sum_{k=1}^{\infty} h_{j,k}\) and \(f = \sum_{j,k} h_{j,k}\) in \(\mathcal{S}'(\mathbb{H}^n)\). Moreover, \(h_{j,k}\) is supported on the ball \(B(z_{j,k}, T_2 r_{j,k}) =: T_2 B_{j,k}\), \(| h_{j,k} (z)| \leq C 2^{j}\) for all \(z\), and \(\int h_{j,k}(z) z^{I} dz = 0\) for all \(d(I) \leq m\). Putting \[\label{ajk32and32lambdajk} a_{j,k} := \frac{h_{j,k}}{\lambda_{j,k}}, \,\,\,\, \text{where} \,\, \lambda_{j,k} := C 2^{j} \| \chi_{T_2 B_{j,k}} \|_{\Phi},\tag{10}\] we have that the \(a_{j,k}\)’s are \((\Phi, \infty, m)\) - atoms and \(f = \sum_{j,k} \lambda_{j,k} a_{j,k}\) in \(\mathcal{S}'(\mathbb{H}^n)\). By (w1), (w4) and (10 ), it results \[\left\| \left\{ \sum_{j, k} \left( \frac{\lambda_{j,k} \chi_{T_2 B_{j,k}}}{\| \chi_{T_2 B_{j,k}} \|_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\|_{\Phi} \approx \left\| \left\{ \sum_{j=-\infty}^{+\infty} 2^{j \theta} \chi_{O_j} \right\}^{1/\theta} \right\|_{\Phi}\] Since \(O_j \supseteq O_{j+1}\) for all \(j \in \mathbb{Z}\) and \(0 < \theta \leq 1\), we have that \[\sum_{j=-\infty}^{+\infty} \left( 2^j \chi_{O_j}(z) \right)^{\theta} \approx \left( \sum_{j=-\infty}^{+\infty} 2^j \chi_{O_j}(z) \right)^{\theta} \approx \left( \sum_{j=-\infty}^{+\infty} 2^j \chi_{O_j \setminus O_{j+1}}(z) \right)^{\theta},\] for all \(z \in \mathbb{H}^n\). So, \[\left\| \left\{ \sum_{j, k} \left( \frac{\lambda_{j,k} \chi_{T_2 B_{j,k}}}{\| \chi_{T_2 B_{j,k}} \|_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\|_{\Phi} \lesssim \left\| \sum_{j=-\infty}^{+\infty} 2^j \chi_{O_j \setminus O_{j+1}} \right\|_{\Phi}\] Now, taking into account that the family of sets \(\{ O_j \setminus O_{j+1} \}_{j \in \mathbb{Z}}\) is disjoint, for any \(\mu > 0\) we have that \[\int_{\mathbb{H}^n} \Phi\left( \sum_{j=-\infty}^{+\infty} \frac{2^j \chi_{O_j \setminus O_{j+1}}(z)}{\mu} \right) dz = \sum_{j=-\infty}^{+\infty} \int_{O_j \setminus O_{j+1}} \Phi\left( \frac{2^j}{\mu} \right) dz \approx \int_{\mathbb{H}^n} \Phi\left( \frac{\mathcal{M}_N f(z)}{\mu} \right) dz,\] which leads to \[\left\| \left\{ \sum_{j, k} \left( \frac{\lambda_{j,k} \chi_{T_2 B_{j,k}}}{\| \chi_{T_2 B_{j,k}} \|_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\|_{\Phi} \lesssim \| f \|_{\Phi}.\] Proceeding as in the proof of [9], for \(s \in (I(\Phi),\infty)\), one obtains that \(H^{\Phi}(\mathbb{H}^n) \cap L^{s}(\mathbb{H}^n)\) is dense in \(H^{\Phi}(\mathbb{H}^n)\). From this and an argument similar to [19], we have an atomic decomposition for any \(f \in H^{\Phi}(\mathbb{H}^n)\) satisfying (?? ). ◻
The following auxiliary result will be useful to get our main theorem of Section 6.
Proposition 17. Let \(\Phi\) be an Orlicz function such that \(0 < i(\Phi) \leq I(\Phi) < \infty\), and let \(\max\{ 1, I(\Phi) \} < p_0 < \infty\) and \(0 < \theta < \min\{ 1, i(\Phi) \}\). Then, for any non-negative sequence \(\{ k_j \}_{j=1}^{\infty}\), \(r \geq 1\), and any sequence \(\{ a_j \}_{j=1}^{\infty}\) of \((\Phi, p_0, m)\)-atoms such that every atom \(a_j\) is supported on the \(\rho\) - ball \(B_j\), we have \[\left\| \sum_{j=1}^{\infty} k_j \chi_{r B_j} M a_j \right\|_{\Phi} \lesssim \left\| \left\{ \sum_{j=1}^{\infty} \left( \frac{ k_j \chi_{B_j}}{\| \chi_{B_j} \|_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\|_{\Phi},\] where \(M\) is the Hardy-Littlewood maximal operator and the implicit constant does not depend on \(\{ k_j \}_{j=1}^{\infty}\), \(\{ a_j \}_{j=1}^{\infty}\), and \(\{ B_j \}_{j=1}^{\infty}\).
Proof. Let \(0 < \theta < \min\{1, i(\Phi) \}\) be fixed, \(r \geq 1\) and \(p_{0} > \max \{1, I(\Phi) \}\). Being the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{p_0}(\mathbb{H}^n)\) (see [19]), we have \[\label{Mfi} \| (Ma_j)^{\theta} \|_{L^{p_{0}/\theta}(r B_j)} \lesssim \| a_j \|_{p_0}^{\theta} \lesssim \frac{|B_j |^{\theta/p_0}}{\| \chi_{B_j} \|_{\Phi}^{\theta}} \lesssim \frac{ |r B_j |^{\theta/p_0}}{\| \chi_{ B_j} \|_{\Phi}^{\theta}}.\tag{11}\] Now, since \(0 < \theta < 1\), we apply the \(\theta\)-inequality given in [25] and Lemma 1, to obtain \[\left\| \sum_{j} k_j \chi_{r B_j} M a_j \right\|_{\Phi} \leq \left\| \sum_{j} \left(k_j \, \chi_{r B_{j}} \, Ma_j \right)^{\theta} \right\|^{1/\theta}_{\Phi_{1/\theta}} =:J_{\theta}.\] Then, taking into account (11 ), by Proposition 13 with \(b_j = \left( \chi_{r B_{j}} \, (Ma_j)^{\theta} \right)\), \(A_j = \| \chi_{ B_{j}} \|_{\Phi}^{-\theta}\) and \(s= p_0/\theta\), we have that \(J_{\theta}\) \[\label{Jteta} \lesssim \left\| \sum_{j} \left( \frac{k_j}{\left\| \chi_{ B_{j}} \right\|_{\Phi}} \right)^{\theta} \chi_{r B_{j}} \right\|^{1/\theta}_{\Phi_{1/\theta}}.\tag{12}\] It is easy to check that \(\chi_{r B_{j}} \leq (M\chi_{B_j})^{2}\). From this pointwise inequality, the inequality (12 ), Theorem 12 and Lemma 1, we have \[\begin{align} \left\| \sum_{j} k_j \chi_{r B_j} M a_j \right\|_{\Phi} &\lesssim& \left\| \left\{ \sum_{j} \left( \frac{k_j^{\theta/2}}{\left\| \chi_{B_{j}} \right\|^{\theta/2}_{\Phi}} (M\chi_{B_j}) \right)^{2} \right\}^{1/2} \right\|^{2/\theta}_{\Phi_{2/\theta}} \\ &\lesssim& \left\| \left\{ \sum_{j} \left( \frac{ k_j \chi_{B_j}}{\| \chi_{B_j} \|_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\|_{\Phi}. \end{align}\] This concludes the proof. ◻
We recall that \(\mathcal{P}_{k}\) is the subspace formed by all the polynomials of homogeneous degree at most \(k \geq 0\). Now, the argument used to prove [26] works on \(\mathbb{H}^n\) as well. Then, its analogous on \(\mathbb{H}^n\), it is as follows.
Lemma 9. Given a integer \(k \geq 0\), there exists \(\varphi \in C^{\infty}(\mathbb{H}^n)\) with support on the \(\rho\)-ball \(B(e,1)\) such that for every \(\lambda > 0\) and every polynomial \(p\) of homogeneous degree at most \(k\), \[p(z) = \int \lambda^{Q} \varphi(\lambda \cdot (w^{-1} \cdot z)) \, p(w) dw = (p \ast \varphi_{\lambda^{-1}})(z)\] holds.
Remark 18. From this Lemma, one has that \(X^{I}p = p \ast X^{I}(\varphi_{\lambda^{-1}})\) for every multi-index \(I\).
Let \(L^{q}_{loc}(\mathbb{H}^{n})\), \(1 < q < \infty\), be the space of all measurable functions \(g\) on \(\mathbb{H}^{n}\) that belong locally to \(L^{q}\) for compact sets of \(\mathbb{H}^{n}\). We endowed \(L^{q}_{loc}(\mathbb{H}^{n})\) with the topology generated by the seminorms \[|g|_{q, \, B} = \left( |B|^{-1} \int_{B} \, |g(w)|^{q}\, dw \right)^{1/q},\] where \(B\) is a \(\rho\)-ball in \(\mathbb{H}^{n}\) and \(|B|\) denotes its Haar measure.
For \(g \in L^{q}_{loc}(\mathbb{H}^{n})\), we define a maximal function \(\eta_{q, \, \gamma}(g; z)\) as \[\eta_{q, \, \gamma}(g; \, z) = \sup_{r > 0} r^{-\gamma} |g|_{q, \, B(z, r)},\] where \(\gamma\) is a positive real number and \(B(z, r)\) is the \(\rho\)-ball centered at \(z\) with radius \(r\).
We denote by \(E^{q}_{k}\) the quotient space of \(L^{q}_{loc}(\mathbb{H}^{n})\) by \(\mathcal{P}_{k}\). If \(G \in E^{q}_{k}\), we define the seminorm \(\| G \|_{q, \, B} = \inf \left\{ |g|_{q, \, B} : g \in G \right\}\). The family of all these seminorms induces on \(E^{q}_{k}\) the quotient topology.
Given a positive real number \(\gamma\), we can write \(\gamma = k + t\), where \(k\) is a non negative integer and \(0 < t \leq 1\). This decomposition is unique.
For \(G \in E^{q}_{k}\), we define a maximal function \(N_{q, \, \gamma}(G; z)\) as \[N_{q, \, \gamma}(G; z) = \inf \left\{ \eta_{q, \, \gamma}(g; z) : g \in G \right\}.\]
Lemma 10. ([1]) The maximal function \(z \to N_{q; \, \gamma}(G; z)\) associated with a class \(G\) in \(E_{k}^{q}\) is lower semicontinuous.
Definition 10. Let \(\Phi\) be an Orlicz function, we say that an element \(G \in E^{q}_{k}\) belongs to the Orlicz-Calderón Hardy space \(\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})\) if the maximal function \(N_{q, \, \gamma}(G; \, \cdot \,) \in L^{\Phi}(\mathbb{H}^{n})\). In this case, for any \(G \in \mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})\), we set \[\| G \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})} := \| N_{q, \, \gamma}(G; \, \cdot \,) \|_{\Phi}.\]
If \(\Phi\) is an Orlicz function with positive lower type \(p_{\Phi}^{-}\) and positive upper type \(p_{\Phi}^{+}\), by Proposition 6, it follows that the couple \(\left( \mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n}), \| \cdot \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})} \right)\) results a quasi-normed space.
Now, taking into account Lemma 9 and Remark 18, to follow the proof of [27], it obtains the following result.
Lemma 11. Let \(g_1\) and \(g_2\) be two representatives of an element \(G \in E^q_k\) and \(p = g_1 - g_2\) (\(p\) is a polynomial of homogeneous degree at most \(k\)). Then, for every multi-index \(I\) with \(0 \leq d(I) \leq k\), there exists a positive constant \(C = C_{I}\) such that \[|X^{I}p(z)| \leq C \left[ \eta_{q, \gamma}(g_1; z_1) + \eta_{q, \gamma}(g_2; z_2) \right] \left( \rho(z_1^{-1} \cdot z) + \rho(z_2^{-1} \cdot z) \right)^{k+1 - d(I)}\] holds for every \(z_1\), \(z_2\) and \(z\) in \(\mathbb{H}^n\).
Lemma 12. Let \(G \in E^{q}_{k}\) with \(N_{q, \, \gamma}(G; z_0) < \infty,\) for some \(z_0 \in \mathbb{H}^{n}\). Then:
\((i)\) There exists a unique \(g \in G\) such that \(\eta_{q, \, \gamma} (g; z_0) < \infty\) and, therefore, \(\eta_{q, \, \gamma} (g; z_0) = N_{q, \, \gamma}(G; z_0)\).
\((ii)\) For any \(\rho\)-ball \(B\), there is a constant \(c\) depending on \(z_0\) and \(B\) such that if \(g\) is the unique representative of \(G\) given in \((i)\), then \[\|G\|_{q, \, B} \leq |g|_{q, \, B} \leq c \, \eta_{q, \, \gamma} (g; z_0) = c \, N_{q, \, \gamma}(G; z_0).\]
The constant \(c\) can be chosen independently of \(z_0\) provided that \(z_0\) varies in a compact set.
Proof. To prove (i), we assume that \(g_1\) and \(g_2\) belong to \(G\) and both \(\eta_{q, \gamma}(g_1; z_0)\) and \(\eta_{q, \gamma}(g_2; z_0)\) are finite. We call \(p\) to the polynomial \(g_1 - g_2\) of homogeneous degree at most \(k\). Applying Lemma 11 with \(z = z_1 = z_2 = z_0\) we have \(X^{I}p(z_0) = 0\) for every multi-index \(I\) such that \(0 \leq d(I) \leq k\). Since every polynomial of homogeneous degree at most \(k\) can be centered at \(z_0\), with \(z_0\) being an arbitrary point of \(\mathbb{H}^n\) (see the formula that appears in [28] for the Taylor polynomial of a smooth function), we have that \(p \equiv 0\). Then, (i) follows. Finally, (i) implies (ii). ◻
Corollary 3. If \(\{ G_{j} \}\) is a sequence of elements of \(E^{q}_{k}\) converging to \(G\) in \(\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})\), then \(\{ G_{j} \}\) converges to \(G\) in \(E^{q}_{k}\).
Proof. For any \(\rho\)-ball \(B\), by \((ii)\) of Lemma 12, we have \[\| G- G_{j} \|_{q, \, B} \leq c \, \| \chi_{B} \|_{\Phi}^{-1} \| \chi_{B} \,\, N_{q, \, \gamma}(G - G_{j}; \, \cdot \,) \|_{\Phi} \leq c \, \| G - G_{j} \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})},\] which proves the corollary. ◻
Lemma 13. Let \(\{ G_{j} \}\) be a sequence in \(E^{q}_{k}\) such that for a given point \(z_0 \in \mathbb{H}^n\), the series \(\sum_j N_{q, \, \gamma}(G_{j}; \, z_0 )\) is finite. Then
\((i)\) The series \(\sum_j G_j\) converges in \(E_{k}^{q}\) to an element \(G\) and \[N_{q, \, \gamma}(G; \, z_0 ) \leq \sum_j N_{q, \, \gamma}(G_{j}; \, z_0 ).\]
\((ii)\) If \(g_j\) is the unique representative of \(G_j\) satisfying \(\eta_{q, \, \gamma} (g_j; z_0) = N_{q, \, \gamma}(G_j; z_0)\), then \(\sum_j g_j\) converges in \(L^{q}_{loc}(\mathbb{H}^{n})\) to a function \(g\) that is the unique representative of \(G\) satisfying \(\eta_{q, \, \gamma} (g; z_0) = N_{q, \, \gamma}(G; z_0)\)
Proof. The proof is similar to the one given in [3]. ◻
Proposition 19. ([1]) If \(g \in L^{q}_{loc}(\mathbb{H}^{n})\), \(1 < q < \infty\), and there is a point \(z_0 \in \mathbb{H}^{n}\) such that \(\eta_{q, \, \gamma} (g ; z_0) < \infty\), then \(g \in \mathcal{S}'(\mathbb{H}^{n})\).
Proposition 20. Let \(g \in L^q_{loc} \cap \mathcal{S}'(\mathbb{H}^n)\) and \(f = \mathcal{L} g\) in \(\mathcal{S}'(\mathbb{H}^n)\). If \(\phi \in \mathcal{S}(\mathbb{H}^n)\) and \(N > Q+2\), then \[(M_{\phi}^{dis}f)(z) := \sup \left\{ |(f \ast \phi_{2^{j}})(z)| : j \in \mathbb{Z} \right\}\] \[\leq C \| \phi \|_{\mathcal{S}(\mathbb{H}^{n}), N} \,\,\, \eta_{q, 2}(g; \, z)\] holds for all \(z \in \mathbb{H}^n\).
Proof. Since \((M_{\phi}^{dis}f)(z) \leq (M_{\phi} f)(z) := \sup \left\{ |(f \ast \phi_t)(w)| : \rho(w^{-1} \cdot z) < t, \, 0 < t < \infty \right\}\) for all \(z \in \mathbb{H}^n\), the proposition follows from [1]. ◻
Proposition 21. If \(\Phi\) is an Orlicz function with \(0 < i(\Phi) \leq I(\Phi) < \infty\), then the space \(\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})\) is complete.
Proof. Without loss of generality, we may assume that \(\Phi\) is continuous and strictly increasing (see Remark 7). By [16], it is enough to show that \(\mathcal{H}^{\Phi}_{q, \, \gamma}\) has the generalized Riesz-Fisher property with constant \(C_0 \in [1, \infty)\), i.e.: for any sequence \(\{ G_j \}\) in \(\mathcal{H}^{\Phi}_{q, \, \gamma}\) that satisfies \[\sum_{j=1}^{\infty} C_{0}^{j+1} \| G_j \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}} < \infty,\] the series \(\sum_{j} G_j\) converges in \(\mathcal{H}^{\Phi}_{q, \, \gamma}\).
Let \(C_0 = K\), where \(K\) is as in Proposition 6 - (iii), and let \(m \geq 1\) be fixed, then \[\begin{align} \left\| \sum_{j=m}^{k} N_{q, \, \gamma}(G_{j}; \, \cdot \,) \right\|_{\Phi} &\leq& \sum_{j=m}^{k} C_{0}^{j-m+1} \left\| N_{q, \, \gamma}(G_{j}; \, \cdot \,) \right\|_{\Phi} \\ &\leq& \sum_{j=m}^{\infty} C_{0}^{j-m+1} \| G_j \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}} =: \alpha_m < \infty, \end{align}\] for every \(k \geq m\). Being \(\Phi\) an increasing continuous function, by Lemma 2, we obtain \[\int_{\mathbb{H}^{n}} \, \Phi\left( \alpha_m^{-1} \, \left|\sum_{j=m}^{k} N_{q, \, \gamma}(G_{j}; \, z ) \right| \right) \, dz\] \[\leq \int_{\mathbb{H}^{n}} \Phi \left( \left\| \sum_{j=m}^{k} N_{q, \, \gamma}(G_{j}; \, \cdot \,) \right\|_{\Phi} \, \left| \sum_{j=m}^{k} N_{q, \, \gamma}(G_{j}; z ) \right| \right) \, dz \leq 1, \,\,\, \forall \, k \geq m,\] by applying Fatou’s lemma as \(k \rightarrow \infty\) and the continuity of \(\Phi\), we obtain \[\int_{\mathbb{H}^{n}} \, \Phi \left( \alpha_m^{-1} \, \left| \sum_{j=m}^{\infty} N_{q, \, \gamma}(G_{j}; \, z ) \right| \right) \, dz \leq 1,\] so \[\left\| \sum_{j=m}^{\infty} N_{q, \, \gamma}(G_{j}; \, \cdot \,) \right\|_{\Phi} \leq \alpha_m = \sum_{j=m}^{\infty} C_{0}^{j-m+1} \| G_j \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}} < \infty, \,\,\,\, \forall \, m \geq 1 \label{serie}.\tag{13}\] Taking \(m = 1\) in (13 ), it follows that \(\sum_{j} N_{q, \, \gamma}(G_{j}; z)\) is finite a.e. \(z \in \mathbb{H}^{n}\). Then, by \((i)\) of Lemma 13, the series \(\sum_j G_j\) converges in \(E_{k}^{q}\) to an element \(G\). Now \[N_{q, \, \gamma}\left( G - \sum_{j=1}^{k} G_j; \, z \right) \leq \sum_{j=k+1}^{\infty} N_{q, \, \gamma} (G_j; \, z),\] from this and (13 ) we get \[\left\| G - \sum_{j=1}^{k} G_j \right\|_{\mathcal{H}^{\Phi}_{q, \, \gamma}} \leq \sum_{j=k+1}^{\infty} C_{0}^{j-k} \| G_j \|_{\mathcal{H}^{\Phi}_{q, \, \gamma}},\] and since the right-hand side tends to \(0\) as \(k \rightarrow \infty\), the series \(\sum_{j}G_j\) converges to \(G\) in \(\mathcal{H}^{\Phi}_{q, \, \gamma}(\mathbb{H}^{n})\). ◻
Remark 22. We observe that if \(G \in \mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})\), then \(N_{q, \, 2}(G; z_0) < \infty,\) for some \(z_0 \in \mathbb{H}^{n}\). By \((i)\) in Lemma 12 there exists \(g \in G\) such that \(N_{q, \, 2}(G; z_0) = \eta_{q, \, 2}(g; z_0)\); from Proposition 19 it follows that \(g \in \mathcal{S}'(\mathbb{H}^{n})\). So \(\mathcal{L} g\) is well defined in sense of distributions. On the other hand, since any two representatives of \(G\) differ in a polynomial of homogeneous degree at most \(1\), we get that \(\mathcal{L} g\) is independent of the representative \(g \in G\) chosen. Therefore, for \(G \in \mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})\), we define \(\mathcal{L} G\) as the distribution \(\mathcal{L} g\), where \(g\) is any representative of \(G\).
Theorem 23. Let \(\Phi\) be an Orlicz function of positive upper type \(p_{\Phi}^{+}\). If \(G \in \mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\) and \(\mathcal{L} G = 0\), then \(G \equiv 0\).
Proof. Given \(G \in \mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\), let \(\mathcal{O} := \{ z \in \mathbb{H}^n : N_{q, 2}(G; z) > 1 \}\). By Lemma 10, the set \(\mathcal{O}\) is open with \(| \mathcal{O} | < \infty\), and by Lemmas 3 and 4, we obtain \[\begin{align} \int_{\mathbb{H}^n \setminus \mathcal{O}} N_{q, 2}(G; z)^{p_{\Phi}^{+}} dz &\lesssim& \int_{\mathbb{H}^n \setminus \mathcal{O}} \Phi \left( N_{q, 2}(G; z) \right) dz \\ &\leq& \int_{\mathbb{H}^n} \Phi \left( N_{q, 2}(G; z) \right) dz < \infty. \end{align}\] Then, following the proof of [1] and applying [1] with \(h(\cdot) = N_{q, 2}(G; \cdot) \in L^{p_{\Phi}^{+}}(\mathbb{H}^n \setminus \mathcal{O})\) together with Remark 22, the theorem follows. ◻
If \(a\) is a bounded function with compact support, its potential \(b\), defined as \[\label{potencial32b} b(z) := \left( a \ast c_n \, \rho^{-2n} \right)(z) = c_n \int_{\mathbb{H}^{n}} \rho(w^{-1} \cdot z)^{-2n} a(w) dw,\tag{14}\] is a locally bounded function and, by Theorem 4, \(\mathcal{L} b = a\) in the sense of distributions.
In the sequel, \(Q = 2n+2\) and \(\beta\) is the constant in [24], we observe that \(\beta \geq 1\) (see [24]). For the potentials (14 ), we have the following result.
Lemma 14. Let \(a(\cdot)\) be an \((\Phi, p_{0}, m)\) - atom centered at the \(\rho\) - ball \(B(z_0, \delta)\). If \[b(z) = \left( a \ast c_n \, \rho^{-2n} \right)(z),\] then, for \(\rho(z_0^{-1} z) \geq 2 \beta^{2}\delta\) and every multi-index \(I\) there exists a positive constant \(C_{I}\) such that \[\left| (X^{I}b)(z) \right| \leq C_{I} \, \delta^{2+Q} \| \chi_B \|^{-1}_{\Phi} \rho(z_{0}^{-1} \cdot z)^{-Q-d(I)}\] holds.
The following result is crucial to get our main result.
Proposition 24. Let \(a(\cdot)\) be an \((\Phi, p_{0}, m)\) - atom centered at the \(\rho\) - ball \(B=B(z_0, \delta)\). If \(b(z) = (a \ast c_n \rho^{-2n})(z)\), then for all \(z \in \mathbb{H}^{n}\) \[\begin{align} \label{N32estimate} N_{q, 2} \left(\widetilde{b}; \, z \right) &\lesssim& \| \chi_B \|^{-1}_{\Phi} \left[(M \chi_{B})(z) \right]^{\frac{2 + Q/q}{Q}} + \chi_{4 \beta^2 B}(z) (M a)(z) \\ \notag &+& \chi_{4 \beta^2 B}(z) \sum_{d(I)=2} (T^{*}_{I} a)(z), \end{align}\qquad{(6)}\] where \(\widetilde{b}\) is the class of \(b\) in \(E^{q}_{1}\), \(M\) is the Hardy-Littlewood maximal operator and \((T^{*}_{I} a) (z) = \sup_{\epsilon >0} \left|\int_{\rho(w^{-1} \cdot z) > \epsilon} \, (X^{I} \rho^{-2n})(w^{-1} \cdot z) a(w) \, dw \right|\).
Proof. We point out that the argument used in the proof of [1], to obtain the pointwise inequality (4.9) therein, works in this setting as well, but considering now the conditions \(a1)\), \(a2)\) and \(a3)\) given in Definition 9 of \((\Phi, p_0, m)\) - atom. These conditions are similar to those of the atoms in classical context (see p. 71-72 in [24]). Then, this observation and Lemma 14 allow us to get (?? ). ◻
We are now in a position to prove our main results.
Theorem 25. Let \(Q=2n+2\), \(1 < q < \frac{n+1}{n}\) and let \(\Phi\) be an Orlicz function such that \(Q \, (2 + \frac{Q}{q})^{-1} < i(\Phi) \leq I(\Phi) < \infty\). Then the Heisenberg sub-laplacian \(\mathcal{L}\) is a bijective mapping from \(\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\) onto \(H^{\Phi}(\mathbb{H}^{n})\). Moreover, there exist two positive constant \(c_1\) and \(c_2\) such that \[\label{doble32ineq} c_1 \|G \|_{\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})} \leq \| \mathcal{L}G \|_{H^{\Phi}(\mathbb{H}^{n})} \leq c_2 \|G \|_{\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})}\qquad{(7)}\] hold for all \(G \in \mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\).
Proof. The injectivity of the sub-laplacion \(\mathcal{L}\) in \(\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n})\) was proved in Theorem 23. Now, let \(G \in \mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})\), since \(N_{q, 2}(G; z)\) is finite \(\text{a.e.} \,\, z \in \mathbb{H}^{n}\), by \((i)\) in Lemma 12 and Proposition 19 the unique representative \(g\) of \(G\) (which depends on \(z\)), satisfying \(\eta_{q, 2}(g; z) = N_{q, 2}(G; z)\), is a function in \(L^{q}_{loc}(\mathbb{H}^{n}) \cap \mathcal{S}'(\mathbb{H}^{n})\). In particular, for a radial function \(\phi \in \mathcal{S}(\mathbb{H}^n)\) with \(\int \phi = 1\), by Remark 22 and Proposition 20 we get \[M_{\phi}^{dis}(\mathcal{L}G)(z) \leq C \| \phi \|_{\mathcal{S}(\mathbb{H}^{n}), N} \,\,\, N_{q, \, 2}(G; z).\] Then, this inequality and Theorem 14 give \(\mathcal{L}G \in H^{\Phi}(\mathbb{H}^{n})\) and \[\label{continuity} \| \mathcal{L}G \|_{H^{\Phi}(\mathbb{H}^{n})} \leq C \, \| G \|_{\mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})}.\tag{15}\] This proves the continuity of sublaplacian \(\mathcal{L}\) from \(\mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})\) into \(H^{\Phi}(\mathbb{H}^{n})\).
Next, we shall see that the operator \(\mathcal{L}\) is onto. Given \(f \in H^{\Phi}(\mathbb{H}^{n})\), by Theorem 16, there exist a sequence of nonnegative numbers \(\{ \lambda_j \}_{j=1}^{\infty}\) and a sequence of \(\rho\) - balls \(\{B_j \}_{j=1}^{\infty}\) and \((\Phi, \infty, m)\) - atoms \(a_j\) supported on \(B_j\), such that \(f= \sum_{j=1}^{\infty} \lambda_j a_j\) and \[\label{atomic32ineq} \left\Vert \left\{ \sum_{j} \left( \frac{\lambda_j \chi_{B_j}}{\Vert \chi_{B_j} \Vert_{\Phi}} \right)^{\theta} \right\}^{1/\theta} \right\Vert_{\Phi} \lesssim \|f \|_{H^{\Phi}(\mathbb{H}^{n})}, \,\,\,\,\,\,\,\, \forall \,\,\, 0 < \theta < \min \{ 1, i(\Phi) \}.\tag{16}\] For each \(j \in \mathbb{N}\) we put \(b_j(z)= (a_j \ast c_n \rho^{-2n})(z) = \int_{\mathbb{H}^{n}} c_n \rho(w^{-1} \cdot z)^{-2n} a_j(w) dw\), from Proposition 24 we have \[N_{q, 2} \left(\widetilde{b}_j; \, z \right) \lesssim \| \chi_B \|^{-1}_{\Phi} \left[(M \chi_{B_j})(z) \right]^{\frac{2 + Q/q}{Q}} + \chi_{4 \beta^2 B_j}(z) (M a_j)(z)\] \[+ \chi_{4 \beta^2 B_j}(z) \sum_{d(I)=2} (T^{*}_{I} a_j)(z),\] so \[\begin{align} \sum_{j=1}^{\infty} \lambda_j N_{q, 2} \left(\widetilde{b}_j; \, z \right) &\lesssim& \sum_{j=1}^{\infty} \lambda_j \| \chi_B \|^{-1}_{\Phi} \left[(M \chi_{B_j})(z) \right]^{\frac{2 + Q/q}{Q}} \\ &+& \sum_{j=1}^{\infty} \lambda_j \chi_{4 \beta^2 B_j}(z) (M a_j)(z) \\ &+& \sum_{j=1}^{\infty} \lambda_j \chi_{4 \beta^2 B_j}(z) \sum_{d(I)=2} (T^{*}_{I} a_j)(z) \\ &=:& J_1 + J_2 + J_3. \end{align}\] To study \(J_1\), by hypothesis, we have that \(i(\Phi) > Q \, (2 + \frac{Q}{q})^{-1}\). Then, by Lemma 1, \[\begin{align} \|J_1\|_{\Phi} & = & \left\| \sum_{j=1}^{\infty} \frac{\lambda_j}{\| \chi_B \|_{\Phi}} M(\chi_{B_j})(\cdot)^{\frac{2 + Q/q}{Q}} \right\|_{\Phi} \\ & = & \left\| \left\{ \sum_{j=1}^{\infty} \frac{\lambda_j}{\| \chi_B \|_{\Phi}} M(\chi_{B_j})(\cdot)^{\frac{2 + Q/q}{Q}} \right\}^{\frac{Q}{2 + Q/q}} \right\|_{\Phi_{\frac{2 + Q/q}{Q}}}^{\frac{2 + Q/q}{Q}} \end{align}\] \[\begin{align} & \lesssim & \left\| \left\{ \sum_{j=1}^{\infty} \frac{\lambda_j}{\| \chi_B \|_{\Phi}} \chi_{B_j} \right\}^{\frac{Q}{2 + Q/q}} \right\|_{\Phi_{\frac{2 + Q/q}{Q}}}^{\frac{2 + Q/q}{Q}} \\ & = & \left\| \sum_{j=1}^{\infty} \frac{\lambda_j}{\| \chi_B \|_{\Phi}} \chi_{B_j} \right\|_{\Phi} \\ & \leq & \left\| \left\{ \sum_{j=1}^{\infty} \left( \frac{\lambda_j}{\| \chi_B \|_{\Phi}} \chi_{B_j} \right)^{\theta} \right\}^{1/\theta}\right\|_{\Phi} \lesssim \|f \|_{H^{\Phi}(\mathbb{H}^{n})}, \end{align}\] where the first inequality follows from Lemma 6 and Theorem 12, since \(i(\Phi) > Q \, (2 + \frac{Q}{q})^{-1}\) and \((2 + \frac{Q}{q})/Q > 1\), the embedding \(\ell^{\theta}(\mathbb{N}) \hookrightarrow \ell^{1}(\mathbb{N})\) gives the second inequality, and (16 ) gives the last one.
To estimate \(J_2\), we apply Proposition 17 with \(r = 4 \beta^2 \geq 1\), Remark 15 and (16 ). Then, we obtain \[\| J_2 \|_{\Phi} \lesssim \|f \|_{H^{\Phi}(\mathbb{H}^{n})}.\]
To study \(J_3\), by Theorem 3 in [10] and Corollary 2, p. 36, in [19] (see also 2.5, p. 11, in [19]), we have, for every multi-index \(I\) with \(d(I) = 2\), that the operator \(T_{I}^{*}\) is bounded on \(L^{p_0}(\mathbb{H}^n)\) for each \(1 < p_0 < \infty\). So, Proposition 17 holds with the operators \(\{ T_{I}^{*} \}_{d(I) = 2}\) instead of \(M\) and with the \((\Phi, \infty, m)\) - atoms \(a_j\) (see Remark 15). Then, by (16 ), we get \[\| J_3 \|_{\Phi} \lesssim \|f \|_{H^{\Phi}(\mathbb{H}^{n})}.\]
Thus, \[\left\| \sum_{j=1}^{\infty} \lambda_j N_{q, 2} \left( \widetilde{b}_j; \, \cdot \right) \right\|_{\Phi} \lesssim \|f \|_{{H^{\Phi}(\mathbb{H}^n)}}.\] By Lemma 4, \(\kappa_{\Phi}\left( \sum_{j=1}^{\infty} \lambda_j N_{q, 2}(\widetilde{b}_j; \cdot) \right) < \infty\). Hence \[\sum_{j=1}^{\infty} \lambda_j N_{q, 2}(\widetilde{b}_j; \, z) < \infty \,\,\,\,\,\, \text{a.e.} \, z \in \mathbb{H}^{n} \label{Nq}\tag{17}\] and \[\kappa_{\Phi}\left( \sum_{j=M+1}^{\infty} \lambda_j N_{q, 2}(\widetilde{b}_j; \, \cdot) \right) \rightarrow 0, \,\,\,\, \text{as} \,\, M \rightarrow \infty \label{Nq2}.\tag{18}\] From (17 ) and Lemma 13, there exists a function \(G\) such that \(\sum_{j=1}^{\infty} \lambda_j \widetilde{b}_j = G\) in \(E^{q}_{1}\) and \[N_{q, 2} \left( \left(G - \sum_{j=1}^{M} \lambda_j \widetilde{b}_j \right) ; \, z \right) \leq c \, \sum_{j=M+1}^{\infty} \lambda_j N_{q, 2}(\widetilde{b}_j; z).\] This estimate together with (18 ) and Lemma 5 implies \[\left\| G - \sum_{j=1}^{M} \lambda_j \widetilde{b}_j \right\|_{\mathcal{H}^{\Phi}_{q,2}} \rightarrow 0, \,\,\,\, \text{as} \,\, M \rightarrow \infty.\] So \(G \in \mathcal{H}^{\Phi}_{q,2}(\mathbb{H}^{n})\) and, by Proposition 21, \(G = \sum_{j=1}^{\infty} \lambda_j \widetilde{b}_j\) in \(\mathcal{H}^{\Phi}_{q,2}(\mathbb{H}^{n})\). Since \(\mathcal{L}\) is a continuous operator from \(\mathcal{H}^{\Phi}_{q,2}(\mathbb{H}^{n})\) into \(H^{\Phi}(\mathbb{H}^{n})\), we get \[\mathcal{L}G = \sum_j \lambda_j \mathcal{L} \widetilde{b}_j = \sum_j \lambda_j a_j = f,\] in \(H^{\Phi}(\mathbb{H}^{n})\). This shows that \(\mathcal{L}\) is onto \(H^{\Phi}(\mathbb{H}^{n})\). Moreover, \[\label{continuity322} \| G\|_{\mathcal{H}^{\Phi}_{q,2}} = \left\| \sum_{j=1}^{\infty} \lambda_j \widetilde{b}_j \right\|_{\mathcal{H}^{\Phi}_{q,2}} \lesssim \left\| \sum_{j=1}^{\infty} \lambda_j N_{q, 2}(\widetilde{b}_j; \cdot) \right\|_{\Phi} \lesssim \|f \|_{H^{\Phi}} = \| \mathcal{L} G \|_{H^{\Phi}}.\tag{19}\] Finally, (15 ) and (19 ) give (?? ), and so the proof is concluded. ◻
We shall now see that the case \(0 < I(\Phi) < Q \, (2 + \frac{Q}{q})^{-1}\) is trivial.
Theorem 26. If \(1 < q < \frac{n+1}{n}\) and \(\Phi\) is an Orlicz function with \(0 < I(\Phi) < Q \, (2 + \frac{Q}{q})^{-1}\), then \(\mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n}) = \{ 0 \}.\)
Proof. Let \(G \in \mathcal{H}^{\Phi}_{q, \, 2}(\mathbb{H}^{n})\) and assume \(G \neq 0\). Then there exists \(g \in G\) that is not a polynomial of homogeneous degree less or equal to \(1\). It is easy to check that there exist a positive constant \(c \in (0, 1]\) and a \(\rho\) - ball \(B = B(e, r)\) with \(r > 1\) such that \[\int_{B} |g(w) - P(w)|^{q} \, dw \geq c > 0,\] for every \(P \in \mathcal{P}_{1}\).
Let \(z\) be a point such that \(\rho(z) > r\) and let \(\delta = 2 \rho(z)\). Then \(B(e, r) \subset B(z, \delta)\). If \(h \in G\), then \(h = g - P\) for some \(P \in \mathcal{P}_{1}\) and \[\delta^{-2}|h|_{q, B(z, \delta)} \geq c \rho(z)^{-2-Q/q}.\] So \(N_{q,2}(G; \, z) \geq c \, \rho(z)^{-(2+Q/q)}\), for \(\rho(z) > r\). Since \(\Phi\) is an Orlicz function of positive upper type \(p_{\Phi}^{+}\) with \(I(\Phi) \leq p_{\Phi}^{+} < Q(2+Q/q)^{-1}\) and \(0 < c \, \rho(z)^{-(2+Q/q)} < 1\) on \(\rho(z) > r\), by Lemma 3 - (ii), we have that \[\begin{align} \int_{\mathbb{H}^n} \Phi\left( N_{q,2}(G; z) \right) dz &\geq& \, \int_{\rho(z) > r} \Phi\left( c \rho(z)^{-(2+Q/q)} \right) \, dz \\ &\gtrsim& \int_{\rho(z) > r} \rho(z)^{-(2+Q/q)p^{+}_{\Phi}} \, dz = \infty, \end{align}\] which gives a contradiction (see Lemma 4). Thus \(\mathcal{H}^{\Phi}_{q, 2}(\mathbb{H}^{n}) = \{0\}\), if \(0 < I(\Phi) < Q(2+Q/q)^{-1}\). ◻
Pablo Rocha, Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina.
e-mail: pablo.rocha@uns.edu.ar
Keywords: Orlicz-Calderón Hardy type spaces, Orlicz-Hardy type spaces, atomic decomposition, Heisenberg group, sub-laplacian.
2020 Mathematics Subject Classification: 42B25, 42B30, 42B35, 43A80↩︎