July 21, 2024
In this note we prove that the discrete Riesz potential \(I_{\alpha}\) defined on \(\mathbb{Z}^n\) is a bounded operator \(H^p(\mathbb{Z}^n) \to \ell^q(\mathbb{Z}^n)\) for \(0 < p \leq 1\) and \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\), where \(0 < \alpha < n\).
Given \(0 < \alpha < n\), the discrete Riesz potential \(I_{\alpha}\) on \(\mathbb{Z}^n\) is formally defined by \[\label{Riesz32potential} (I_{\alpha}b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{ j \}} \frac{b(i)}{|i-j |^{n - \alpha}}, \,\,\,\,\,\, j \in \mathbb{Z}^n.\tag{1}\] The continuous counterpart of (1 ) is well known in the literature (see [1], [2], [3], [4], [5], [6]). In the discrete setting, Y. Kanjin and M. Satake in [7] studied the operator given in (1 ) for the case \(n=1\) and proved, for \(0 < p \leq 1\) and \(\frac{1}{q} = \frac{1}{p} - \frac{1}{\alpha}\), their \(H^{p}(\mathbb{Z}) - H^{q}(\mathbb{Z})\) boundedness by means of the molecular decomposition of \(H^p(\mathbb{Z})\). The \(H^{p}(\mathbb{Z}) - \ell^{q}(\mathbb{Z})\) boundedness of (1 ), with \(n=1\), was pointed out by the author in [8]. Discrete operators analogous to (1 ) were studied by E. Stein and S. Wainger in [9] and by D. Oberlin in [10].
The theory for Hardy spaces on \(\mathbb{Z}^n\) was developed by S. Boza and M. Carro in [11] (see also [12]). There, the authors gave a variety of distinct approaches to characterize the discrete Hardy spaces \(H^p(\mathbb{Z}^n)\) analogous to the ones given for the Hardy spaces \(H^p(\mathbb{R}^n)\). Ones of these characterizations is as follows: we consider the discrete Poisson kernel on \(\mathbb{Z}^n\), which is defined by \[P_t^d(j) = C_n \frac{t}{(t^2 + |j|^2)^{(n+1)/2}}, \,\,\,\, t > 0, \,\, j \in \mathbb{Z}^n \setminus \{ {\boldsymbol{0}} \}, \,\, P_t^d({\boldsymbol{0}}) = 0,\] where \(C_n\) is a normalized constant depending on the dimension. For \(0 < p < \infty\) and a sequence \(b = \{ b(i) \}_{i \in \mathbb{Z}^n}\) we say that \(b\) belongs to \(\ell^{p}(\mathbb{Z}^n)\) if \[\| b \|_{\ell^p(\mathbb{Z}^n)} :=\left( \sum_{i \in \mathbb{Z}^n} |b(i)|^p \right)^{1/p} < \infty.\] For \(p=\infty\), we say that \(b\) belongs to \(\ell^{\infty}(\mathbb{Z}^n)\) if \[\|b \|_{\ell^\infty(\mathbb{Z}^n)} := \sup_{i \in \mathbb{Z}^n} |b(i)| < \infty.\] Then, for \(0 < p \leq 1\), we define \[H^p(\mathbb{Z}^n) = \left\{ b \in \ell^p(\mathbb{Z}^n) : \sup_{t>0} |(P_t^d \ast_{\mathbb{Z}^n} b)| \in \ell^p(\mathbb{Z}^n) \right\},\] with the "\(p\)-norm" given by \[\| b \|_{H^p(\mathbb{Z}^n)} := \| b \|_{\ell^p(\mathbb{Z}^n)} + \|\sup_{t>0} |(P_t^d \ast_{\mathbb{Z}^n} b)| \|_{\ell^p(\mathbb{Z}^n)}.\] In [11], S. Boza and M. Carro also gave an atomic characterization of \(H^p(\mathbb{Z}^n)\) for \(0 < p \leq 1\). Before establishing this result we recall the definition of \((p, \infty, d_p)\)-atom in \(H^p(\mathbb{Z}^n)\).
Definition 1. Let \(0 < p \leq 1\) and \(d_p := \lfloor n(p^{-1} - 1) \rfloor\). We say that a sequence \(a = \{ a(j) \}_{j \in \mathbb{Z}^n}\) is an \((p, \infty, d_p)\)-atom centered at a discrete cube \(Q \subset \mathbb{Z}^n\) if the following three conditions hold:
(a1) \(\mathop{\rm supp}a \subset Q\),
(a2) \(\| a \|_{\ell^\infty(\mathbb{Z}^n)} \leq (\# Q)^{-1/p}\),
(a3) \(\displaystyle{\sum_{j \in Q}} j^{\beta} a(j) = 0\) for every multi-index \(\beta=(\beta_1, ..., \beta_n) \in \mathbb{N}_0^n\) with \(\beta_1 + \cdot \cdot \cdot + \beta_n \leq d_p\).
The atomic decomposition mentioned for \(H^p(\mathbb{Z}^n)\) is established in the following theorem.
Theorem 1. ([11]) Let \(0 < p \leq 1\), \(d_p = \lfloor n (p^{-1} - 1) \rfloor\) and \(b \in H^{p}(\mathbb{Z}^n)\). Then there exist a sequence of \((p, \infty, d_p)\)-atoms \(\{ a_k \}_{k=0}^{+\infty}\), a sequence of scalars \(\{ \lambda_k \}_{k=0}^{+\infty}\) and a positive constant \(C\), which depends only on \(p\) and \(n\), with \(\sum_{k=0}^{+\infty} |\lambda_k |^{p} \leq C \| b \|_{H^{p}(\mathbb{Z}^n)}^{p}\) such that \(b = \sum_{k=0}^{+\infty} \lambda_k a_k\), where the series converges in \(H^{p}(\mathbb{Z}^n)\).
The main result of this note is contained in the following theorem, which will be proved in Section 3 via the atomic decomposition of \(H^p(\mathbb{Z}^n)\) joint with some auxiliary results of Section 2.
Theorem 8. Let \(0 < \alpha < n\) and let \(I_{\alpha}\) be the discrete Riesz potential given by (1 ). Then, for \(0 < p \leq 1\) and \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\) \[\| I_{\alpha} \, b \|_{\ell^{q}(\mathbb{Z}^n)} \leq C \| b \|_{H^{p}(\mathbb{Z}^n)},\] where \(C\) does not depend on \(b\).
Notation. Throughout this paper, \(C\) will denote a positive real constant not necessarily the same at each occurrence. We set \(\mathbb{N}_0 = \mathbb{N} \cup \{0\}\). For every \(A \subset \mathbb{Z}^n\), we denote by \(\#A\) and \(\chi_{A}\) the cardinality of the set \(A\) and the characteristic sequence of \(A\) on \(\mathbb{Z}^n\) respectively. Given a real number \(s \geq 0\), we write \(\lfloor s \rfloor\) for the integer part of \(s\).
We start recalling some basic facts about multiple series. A multiple series is of the form \[\displaystyle{\sum_{k \in \mathbb{Z}^{n}}} b(k), \label{multi-series}\tag{2}\] where \(b(k) \in \mathbb{C}\) for each \(k \in \mathbb{Z}^{n}\). There are many different ways to define the sum of a multiple series by means of partial sums. In the literature the following two are the most common (see e.g. [13], [14]):
The Nth-quadratic partial sum of the series in (2 ) is defined by \[S_N = \sum_{|k|_{\infty} \leq N} b(k),\] where \(k =(k_1, ..., k_n) \in \mathbb{Z}^{n}\) and \(|k|_{\infty} = \max \{ |k_i| : i=1, ..., n \}\). If \(\displaystyle{\lim_{N \rightarrow \infty}} S_N\) exists we say that the series in (2 ) is quadratically convergent.
Remark 2. Given a nonnegative sequence \(\{ b(k) \}_{k \in \mathbb{Z}^n}\), we have that \[\sum_{|k|_{\infty} \leq N} b(k) \leq \sum_{|k_n| \leq N} \cdot \cdot \cdot \sum_{|k_1| \leq N} b(k_1, ..., k_n), \,\,\,\,\,\, \forall \, N \geq 1.\]
The Nth-circular partial sum of the series in (2 ) is defined by \[\widetilde{S}_N = \sum_{|k| \leq N} b(k),\] where \(k =(k_1, ..., k_n) \in \mathbb{Z}^{n}\) and \(|k| = (k_1^{2} + \cdot \cdot \cdot + k_n^{2})^{1/2}\). The series in (2 ) is circularly convergent if \(\displaystyle{\lim_{N \rightarrow \infty}} \widetilde{S}_N\) exists.
In general, the circular convergence and the quadratic convergence are not equivalent (see [13], p. 7-8). However, if a series is absolutely convergent in the sense circular or quadratic, then both convergence are equivalent and their sums coincide. Since our results only involve absolutely convergent series we can use one or another definition as it suits.
The following result will be useful in the study of the discrete Riesz potential.
Lemma 1. If \(\epsilon > 0\), then the multiple series \[\sum_{k \in \mathbb{Z}^{n} \setminus \{ \boldsymbol{0} \}} \frac{1}{|k|^{n+\epsilon}} \label{serie320}\tag{3}\] converges.
Proof. Let \(S_N\) be the \(N\)th-quadratic partial sum of the series in (3 ). Then, by Remark 2, we obtain \[S_{N} = \sum_{0 < | k |_{\infty} \leq N} \frac{1}{|k|^{n+\epsilon}} \leq 2^{n} \sum_{k_n =0}^{N} \cdot \cdot \cdot \sum_{k_2 =0}^{N} \sum_{k_1 =1}^{N} \frac{1}{(k_1^{2} + k_2^2 + \cdot \cdot \cdot + k_n^{2})^{\frac{n + \epsilon}{2}}}.\] On the other hand, it is clear that \[\label{estimate32norm} |k_1| + |k_2| + \cdot \cdot \cdot + |k_n| \leq n (k_1^{2} + k_2^2 + \cdot \cdot \cdot + k_n^{2})^{1/2},\tag{4}\] by Multinomial Theorem, for every \(k \in \mathbb{Z}^{n} \setminus \{ \boldsymbol{0} \}\), we have that \[\label{multinomial32ineq} (|k_1| + |k_2| + \cdot \cdot \cdot + |k_n|)^{n + \epsilon} \geq \max \{1,|k_1|^{1+ \frac{\epsilon}{n}}\} \cdot \max \{1,|k_2|^{1+ \frac{\epsilon}{n}}\} \cdot \cdot \cdot \max \{ 1, |k_n|^{1+ \frac{\epsilon}{n}} \}.\tag{5}\] So, \[\sum_{0 < | k |_{\infty} \leq N} \frac{1}{|k|^{n+\epsilon}} \leq 2^{n} \sum_{k_n =0}^{N} \cdot \cdot \cdot \sum_{k_2 =0}^{N} \sum_{k_1 =1}^{N} \frac{n^{n + \epsilon}}{ \max \{1,|k_1|^{1+ \frac{\epsilon}{n}}\} \cdot \max \{1,|k_2|^{1+ \frac{\epsilon}{n}}\} \cdot \cdot \cdot \max \{ 1, |k_n|^{1+ \frac{\epsilon}{n}} \}}\] \[\leq 2^{n}n^{n + \epsilon} \left( 1 + \sum_{k_1=1}^{N} \frac{1}{k_1^{1 + \frac{\epsilon}{n}}} \right)^{n}, \,\,\, \forall \, N \geq 2.\] Finally, letting \(N\) tend to infinity, we obtain \[\sum_{k \in \mathbb{Z}^{n} \setminus \{ \boldsymbol{0} \}} \frac{1}{|k|^{n+\epsilon}} := \lim_{N \rightarrow \infty} S_{N} \leq 2^{n}n^{n+\epsilon} \left( 1 + \sum_{k_1=1}^{\infty} \frac{1}{k_1^{1 + \frac{\epsilon}{n}}} \right)^{n} < \infty.\] ◻
A discrete cube \(Q\) centered at \(j =(j_1, ..., j_n) \in \mathbb{Z}^n\) is of the form \(Q = \prod_{1 \leq l \leq n} [j_l -m, j_l +m]\), where for each \(l=1, ..., n\), \([j_l -m, j_l +m] = \{ j_l - m, ..., j_l, ..., j_l + m \}\) with \(m \in \mathbb{N}_0\). It is clear that \(\# Q = (2m+1)^n\).
Let \(0 \leq \alpha < n\), given a sequence \(b = \{ b(i) \}_{i \in \mathbb{Z}^n}\) we define the centered fractional maximal sequence \(M_{\alpha} b\) by \[(M_{\alpha} b)(j) = \sup_{Q \ni j} \frac{1}{\# Q^{1 - \frac{\alpha}{n}}} \sum_{i \in Q} |b(i)|, \,\,\,\,\,\, j \in \mathbb{Z}^n,\] where the supremum is taken over all discrete cubes \(Q\) centered at \(j\). We observe that if \(\alpha = 0\), then \(M_0 = M\) where \(M\) is the centered discrete maximal operator.
The following result is a consequence of the harmonic analysis on spaces of homogeneous type applied to the space \((\mathbb{Z}^n, \mu, | \cdot |)\) where \(\mu\) is the counting measure and \(| \cdot |\) is the usual distance in \(\mathbb{Z}^n\) (see [15] or [16]). We omit its proof.
Theorem 3. Let \(b = \{ b(i) \}_{i \in \mathbb{Z}^n}\) be a sequence.
If \(b \in \ell^{1}(\mathbb{Z}^n)\), then for every \(\alpha > 0\) \[\#\{ j \in \mathbb{Z}^n : (Mb)(j) > \alpha \} \leq \frac{C}{\alpha} \| b \|_{\ell^{1}(\mathbb{Z}^n)},\] where \(C\) is a positive constant which does not depend on \(\alpha\) and \(b\).
If \(b \in \ell^{p}(\mathbb{Z}^n)\), \(1 < p \leq \infty\), then \(Mb \in \ell^{p}(\mathbb{Z}^n)\) and \[\| Mb \|_{\ell^{p}(\mathbb{Z}^n)} \leq C \| b \|_{\ell^{p}(\mathbb{Z}^n)},\] where \(C\) depends only on \(p\) and \(n\).
Next, we consider \(0 < \alpha < n\), \(1 < p < \frac{n}{\alpha}\) and \(q\) defined by \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\). Let \(Q\) be a discrete cube centered at \(j \in \mathbb{Z}^n\). By taking into account that \(\frac{p}{q} + \frac{\alpha p}{n} = 1\), to apply the Hölder inequality with \(\frac{n-\alpha}{n} + \frac{\alpha}{n} = 1\), we have \[\begin{align} \frac{1}{\# Q^{1 - \frac{\alpha}{n}}} \sum_{i \in Q} |b(i)| &=& \frac{1}{\# Q^{1 - \frac{\alpha}{n}}} \sum_{i \in Q} |b(i)|^{\frac{p}{q}} |b(i)|^{\frac{\alpha p}{n}} \\ &\leq& \left( \frac{1}{\# Q} \sum_{i \in Q} |b(i)|^{\frac{p}{q}(\frac{n}{n-\alpha})} \right)^{\frac{n-\alpha}{n}} \left( \sum_{i \in \mathbb{Z}^n} |b(i)|^{p} \right)^{\frac{\alpha}{n}}, \end{align}\] to take the supremum over all cubes \(Q\) centered at \(j\) we obtain \[\label{fract32max} (M_{\alpha} b)(j) \leq \left[ M \left(|b|^{\frac{p}{q}(\frac{n}{n-\alpha})} \right)(j) \right]^{\frac{n-\alpha}{n}} \left( \sum_{i \in \mathbb{Z}^n} |b(i)|^p \right)^{\frac{\alpha}{n}}, \,\,\,\,\,\, j \in \mathbb{Z}^n.\tag{6}\] Now, the pointwise estimate in (6 ) and Theorem 3 lead to the following result.
Proposition 4. Let \(0 < \alpha < n\). If \(1 < p < \frac{n}{\alpha}\) and \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\), then \[\| M_{\alpha} b \|_{\ell^q(\mathbb{Z}^n)} \leq C \| b \|_{\ell^p(\mathbb{Z}^n)}, \,\,\,\, \forall \,\, b \in \ell^p(\mathbb{Z}^n).\]
We conclude these preliminaries with the following supporting result on \(\mathbb{Z}\).
Proposition 5. Given \(0 < \gamma <1\) and a sequence \(b = \{ b(i) \}_{i \in \mathbb{Z}}\), let \(J_{\gamma}\) be the operator defined by \[(J_{\gamma}b)(j) = \sum_{i \in \mathbb{Z}} \frac{b(i)}{\max \{ 1, |i-j|^{1 - \gamma} \}}, \,\,\,\,\, j \in \mathbb{Z}.\] Then, for \(1 < p < \gamma^{-1}\) and \(\frac{1}{q} = \frac{1}{p} - \gamma\) \[\| J_{\gamma} b \|_{\ell^q(\mathbb{Z})} \leq C \| b \|_{\ell^p(\mathbb{Z})}, \,\,\,\, \forall \, b \in \ell^p(\mathbb{Z}).\]
Proof. It is easy to check that \((J_{\gamma}b)(j) = b(j) + (I_{\gamma}b)(j)\) for every \(j \in \mathbb{Z}\), where \(I_{\gamma}\) is the discrete Riesz potential on \(\mathbb{Z}\). Then, the proposition follows from the fact that \(\ell^p(\mathbb{Z}) \subset \ell^q(\mathbb{Z})\) embeds continuously and that \(I_{\gamma}\) is a bounded operator \(\ell^p(\mathbb{Z}) \to \ell^q(\mathbb{Z})\) for \(1 < p < \gamma^{-1}\) and \(\frac{1}{q} = \frac{1}{p} - \gamma\) (see [17], p. 288). ◻
In this section we establish the \(H^p(\mathbb{Z}^n) - \ell^q(\mathbb{Z}^n)\) boundedness of the discrete Riesz potential \(I_{\alpha}\) on \(\mathbb{Z}^n\). For them, we first start studying the \(\ell^p(\mathbb{Z}^n) - \ell^q(\mathbb{Z}^n)\) boundedness of \(I_{\alpha}\).
Theorem 6. For \(0 < \alpha < n\), let \(I_{\alpha}\) be the discrete Riesz potential given by (1 ) . If \(1 < p < \frac{n}{\alpha}\), \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\) and \(b \in \ell^p(\mathbb{Z}^n)\), then \[\label{pointwise32estim} | (I_{\alpha} b)(j) | < \infty, \,\,\,\, \forall \,\, j \in \mathbb{Z}^n,\tag{7}\] and \[\label{lplq32estim32for32Riesz} \| I_{\alpha} b \|_{\ell^q(\mathbb{Z}^n)} \leq C \| b \|_{\ell^p(\mathbb{Z}^n)}.\tag{8}\]
Proof. Let \(b \in \ell^p(\mathbb{Z}^n)\) with \(1 < p < \frac{n}{\alpha}\), then \(p' > \frac{n}{n-\alpha}\) and so \((n-\alpha) p' - n >0\). To apply the Hölder inequality and Lemma 1 with \(\epsilon := (n-\alpha) p' - n\), we obtain \[| (I_{\alpha} b)(j) | \leq \| b\|_{\ell^p(\mathbb{Z}^n)} \|\{ |i|^{-(n-\alpha)} \}\|_{\ell^{p'}(\mathbb{Z}^n \setminus \{ {\boldsymbol{0}}\})} < \infty, \,\,\,\, \forall \,\, j \in \mathbb{Z}^n.\] Then, (7 ) follows.
From the inequalities (4 ) and (5 ), with \(n - \alpha\) instead of \(n + \epsilon\), and Remark 2, we have for \(j=(j_1, ..., j_n) \in \mathbb{Z}^n\) \[|(I_{\alpha}b)(j)| \leq \sum_{i_n \in \mathbb{Z}} \cdot \cdot \cdot \sum_{i_1 \in \mathbb{Z}} \frac{|b(i_1, ..., i_n)|}{\max \{ 1, |i_1-j_1|^{1 - \alpha/n}\} \cdot \cdot \cdot \max \{ 1, |i_n-j_n|^{1 - \alpha/n}\}}.\] Now, Remark 2 leads to \[\label{estim32Iriesz} \left(\sum_{j \in \mathbb{Z}^n}|(I_{\alpha}b)(j)|^q \right)^{1/q} \leq\tag{9}\] \[\left[ \sum_{j_n \in \mathbb{Z}} \cdot \cdot \cdot \sum_{j_1 \in \mathbb{Z}} \left(\sum_{i_n \in \mathbb{Z}} \cdot \cdot \cdot \sum_{i_1 \in \mathbb{Z}} \frac{|b(i_1, ..., i_n)|}{\max \{ 1, |i_1-j_1|^{1 - \alpha/n}\} \cdot \cdot \cdot \max \{ 1, |i_n-j_n|^{1 - \alpha/n}\}} \right)^q \right]^{1/q}.\] Finally, (8 ) follows from Proposition 5 with \(\gamma = \frac{\alpha}{n}\), the Minkowski’s inequality for integrals on the \(\sigma\)-finite product measure space \(\mathbb{Z} \times \mathbb{Z}\) with the counting measure, and an iterative argument applied on the right-hand side of the inequality that appears in (9 ). ◻
Theorem 8. Let \(0 < \alpha < n\) and let \(I_{\alpha}\) be the discrete Riesz potential given by (1 ). Then, for \(0 < p \leq 1\) and \(\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}\) \[\| I_{\alpha} \, b \|_{\ell^{q}(\mathbb{Z}^n)} \leq C \| b \|_{H^{p}(\mathbb{Z}^n)},\] where \(C\) does not depend on \(b\).
Proof. We take \(p_0\) such that \(1 < p_0 < \frac{n}{\alpha}\). By Theorem 1, given \(b \in H^{p}(\mathbb{Z}^n)\) we can write \(b = \sum_k \lambda_k a_k\) where the \(a_k\)’s are \((p, \infty, d_p)\) atoms, the scalars \(\lambda_k\) satisfies \(\sum_{k} |\lambda_k |^{p} \leq C \| b \|_{H^{p}(\mathbb{Z}^n)}^{p}\) and the series converges in \(H^{p}(\mathbb{Z}^n)\) and so in \(\ell^{p_0}(\mathbb{Z}^n)\) since \(H^{p}(\mathbb{Z}^n) \subset \ell^{p}(\mathbb{Z}^n) \subset \ell^{p_0}(\mathbb{Z}^n)\) embed continuously. For \(\frac{1}{q_0} = \frac{1}{p_0} - \frac{\alpha}{n}\), by Theorem 6, \(I_{\alpha}\) is a bounded operator \(\ell^{p_0}(\mathbb{Z}^n) \to \ell^{q_0}(\mathbb{Z}^n)\). Since \(b = \sum_k \lambda_k a_k\) in \(\ell^{p_0}(\mathbb{Z}^n)\), we have that \((I_{\alpha} \, b)(j) = \sum_{k} \lambda_k (I_{\alpha} \, a_k)(j)\) for all \(j \in \mathbb{Z}^n\), and thus \[|(I_{\alpha} \, b)(j)| \leq \sum_{k} |\lambda_k| |(I_{\alpha} \, a_k)(j)|, \,\,\,\, \forall \, j \in \mathbb{Z}^n. \label{puntual}\tag{10}\] Then for \(1 \leq q\), by (10 ) and Minkowski’s integral inequality on \(\sigma\)-finite measure spaces, we have \[\label{mink32ineq} \|I_{\alpha} \, b \|_{\ell^{q}(\mathbb{Z}^n)} \leq \sum_{k} |\lambda_k| \|I_{\alpha} \, a_k \|_{\ell^{q}(\mathbb{Z}^n)};\tag{11}\] now for \(0 < q < 1\), from (10 ), it is easy to check that \[\label{q32ineq} \|I_{\alpha} \, b \|_{\ell^{q}(\mathbb{Z}^n)}^q \leq \sum_{k} |\lambda_k|^q \|I_{\alpha} \, a_k \|_{\ell^{q}(\mathbb{Z}^n)}^q.\tag{12}\] Thus, if for \(0 < p \leq 1\) and \(\frac{1}{q}= \frac{1}{p} - \frac{\alpha}{n}\) we see that \(\|I_{\alpha} \, a_k \|_{\ell^{q}(\mathbb{Z}^n)} \leq C\), with \(C\) independent of the \((p, \infty, d_p)\)-atom \(a_k\), then the estimate (11 ) or (12 ) according to the case and the fact that \(\sum_{k} |\lambda_k |^{p} \leq C \| b \|_{H^{p}(\mathbb{Z}^n)}^{p}\) lead to \[\|I_{\alpha} \, b \|_{\ell^{q}(\mathbb{Z}^n)} \leq C \left( \sum_{k} |\lambda_k|^{\min\{1, q \}} \right)^{\frac{1}{\min\{1, q \}}} \leq C \left( \sum_{k} |\lambda_k |^{p} \right)^{1/p} \leq C\| b \|_{H^{p}(\mathbb{Z}^n)}.\] Being \(b\) an arbitrary element of \(H^{p}(\mathbb{Z}^n)\), the theorem follows.
To conclude the proof we will prove that for \(0 < p \leq 1\) and \(\frac{1}{q}= \frac{1}{p} - \frac{\alpha}{n}\) there exists an universal constant \(C >
0\), which depends on \(\alpha\), \(n\), \(p\) and \(q\) only, such that
\[\label{uniform32estimate}
\|I_{\alpha} \, a \|_{\ell^{q}(\mathbb{Z}^n)} \leq C, \,\,\,\, \textit{for all} \,\, (p, \infty, d_p) - \textit{atom} \,\, a=\{ a(i) \}.\tag{13}\] To prove (13 ), let \(a(\cdot)\) be an atom centered at the cube \(Q_{k^0}= \prod_{1 \leq l \leq n}[ k^0_l - m, k^0_l + m ]\). We put \(4\lfloor \sqrt{n} \rfloor Q_{k^0}=\prod_{1 \leq l \leq
n} \left[ k^0_l - 4\lfloor \sqrt{n} \rfloor m, k^0_l + 4\lfloor \sqrt{n} \rfloor m \right]\). So \[\label{sum2}
\sum_{j \in \mathbb{Z}^n} |(I_{\alpha} \, a)(j)|^{q} = \sum_{j \in 4\lfloor \sqrt{n} \rfloor Q_{k^0}} |(I_{\alpha} \, a)(j)|^{q} +
\sum_{j \in \mathbb{Z}^n \setminus 4\lfloor \sqrt{n} \rfloor Q_{k^0}} |(I_{\alpha} \, a)(j)|^{q}.\tag{14}\] To estimate the first sum, we apply Hölder inequality with respect the exponent \(\frac{q_0}{q}\),
then from Theorem 6, the size condition (a2) on the atom \(a(\cdot)\), and since \(\frac{1}{p} - \frac{1}{q} =
\frac{1}{p_0} - \frac{1}{q_0} = \frac{\alpha}{n}\) we have \[\label{estim32C}
\sum_{j \in 4\lfloor \sqrt{n} \rfloor Q_{k^0}}|(I_{\alpha} \, a)(j)|^{q} \leq
\left(\frac{8 \lfloor \sqrt{n} \rfloor + 1}{2} \right)^{n(q_0 - q)/q_0} \cdot
\left( \sum_{j \in \mathbb{Z}^n} |(I_{\alpha} \, a)(j)|^{q_0}\right)^{q/q_0} \cdot (\# Q_{k^0})^{(q_0-q)/q_0}\tag{15}\] \[\leq C \left( \sum_{j \in Q_{k^0}}|a (j)|^{p_0}\right)^{q/p_0} \cdot (\#
Q_{k^0})^{(q_0-q)/q_0}\] \[\leq C \, (\# Q_{k^0})^{-q/p} \cdot (\# Q_{k^0})^{q/p_0} \cdot (\# Q_{k^0})^{(q_0-q)/q_0} = C,\]
with \(C\) independent of \(k^0\) and \(m\).
To estimate the second sum in (14 ), we put \(N - 1 = \lfloor n(p^{-1} - 1) \rfloor\). In view of the moment condition (a3) of \(a(\cdot)\) we have, for \(j \in \mathbb{Z}^n \setminus 4 \lfloor \sqrt{n} \rfloor Q_{k^0}\), that \[(I_{\alpha} \, a)(j) = \sum_{i \in Q_{k^0}} |i-j|^{\alpha-n} \, a(i) = \sum_{i \in Q_{k^0}} [|i-j|^{\alpha-n} - q_{N}(i,j)] \, a(i),\] where \(q_{N}(\, \cdot \,, j)\) is the degree \(N - 1\) Taylor polynomial of the function \(x \rightarrow |x-j|^{\alpha-n}\) expanded around \(k^0\). By the standard estimate of the remainder term in the Taylor expansion there exists \(\xi\) between \(i\) and \(k^0\) such that \[| |i-j|^{\alpha-n} - q_{N}(i, j) | \leq C |i - k^0 |^{N} | j - \xi|^{\alpha-n-N},\] for any \(i \in Q_{k^0}\) and any \(j \notin 4 \lfloor \sqrt{n} \rfloor Q_{k^0}\). Since \(|j - \xi| \geq \displaystyle{\frac{|j - k^0|}{2}}\), we get \[| |i-j|^{\alpha-n} - q_{N}(i, j) | \leq C (2m+1)^{N} | j - k^0|^{\alpha-n-N}.\] This inequality and the condition (a2) of the atom \(a(\cdot)\) allow us to conclude that \[|(I_{\alpha}a)(j)| \leq C \frac{(2m+1)^{n+N}}{(\# Q_{k^0})^{1/p}} | j - k^0|^{\alpha-n-N} \leq \frac{C}{(\# Q_{k^0})^{1/p}} \left[ M_{\frac{\alpha n}{n+N}} (\chi_{Q_{k^0}})(j) \right]^{\frac{n+N}{n}},\] for all \(j \notin 4 \lfloor \sqrt{n} \rfloor Q_{k^0}\). Thus, \[\label{cota32afuera} \sum_{j \in \mathbb{Z}^n \setminus 4\lfloor \sqrt{n} \rfloor Q_{k^0}} |(I_{\alpha}a)(j)|^q \leq \frac{C}{(\# Q_{k^0})^{q/p}} \sum_{j \in \mathbb{Z}^n} \left[ M_{\frac{\alpha n}{n+N}} (\chi_{Q_{k^0}})(j) \right]^{q \frac{n+N}{n}}.\tag{16}\] Since \(N-1= \lfloor n(\frac{1}{p}-1) \rfloor\), we have \(q \frac{n+N}{n} > 1\). We write \(\widetilde{q} = q \frac{n+N}{n}\) and let \(\frac{1}{\widetilde{p}} = \frac{1}{\widetilde{q}} + \frac{\alpha}{n+N}\), so \(\frac{\widetilde{p}}{\widetilde{q}} = \frac{p}{q}\). From Proposition 4, we obtain \[\label{cota32afuera322} \sum_{j \in \mathbb{Z}^n} \left[ M_{\frac{\alpha n}{n+N}} (\chi_{Q_{k^0}})(j) \right]^{q \frac{n+N}{n}} \leq C \left( \sum_{j \in \mathbb{Z}^n} \chi_{Q_{k^0}}(j) \right)^{q/p} = C (\# Q_{k^0})^{q/p}.\tag{17}\] Now, (16 ) and (17 ) give \[\label{cota32afuera323} \sum_{j \in \mathbb{Z}^n \setminus 4\lfloor \sqrt{n} \rfloor Q_{k^0}} |(I_{\alpha}a)(j)|^q \leq C.\tag{18}\] Finally, (15 ) and (18 ) lead to (13 ). Thus the proof is concluded. ◻
Remark 9. In [7] for \(n=1\), \(0 < p \leq 1\), \(1/q = 1/p - \alpha\) and \(0 < \alpha < 1\), the \(H^p(\mathbb{Z}) \to H^q(\mathbb{Z})\) boundedness of \(I_{\alpha}\) was obtained. Recently, in [18] we generalize this result on \(\mathbb{Z}^n\). More precisely, we prove the \(H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)\) boundedness of \(I_{\alpha}\) for \(n \geq 1\), \(\frac{n-1}{n} < p \leq 1\), \(1/q = 1/p - \alpha/n\) and \(0 < \alpha < n\). To prove this result, as in [7], we furnish a molecular decomposition for the elements of \(H^{p}(\mathbb{Z}^n)\) on the range \(\frac{n-1}{n} < p \leq 1\). This decomposition joint with some results and ideas of the present work allow us to obtain such estimate.
Acknowledgements. I express my thanks to the referees for their useful suggestions and comments.
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