October 16, 2025
We introduce new classes of general monotone sequences and study their properties. For functions whose Fourier coefficients belong to these classes, we establish Hardy–Littlewood-type theorems.
In this paper, we explore the relationship between an integrable function \(f(x)\) defined on the interval \([-\pi,\pi]\) and its Fourier coefficients sequence. One such result is the well-known Hardy-Littlewood theorem [1], [2].
Theorem A 1. Let \(\{a_k\}_{k=0}^{\infty}\), \(\{b_k\}_{k=1}^{\infty}\) be nonincreasing, nonnegative sequences, and \(f(x) \sim \frac{a_0}{2} + \sum\limits_{k=1}^{\infty} (a_k \cos kx + b_k \sin kx)\). Then, for any \(1 < p < \infty\), \[\label{HL1} \|f\|_{L_p([-\pi,\pi])} \asymp \frac{a_0}{2} + \left(\sum_{k=1}^{\infty} k^{p-2}(a_k^p + b_k^p) \right)^{\frac{1}{p}}.\qquad{(1)}\]
The corresponding result for the function \(f(x) \sim \sum\limits_{k=-\infty}^{\infty} c_k e^{i kx}\), where \(\{c_k\}_{k=0}^{\infty}\), \(\{c_{-k}\}_{k=0}^{\infty}\) are nonincreasing, nonnegative sequences, can be stated as follows: \[\label{HL2} \|f\|_{L_p([-\pi, \pi])} \asymp \left(\sum_{k = -\infty}^{\infty} (|k|+1)^{p-2}|c_k|^p\right)^{\frac{1}{p}}, \quad 1 < p < \infty.\tag{1}\]
Equivalences ?? , 1 were extended in many papers, e.g., see [3]–[14] and bibliographies therein. Mainly, authors of extensions of the Hardy-Littlewood theorem impose weaker conditions on Fourier coefficients than the monotonicity condition.
In [8], [14], relation ?? was proved for functions with nonnegative Fourier coefficients \(\{a_k\}_{k=1}^{\infty}\), \(\{b_k\}_{k=1}^{\infty}\) from class of general monotone sequences defined as follows:
Definition 1 ([15]). We will say that a sequence of complex numbers \(\{a_k\}_{k=1}^{\infty}\) belongs to class \(\textrm{GM}\), if there exist constants \(C > 0\) and \(\lambda >1\) such that, for any \(n \geqslant 1\), \[\sum_{k=n}^{2n}|a_k - a_{k+1}| \leqslant\frac{C}{n}\sum_{k = \frac{n}{\lambda}}^{\lambda n} |a_k|.\]
In [4], equivalence ?? was obtained for functions with real-valued Fourier coefficients \(\{a_k\}_{k=1}^{\infty}\), \(\{b_k\}_{k=1}^{\infty}\) from \(\textrm{GM}\) without assumption on nonnegativity.
In [10], equivalence 1 was derived for functions \(f(x) \sim \sum\limits_{k=0}^{\infty} c_k e^{ikx}\), where \(\{c_k\}_{k=1}^{\infty} \in \textrm{GM}\) is a sequence of complex numbers \(c_k \in S_{\alpha, \beta} = \{z \in \mathbb{C} : |\arg z -\alpha| \leqslant\beta \}, k \geqslant 1\), \(0 \leqslant\alpha < 2\pi\), \(0 \leqslant\beta < \frac{\pi}{2}\).
In [11], equivalence 1 was established, in case when \(2 \leqslant p < \infty\), for functions \(f(x) \sim \sum\limits_{k=1}^{\infty} c_k e^{ikx}\), where the sequence \(\{c_k\}_{k=1}^{\infty}\) of complex numbers belongs to class of weak monotone sequences.
Definition 2. We will say that a sequence \(\{a_{n}\}_{n=1}^{\infty}\) of complex numbers belongs to the class of weak monotone sequences \(\textrm{WM}\), if there exists \(C > 0\) such that, for any \(n \geqslant 1\), \[|a_n| \leqslant\frac{C}{n} \left|\sum_{j = 1}^{n} a_{j}\right|.\]
Remark 1. The notions of general monotonicity and weak monotonicity were introduced by S. Tikhonov (see [15], [16]). Various classes of general monotone and weak monotone sequences (or functions) have been studied in the context of many problems of Fourier analysis and approximation theory; see, for instance, [3], [17].
Analogues of the Hardy-Littlewood theorem for series with complex coefficients satisfying another general monotonicity conditions can be found in [3], [5].
Another generalizations of the Hardy-Littlewood theorem can be found in [13], [18], [19].
The main aim of this paper is to define the class of sequences \(\{a_k\}_{k=-\infty}^{\infty}\) of complex numbers such that, for any \(1< p < \infty\), the relation \[\|f\|_{L_p([-\pi,\pi])} \asymp J_p(f)\] holds, where \(f(x) \in L_1([-\pi, \pi])\) is a function with Fourier series \(\sum\limits_{k= -\infty}^{\infty} a_k e^{ikx}\), and \(J_p(f)\) is Paley type functional, defined as follows \[J_p(f):= \left(\sum_{k= -\infty}^{\infty} (|k|+1)^{p-2} |a_k|^p\right)^{\frac{1}{p}}.\]
The similar question is considered in the setting of the following Paley type functional \[J_p^*(f) = \left(\sum\limits_{k=-\infty}^{\infty} (|k|+1)^{p-2} (a_k^*)^p\right)^{\frac{1}{p}},\] where \(\{a_k^*\}_{k=-\infty}^{\infty}\) is symmetric nonincreasing rearrangement of sequence \(\{a_k\}_{k=-\infty}^{\infty}\), i.e., \(a_0^* \geqslant a_{-1}^* \geqslant a_1^* \geqslant a_{-2}^* \geqslant a_2^* \geqslant\ldots\).
For the sequence \(\{a_k\}_{k=-\infty}^{\infty}\), for the sake of symmetry, we put \[|\Delta a_k| := \begin{cases} |a_k - a_{k+1}|, & k > 0,\\ |a_k - a_{k-1}|, & k < 0,\\ |a_0 - a_1| + |a_0 - a_{-1}|, & k = 0. \end{cases}\]
We will call by the interval in \(\mathbb{Z}\) any finite arithmetic progression with difference equal to 1. By \(\mathcal{W}\) we denote the set of all intervals in \(\mathbb{Z}\). Let us define new two classes of general monotone sequences \(\textrm{GM}^*\) and \(\overline{\textrm{GM}}\).
Definition 3. We will say that a sequence \(\{a_{m}\}_{m=-\infty}^{\infty}\) of complex numbers belongs to the class \(\textrm{GM}^*\), if there exists \(C > 0\) such that, for any \({n} \geqslant 0\), \[\sum_{[2^{n-1}]\leqslant|m| < 2^{n}}\left|\Delta a_{m}\right|\leqslant C \sup_{k \in \mathbb{N}_0} \min (1, 2^{k-n}) \widetilde{a}_{2^k},\] where \[\widetilde{a}_{2^k} = \sup_{\substack{{w} \in \mathcal{W} \\|{w}|\geqslant 2^{k}}} \frac{1}{|{w}|} \left|\sum_{{j} \in {w}} a_{j}\right|, \quad k \geqslant 0,\] and \([\cdot]\) is the floor function, and \(|w|\) is the cardinality of the set \(w \in \mathcal{W}\).
Remark 2. Note that \[[2^{n-1}] = \begin{cases} 2^{n-1}, & if \;\; n \in \mathbb{N},\\ 0, & if \;\;n = 0. \end{cases}\]
Definition 4. We will say that a sequence \(\{a_{m}\}_{m=-\infty}^{\infty}\) of complex numbers belongs to the class \(\overline{\textrm{GM}}\), if there exists \(C > 0\) such that, for any \({n} \geqslant 0\), \[\sum_{[2^{n-1}] \leqslant|m| < 2^n}\left|\Delta a_{m}\right|\leqslant C \sup_{k \in \mathbb{N}_0} \min (1, 2^{k-n}) \widehat{a}_{2^k},\] where \[\widehat{a}_{2^k} = \sup_{2^{k}\leqslant|m| < 2^{k+1}} \frac{1}{|m|+1} \left|\sum_{j=0}^{m} a_j\right|, \quad k \geqslant 0.\]
Remark 3. It is easy to see that \(\overline{\textrm{GM}} \subseteq \textrm{GM}^*\).
The main results of this paper is the following theorems.
Theorem 1. Let \(1<p<\infty\) and \(f \in L_1([-\pi, \pi])\) be a function with Fourier series \(\sum\limits_{k=-\infty}^{\infty} a_{k} e^{i{kx}}\). Let also \(a=\{a_{k}\}_{k=-\infty}^{\infty} \in \textrm{GM}^*\), then \[\label{f4} \|f\|_{L_p([-\pi, \pi])} \asymp J_p^*(f).\tag{2}\]
Theorem 2. Let \(1<p<\infty\) and \(f \in L_1([-\pi, \pi])\) be a function with Fourier series \(\sum\limits_{k=-\infty}^{\infty} a_{k} e^{i{kx}}\). Let also \(a=\{a_{k}\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\), then \[\label{f5} \|f\|_{L_p([-\pi, \pi])} \asymp J_p(f).\tag{3}\]
I. One can see that the conditions stated in Theorems 1 and 2 are presented in their most general form. In particular, in Theorems 1 and 2, the following conditions hold:
The Fourier coefficients of the function are allowed to be complex;
No additional restrictions are imposed on the coefficients, other than the general monotonicity condition;
The equivalence of function’s \(L_p\)-norm to the Paley type functional holds for the entire range \(1 < p < \infty\).
II. Note that the averages \(\widetilde{a}_{2^k}\) and \(\widehat{a}_{2^k}\) from Definitions 3 and 4 involve both elements of the sequence \(\{a_n\}_{n=-\infty}^{\infty}\) with positive indices and its elements with non-positive indices. This feature of Definitions 3 and 4 has a certain compensatory effect, which made it possible to extend the classes of sequences for which the Hardy–Littlewood theorem remains valid in a different way. For example, in previously obtained generalizations of the Hardy–Littlewood theorem, for series of the form \(\sum\limits_{k=1}^{\infty} (a_k \cos kx + b_k \sin kx),\) general monotonicity condition is required for both sequences \(\{a_k\}_{k=1}^{\infty}\) and \(\{b_k\}_{k=1}^{\infty}\). Similarly, for series of the form \(\sum\limits_{k=-\infty}^{\infty} c_k e^{i kx},\) previously obtained generalizations require general (or weak) monotonicity condition for both sequences \(\{c_k\}_{k=0}^{\infty}\) and \(\{c_{-k}\}_{k=0}^{\infty}\). In turn, general monotonicity conditions in classes \(\textrm{GM}^*\) and \(\overline{\textrm{GM}}\) allow certain parts of sequences to behave poorly. For instance, Section 6 presents an example of a sequence \(\{c_k\}_{k=-\infty}^{+\infty} \in \overline{\textrm{GM}}\) such that \(\{c_k\}_{k=0}^{\infty} \notin \overline{\textrm{GM}}\).
III. Developing the idea of extending the classes of Fourier coefficient sequences for which the equivalence \(\|f\|_{L_p} \asymp J_p(f)\) remains valid, we consider alternating series. For an alternating sequence \(\{c_k\}_{k=-\infty}^{\infty}\), the condition of general monotonicity does not hold, even if the sequence \(\{|c_k|\}_{k=-\infty}^{\infty}\) is monotone, because \(\sum\limits_{k=n}^{2n} |\Delta c_k| \asymp \sum\limits_{k=n}^{2n} |c_k|.\) Nevertheless, despite these restrictions, we have succeeded in establishing the equivalence \(\|f\|_{L_p} \asymp J_p(f)\) for alternating series (see Corollary 1), as well as for other cases, by using Fourier \(L_p\)-multipliers.
The paper is organized as follows. In Section 2, we introduce the net spaces and give some auxiliary results. In Section 3, we obtain some Fourier type inequalities in the setting of the net spaces. In Section 4, we prove Theorems 1 and 2. In Section 5, we compare some classes of general monotone sequences. In particular, we show that \(\textrm{GM}_{\mathbb{R}} \subsetneq \overline{\textrm{GM}}\), where \(\textrm{GM}_{\mathbb{R}} = \left\{\{a_k\}_{k=1}^{\infty} \in \textrm{GM}: \;\;a_k \in \mathbb{R}, \;\;k \geqslant 1\right\}\). In Section 6, we provide examples of sequences that illustrate the compensatory effect of the general monotonicity condition in the class \(\overline{\textrm{GM}}\). In Section 7, we establish the equivalence \(\|f\|_{L_p} \asymp J_p(f)\) for functions \(f(x) \sim \sum\limits_{k=-\infty}^{\infty} c_k e^{ikx}\) satisfying condition \(\{\lambda_k c_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\) with some Fourier idempotent multiplier \(\{\lambda_k\}_{k=-\infty}^{\infty}\).
Throughout the paper, the expression \(L \lesssim R\) means that \(L \leqslant CR\) for some constant \(C > 0\). Moreover, \(L \asymp R\) stands for \(L \lesssim R \lesssim L\).
Let \((\Omega, \mu)\) be a measurable space, and let \(f(x)\) be a \(\mu\)-measurable on \(\Omega\) function. By \(f^{*}(t)\) we denote the nonincreasing rearrangement of \(f(x)\), i.e., \[f^*(t) = \inf\{\sigma: \mu\{x \in \Omega: |f(x)|> \sigma\}\leqslant t\}.\]
For \(0 < p \leqslant\infty\), \(0 < q \leqslant\infty\), the Lorentz space \(L_{p,q}(\Omega)\) is the set of \(\mu\)-measurable functions for which, the functional \[\|f\|_{L_{p,q}} = \begin{cases} \left(\int\limits_0^{\mu(\Omega)} t^{\frac{q}{p}-1} \left(f^{*}(t)\right)^q dt\right)^{\frac{1}{q}}, & \text{for} \;0 < p <\infty \;\text{and} \;0 < q < \infty, \\ \sup\limits_{t \in [0,\, \mu(\Omega)]} t^{\frac{1}{p}}f^*(t), & \text{for} \;0 < p \leqslant\infty \;\text{and} \;q = \infty, \end{cases}\] is finite.
Remark 4.
In case when \(\Omega = \mathbb{Z}\) and \(\mu\) is a counting measure, corresponding discrete Lorentz spaces are denoted as \(l_{p,q}(\mathbb{Z})\).
In case when \(p=q\), \(L_{p,p}(\Omega) = L_p(\Omega)\).
Now, let’s describe the main tool of this paper: net spaces. The net spaces was introduced by E. Nursultanov in [11]. Net space methods have been used to solve many problems in Fourier analysis (see [5], [11], [12], [20]–[22]), approximation theory (see [23], [24]), and Fourier multiplier theory (see [25]).
To define the discrete net spaces we need introduce the average of the sequence.
For any sequence \(\{a_{m}\}_{m=-\infty}^{\infty}\) of complex numbers, we will define the average \(\{\widetilde{a}_{k}\}_{k=1}^{\infty}\) over \(\mathcal{W}\) as follows: \[{\widetilde{a}}_{k} = {\widetilde{a}}_{k}(\mathcal{W})=\sup_{\substack{{w} \in \mathcal{W}\\ |w|\geqslant k}} \frac{1}{|w|}\left|\sum_{m \in w} a_{m}\right|, \quad k \in \mathbb{N}.\]
Definition 5 ([11]). Let \(1 < p < \infty\), \(1 \leqslant q \leqslant\infty\). Discrete net space \(n_{p,q}\) is the set of sequences of complex numbers \(\{a_{m}\}_{m=-\infty}^{\infty}\) for which, the functional \[\|a\|_{n_{p,q}} = \begin{cases}\left(\sum\limits_{k=1}^\infty k^{\frac{q}{p}-1}{\widetilde{a}}_{k}^q\right)^{1/q}, & \textrm{for} \;1 < p < \infty \;\textrm{and} \;1 < q < \infty,\\ \sup\limits_{k \geqslant 1} k^{\frac{1}{p}} \widetilde{a}_k,& \textrm{for} \;1 < p \leqslant\infty \;\textrm{and} \; q = \infty, \end{cases}\] is finite.
Lemma 1. Let \(\{a_n\}_{n=-\infty}^{\infty}\) be a sequence such that \(\widetilde{a}_1 < \infty\). Then, for any integer \(k \geqslant 0\), \[\label{almost95mon} \widetilde{a}_{2^k} \leqslant 5 \widetilde{a}_{2^{k+1}}.\tag{4}\]
Proof. Fix an integer \(k \geqslant 0\) and consider an interval \(w_1 \subset \mathbb{Z}\) with \(|w_1| \geqslant 2^k\). Define an interval \(w \subset \mathbb{Z}\) satisfying the following conditions:
\(w_1 \subset w\);
\(|w| = 3 |w_1|\);
the set \(w_2 := w \setminus w_1\) is an interval of integer numbers.
For these intervals, observe that \[\left|\sum_{m \in w_1} a_m\right| = \left|\sum_{m \in w} a_m - \sum_{m \in w_2} a_m\right| \leqslant\left|\sum_{m \in w} a_m\right| + \left|\sum_{m \in w_2} a_m\right|.\] Dividing both sides of this inequality by \(|w_1|\), we obtain \[\begin{align} \frac{1}{|w_1|}\left|\sum_{m \in w_1} a_m\right| & \leqslant\frac{1}{|w_1|}\left|\sum_{m \in w} a_m\right| + \frac{1}{|w_1|}\left|\sum_{m \in w_2} a_m\right| = \frac{3}{|w|}\left|\sum_{m \in w} a_m\right| + \frac{2}{|w_2|}\left|\sum_{m \in w_2} a_m\right|. \end{align}\] Taking the supremum over all intervals \(w_1\) with \(|w_1| \geqslant 2^k\), we get \[\begin{align} \widetilde{a}_{2^k} & = \sup_{|w_1|\geqslant 2^k}\frac{1}{|w_1|}\left|\sum_{m \in w_1} a_m\right| \leqslant\sup_{|w_1|\geqslant 2^k}\left( \frac{3}{|w|}\left|\sum_{m \in w} a_m\right| + \frac{2}{|w_2|}\left|\sum_{m \in w_2} a_m\right|\right) \\ & \leqslant\sup_{|w|\geqslant 3\cdot 2^k} \frac{3}{|w|}\left|\sum_{m \in w} a_m\right| + \sup_{|w_2|\geqslant 2\cdot 2^k}\frac{2}{|w_2|}\left|\sum_{m \in w_2} a_m\right| \leqslant\sup_{|w_2|\geqslant 2^{k+1}}\frac{5}{|w_2|}\left|\sum_{m \in w_2} a_m\right| = 5 \widetilde{a}_{2^{k+1}}. \end{align}\] Thus, the desired inequality is established. ◻
Using standard arguments, along with the monotonicity of the sequence \(\{\widetilde{a_k}\}_{k=1}^{\infty}\) and inequality [alm95mon], we arrive at the following lemma.
Lemma 2. For any \(1 < p < \infty\), \(1 \leqslant q \leqslant\infty\), the following equivalence holds: \[\|a\|_{n_{pq}} \asymp \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p}}\widetilde{a}_{2^{k}}\right)^q\right)^{\frac{1}{q}}.\]
Lemma 3. Let \(\{a_k\}_{k=-\infty}^{\infty}\) be a sequence of complex numbers, and define a new sequence \(\{b_k\}_{k=-\infty}^{\infty}\) as follows: \[b_k = \begin{cases} a_k, & k \neq 0;\\ \frac{1}{2}a_0, & k = 0. \end{cases}\] Then, for any integer \(r \geqslant 0\), the following inequality holds: \[\widetilde{b}_{2^r} \leqslant 6\widetilde{a}_{2^r}.\]
Proof. Let \(r \geqslant 0\), and let \(w \subset \mathbb{Z}\) be an interval of integers such that \(|w| \geqslant 2^r\), \(|w| > 1\), and \(0 \in w\). Then, we have \[\label{ab-1} \begin{align} \frac{1}{|w|} \left|\sum_{m\in w} b_m\right| & = \frac{1}{|w|} \left|b_0 + \sum_{m\in w \setminus{\{0\}}} b_m\right| = \frac{1}{|w|} \left|\frac{1}{2}a_0 + \sum_{m\in w \setminus{\{0\}}} a_m\right| \\ & = \frac{1}{2|w|} \left|a_0 + 2\sum_{m\in w \setminus{\{0\}}} a_m\right| \leqslant\frac{1}{2|w|} \left|\sum_{m\in w } a_m\right| + \frac{1}{2|w|} \left|\sum_{m\in w \setminus{\{0\}}} a_m\right|\\ & = \frac{1}{2|w|} \left|\sum_{m\in w } a_m\right| + \frac{|w|-1}{2|w|} \frac{1}{|w|-1} \left|\sum_{m\in w \setminus{\{0\}}} a_m\right|\\ & < \frac{1}{|w|} \left|\sum_{m\in w } a_m\right| + \frac{1}{|w|-1} \left|\sum_{m\in w \setminus{\{0\}}} a_m\right|. \end{align}\tag{5}\] Now, consider the family of intervals \(\Omega_r = \{w \in \mathcal{W} \mid |w| \geqslant 2^r\}\). Divide this family into two subfamilies: \(\Omega_{r,1} = \{w \in \Omega_r \mid 0 \in w\}\) and \(\Omega_{r,2} = \{w \in \Omega_r \mid 0 \notin w\}\).
Let \(r \geqslant 1\), then inequalities 4 and 5 imply that \[\begin{align} \widetilde{b}_{2^r} & = \sup_{w \in \Omega_r}\frac{1}{|w|}\left|\sum_{m \in w} b_m\right| = \max\left\{\sup_{w\in \Omega_{r,1}}\frac{1}{|w|}\left|\sum_{m \in w} b_m\right|, \sup_{w\in \Omega_{r,2}}\frac{1}{|w|}\left|\sum_{m \in w} b_m\right|\right\} \\ & \leqslant\max\left\{\sup_{w\in \Omega_{r,1}}\left(\frac{1}{|w|} \left|\sum_{m\in w } a_m\right| + \frac{1}{|w|-1} \left|\sum_{m\in w \setminus{\{0\}}} a_m\right|\right), \sup_{w\in \Omega_{r,2}}\frac{1}{|w|}\left|\sum_{m \in w} a_m\right|\right\}\\ & \leqslant\max\left\{\sup_{w\in \Omega_{r}}\frac{1}{|w|} \left|\sum_{m\in w } a_m\right| + \sup_{w\in \Omega_{r-1}}\frac{1}{|w|} \left|\sum_{m\in w } a_m\right|, \sup_{w\in \Omega_{r}}\frac{1}{|w|}\left|\sum_{m \in w} a_m\right|\right\}\\ & \leqslant\max\left\{\sup_{w\in \Omega_{r}}\frac{1}{|w|} \left|\sum_{m\in w } a_m\right| + 5\sup_{w\in \Omega_{r}}\frac{1}{|w|} \left|\sum_{m\in w } a_m\right|, \sup_{w\in \Omega_{r}}\frac{1}{|w|}\left|\sum_{m \in w} a_m\right|\right\} = 6\widetilde{a}_{2^r}. \end{align}\] By similar arguments, we obtain \(\widetilde{b}_1 \leqslant 2\widetilde{a}_1\). This completes the proof. ◻
The following lemma follows from [11].
Lemma 4. Let \(1 < p \leqslant\infty\), \(1 < q \leqslant\infty\). Then \[\|a\|_{n_{p,q}} \lesssim \|a\|_{l_{p,q}}.\]
We also need the following known Hardy type inequalities (see, for instance, [26]).
Lemma 5 (Hardy inequalities). Let \(1 < q < \infty\), \(0 < \alpha < 1\) and let \(\{a_n\}_{n=0}^{\infty}\) be a nonnegative sequence. Then \[\label{Hardy1} \left(\sum_{k=0}^{\infty} \left(2^{\alpha k} \sum_{m=k}^{\infty} a_m\right)^q\right)^{\frac{1}{q}} \lesssim \left(\sum_{k=0}^{\infty} \left(2^{\alpha k} a_k\right)^q\right)^{\frac{1}{q}},\tag{6}\] \[\label{Hardy2} \left(\sum_{k=0}^{\infty} \left(2^{(\alpha - 1) k} \sum_{m=0}^{k} 2^m a_m\right)^q\right)^{\frac{1}{q}} \lesssim \left(\sum_{k=0}^{\infty} \left(2^{\alpha k} a_k\right)^q\right)^{\frac{1}{q}}.\tag{7}\]
The following Fourier inequality for the case \(p \geqslant 2\) was established in [11]. For the reader’s convenience, we provide a full proof of this Theorem.
Theorem 3. Let \(1<p<\infty\), \(p'=p/(p-1)\) and \(f\in L_p([-\pi, \pi])\) be a function with Fourier series \(\sum\limits_{k=-\infty}^{\infty} a_{k} e^{ikx}\). Then \[\label{nurs-ineq} \|a\|_{n_{ p',p}} \lesssim \|f\|_{L_p([-\pi, \pi])}.\tag{8}\]
Proof. For an arbitrary interval \(w \subset \mathbb{Z}\), let \(D_w(x) = \sum\limits_{m \in w} e^{-imx}\) denote the corresponding Dirichlet kernel.
Let \(1 < q < \infty\), then the standard calculation gives \[\begin{align} \frac{1}{|w|^{\frac{1}{q}}} \left|\sum_{m \in w} a_m\right| & = \frac{1}{2\pi|w|^{\frac{1}{q}}} \left|\sum_{m \in w} \int_{-\pi}^{\pi} f(x) e^{-imx} dx\right|\\ & \leqslant\frac{1}{2\pi|w|^{\frac{1}{q}}} \int_{-\pi}^{\pi} |f(x)||D_w(x)|dx \leqslant\frac{1}{2\pi|w|^{\frac{1}{q}}} \int_{-\pi}^{\pi} |f(x)|\left|\frac{2\sin \frac{|w| x}{2}}{\sin \frac{x}{2}}\right|dx\\ & \lesssim \frac{1}{|w|^{\frac{1}{q}}} \int_{-\pi}^{\pi} |f(x)|\min \left(|w|, \frac{1}{|x|}\right) dx = \int_{-\pi}^{\pi} |f(x)|\varphi_w(x)dx. \end{align}\] Let us estimate function \(\varphi_w(x)\). If \(|w| \geqslant\frac{1}{|x|}\), then \[\varphi_w(x) = \frac{1}{|x| |w|^{\frac{1}{q}}} = \frac{1}{|x|^{\frac{1}{q'}} (|x||w|)^{\frac{1}{q}}} \leqslant\frac{1}{|x|^{\frac{1}{q'}}}.\] If \(|w| < \frac{1}{|x|}\), then \[\varphi_{w}(x) = \frac{|w|}{|w|^{\frac{1}{q}}} = |w|^{\frac{1}{q'}} \leqslant\frac{1}{|x|^{\frac{1}{q'}}}.\] Therefore, by using the rearrangement inequality, we get \[\label{in1} \frac{1}{|w|^{\frac{1}{q}}} \left|\sum_{m \in w} a_m\right| \leqslant\int_{-\pi}^{\pi} |f(x)| |x|^{-\frac{1}{q'}} dx \leqslant\int_0^{2\pi} f^*(t) t^{-\frac{1}{q'}} dt = \|f\|_{L_{q,1}([-\pi,\pi])}.\tag{9}\] On the other hand, we have \[\label{in2} \|a\|_{n_{q', \infty}} = \sup_{k \in \mathbb{N}} k^{\frac{1}{q'}} \widetilde{a}_k \leqslant \sup_{k \in \mathbb{N}} \sup_{\stackrel{|w| \geqslant k}{w \in \mathcal{W}}} \frac{1}{|w|^{\frac{1}{q}}}\left|\sum_{m \in w} a_m\right| = \sup_{w \in \mathcal{W}} \frac{1}{|w|^{\frac{1}{q}}}\left|\sum_{m \in w} a_m\right|.\tag{10}\] Combining inequalities 9 , 10 , we conclude that \[\label{in3} \|a\|_{n_{q', \infty}} \leqslant\|f\|_{L_{q,1}([-\pi,\pi])}.\tag{11}\] Consider operator \(T: f \mapsto \{a_k\}_{k \in \mathbb{Z}}\), where \(a_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-ikx} dx\). Inequality 11 implies the boundedness \[T : L_{q,1}([-\pi,\pi]) \to n_{q', \infty}, \quad 1 < q < \infty.\] Let now \(p \in (1, \infty)\), then for \(1 < p_0 < p < p_1 < \infty\), we have the boundedness \[T : L_{p_i,1}([-\pi,\pi]) \to n_{p_i^{'}, \infty}, \quad i = 0,1.\] By interpolation, using Marcinkiewicz’s theorem [27] and the interpolation theorem of the net spaces [11], we obtain the boundedness \[T : L_{p, \tau}([-\pi,\pi]) \to n_{p', \tau}, \quad 1 < \tau < \infty,\] i.e., \[\|a\|_{n_{p', \tau}} \leqslant C\|f\|_{L_{p, \tau}([-\pi,\pi])}, \quad 1 < \tau < \infty,\] in particular, this gives the estimate \[\|a\|_{n_{p', p}} \leqslant C\|f\|_{L_{p}([-\pi,\pi])}.\] ◻
Remark 5. Inequality 8 was first obtained by E. Nursultanov in [11]. The main advantage of this inequality, compared to other Fourier inequalities, is its validity for all \(1 < p < \infty\) without imposing additional conditions on the function or its Fourier coefficients. Later, similar inequalities were derived in many papers; see, for instance, [5], [12], [19]–[22].
A weaker form of this inequality was earlier obtained by Y. Sagher in [28] (see also [4], [10], [29]).
In the case when \(1 < p < 2\), inequality 8 follows from the inequality \(\|a\|_{l_{p'p}} \lesssim \|f\|_{L_p}\) (see [30]) and Lemma 4.
Let \(1 < p < \infty\), for an integrable function \(f(x)\sim \sum\limits_{k=-\infty}^{\infty} a_{k} e^{ikx}\), define \[I_p(f) := \left(\sum_{k=0}^\infty \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p\right)^{1/p},\] where \[\Theta_{k}(f) = \sum\limits_{[2^{k-1}] \leqslant|m| < 2^{k}}\left|\Delta a_{m}\right|, \quad k \geqslant 0.\]
Lemma 6. Let \(f({x})\) be an integrable on \([-\pi,\pi]\) function with Fourier series \(\sum\limits_{k=-\infty}^{\infty} a_{k} e^{ikx}\). Let also \[S_N(f) =\sum_{m=-2^N}^{2^N} a_{m}e^{imx}, \quad N \in \mathbb{N}.\] Then, for any \(1 < p < \infty\) and \(N \in \mathbb{N}\), \[I_p(S_N(f)) \leqslant 2^{\frac{1}{p'}}I_p(f).\]
Proof. For any \(N\in \mathbb{N}\), we get \[\begin{align} & \left(I_p(S_N(f))\right)^p = \sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \left(2^{\frac{N+1}{p'}}\left(\left| a_{2^N}\right| + \left| a_{-2^N}\right|\right)\right)^p \\ & = \sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \left(2^{\frac{N+1}{p'}}\left(\left| \sum_{k=2^N}^\infty\Delta a_k\right| + \left| \sum_{k=-\infty}^{-2^N}\Delta a_k\right|\right)\right)^p\\ & \leqslant\sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \left(2^{\frac{N+1}{p'}}\left( \sum_{k=2^N}^\infty\left|\Delta a_k\right| + \sum_{k=-\infty}^{-2^N}\left|\Delta a_k\right|\right)\right)^p \\ & = \sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \left(2^{\frac{N+1}{p'}}\left( \sum_{k=N}^\infty\sum_{m=2^k}^{2^{k+1}-1}\left|\Delta a_m\right| + \sum_{k=N}^{\infty}\sum_{m=-(2^{k+1} -1)}^{-2^k}\left|\Delta a_m\right|\right)\right)^p \\ & = \sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \left(2^{\frac{N+1}{p'}} \sum_{k=N+1}^\infty \Theta_k(f)\right)^p. \\ \end{align}\]
Using Hölder’s inequality, we obtain \[\begin{align} \left(I_p(S_N(f))\right)^p & \leqslant\sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + \\ & + \left(2^{\frac{N+1}{p'}} \left(\sum_{k=N+1}^\infty \left(2^{\frac{k}{p'}}\Theta_k(f)\right)^p\right)^{\frac{1}{p}}\left(\sum_{k=N+1}^\infty \left(2^{-\frac{k}{p'}}\right)^{p'}\right)^{\frac{1}{p'}}\right)^p\\ & = \sum_{k=0}^{N} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p + 2^{\frac{p}{p'}} \sum_{k=N+1}^\infty \left(2^{\frac{k}{p'}}\Theta_k(f)\right)^p \leqslant 2^{\frac{p}{p'}} \sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p.\\ \end{align}\]
Therefore, \[I_p(S_N(f)) \leqslant 2^{\frac{1}{p'}}I_p(f).\] ◻
Theorem 4. Let \(1<p<\infty\), and \(f \in L_1([-\pi,\pi])\) with Fourier series \(\sum\limits_{k=-\infty}^{\infty} a_{k} e^{ikx}\). Let \[I_p(f) < \infty,\] then \(f\in L_p([-\pi, \pi])\) and, moreover, the following inequality holds: \[\label{f2} \|f\|_{L_p} \lesssim I_p(f).\tag{12}\]
Proof. First, consider case where \(f\) is a trigonometric polynomial, i.e., \(f({x})= \sum\limits_{k=-\infty}^{\infty} a_{k} e^{ikx}\), where only a finite number of coefficients are nonzero. Using Parseval’s equality \[\label{in11} \|f\|_{L_p} =\sup_{\|g\|_{L_{p'}}=1}\left|\int_{[-\pi,\pi]}f({x})\overline{g({x})}{dx} \right|= \sup_{\|g\|_{L_{p'}}=1}\left|\sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k}\right|,\tag{13}\] where \(\{b_{k}\}_{k=-\infty}^{\infty}\) is the sequence of Fourier coefficients of the function \(g\), \(\overline{g({x})}\) is the complex conjugate of \(g(x)\), \(\{\overline{b}_{k}\}_{k=-\infty}^{\infty}\) is the complex conjugate of the sequence \(\{b_{k}\}_{k=-\infty}^{\infty}\). Let us define \[d_k = \begin{cases} b_k, & \text{for} \;k \neq 0;\\ \frac{1}{2}b_0, & \text{for} \;k = 0. \end{cases}\]
Since the sequence \(\{a_k\}_{k=-\infty}^{\infty}\) contains only finitely many nonzero elements, we have \[\begin{align} \sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k} & = \sum_{k=-\infty}^{-1} a_{k}\overline{b}_{k} + \frac{1}{2}a_0\overline{b}_0 + \frac{1}{2}a_0\overline{b}_0 + \sum_{k=1}^{\infty} a_{k}\overline{b}_{k} = \sum_{k=-\infty}^{0} a_{k}\overline{d}_{k} + \sum_{k=0}^{\infty} a_{k}\overline{d}_{k}\\ & =\sum_{k=-\infty}^{0}\overline{d}_{k}\sum_{m= -\infty}^{k}(a_m - a_{m-1}) + \sum_{k=0}^{\infty}\overline{d}_{k}\sum_{m=k}^{\infty}(a_m - a_{m+1})=\\ & =\sum_{m=-\infty}^{0}(a_{m} - a_{m-1}) \overline{\sum_{k=m}^{0}d_k} + \sum_{m=0}^{\infty}(a_{m} - a_{m+1}) \overline{\sum_{k=0}^{m}d_k}. \\ \end{align}\]
Therefore, \[\begin{align} \left|\sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k}\right| &\leqslant\sum_{m=-\infty}^{0}|a_{m} - a_{m-1}| \left|\sum_{k=m}^{0}d_k\right| + \sum_{m=0}^{\infty}|a_{m} - a_{m+1}| \left|{\sum_{k=0}^{m}d_k}\right|\\ & \leqslant\sum_{r=0}^\infty\sum_{m=-(2^{r}-1)}^{- [2^{r-1}]}\left|\Delta a_{m}\right|\left|\sum_{k = m}^{0}d_{k}\right| + \sum_{r=0}^\infty\sum_{m=[2^{r-1}]}^{2^{r}-1}\left|\Delta a_{m}\right|\left|\sum_{k=0}^m d_{k}\right|\\ & = \sum_{r=0}^\infty\sum_{m=-(2^{r}-1)}^{- [2^{r-1}]}(|m|+1)\left|\Delta a_{m}\right|\frac{1}{|m|+1}\left|\sum_{k = m}^{0}d_{k}\right| \\ &+ \sum_{r=0}^\infty\sum_{m=[2^{r-1}]}^{2^{r}-1}(|m|+1)\left|\Delta a_{m}\right|\frac{1}{|m|+1}\left|\sum_{k=0}^m d_{k}\right|\\ & \leqslant\sum_{r=0}^\infty\sum_{m=-(2^{r}-1)}^{- [2^{r-1}]}(|m|+1)\left|\Delta a_{m}\right|\widetilde{d}_{|m|+1} + \sum_{r=0}^\infty\sum_{m=[2^{r-1}]}^{2^{r}-1}(|m|+1)\left|\Delta a_{m}\right|\widetilde{d}_{|m|+1}. \end{align}\] Using monotonicity of the sequence \(\{\widetilde{d}_m\}_{m=1}^{\infty}\) and inequality [alm95mon], we obtain
\[\begin{align} \left|\sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k}\right| & \leqslant\sum_{r=0}^\infty 2^r\widetilde{d}_{[2^{r-1}]+1} \sum_{[2^{r-1}] \leqslant|m| < 2^{r}-1}\left|\Delta a_{m}\right|\\ & \leqslant\widetilde{d}_{1} |\Delta a_0| + 5\sum_{r=1}^\infty 2^r\widetilde{d}_{2^{r}} \sum_{[2^{r-1}] \leqslant|m| < 2^{r}-1}\left|\Delta a_{m}\right|\\ \\ & \leqslant 5\sum_{r=0}^\infty 2^r\widetilde{d}_{2^{r}} \sum_{[2^{r-1}] \leqslant|m| < 2^{r}-1}\left|\Delta a_{m}\right|. \end{align}\] By using Lemma 3, we derive \[\left|\sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k}\right| \lesssim \sum_{r=0}^\infty 2^{r} \widetilde{b}_{2^{r}} \Theta_{r}(f).\]
Applying Hölder’s inequality and Theorem 3, we get \[\label{in13} \begin{align} \left|\sum_{k=-\infty}^{\infty} a_{k}\overline{b}_{k}\right| & \leqslant \sum_{r=0}^\infty 2^{\frac{r}{p}} \widetilde{b}_{2^{r}} \cdot 2^{\frac{r}{p'}}\Theta_{r}(f) \leqslant\left(\sum_{r=0}^\infty \left(2^{\frac{r}{p}} \widetilde{b}_{2^{r}}\right)^{p'} \right)^{\frac{1}{p'}} \left(\sum_{r=0}^\infty \left(2^{\frac{r}{p'}}\Theta_{r}(f)\right)^{p} \right)^{\frac{1}{p}} \\ & \lesssim \|b\|_{n_{pp'}}\cdot I_p(f) \lesssim \|g\|_{L_{p'}}\cdot I_p(f). \end{align}\tag{14}\] Combining 13 and 14 , we conclude \[\|f\|_{L_p} \lesssim I_p(f).\] Now consider general case where \(f\) satisfies conditions of Theorem. Let \[S_N(f)=\sum_{k=-2^N}^{2^N}a_{k}e^{ikx}.\]
From the above, we know that, for any \(N \in \mathbb{N}\), \[\|S_N(f)\|_{L_p} \lesssim I_p(S_N(f)).\] From Lemma 6 it follows that, for any \(N \in \mathbb{N}\), \[I_p(S_N(f))\leqslant 2^{\frac{1}{p'}}I_p(f).\] Since, for any \(1 < p <\infty\), \(\lim\limits_{N\to \infty} \|f - S_N(f)\|_{L_p} = 0\), we get \[\|f\|_{L_p([-\pi,\pi])} \lesssim I_{p}(f).\] ◻
Remark 6. The estimate 12 in the non-periodic case was obtained in [31]. An estimate of this type was obtained in [6] (see also [7]).
Proof of Theorem 1. By using the condition of general monotonicity in \(\textrm{GM}^*\), we derive \[\begin{align} I_p(f) & =\left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p\right)^{1/p} \lesssim \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{l \in \mathbb{N}_0} \min (1, 2^{l-k}) \widetilde{a}_{2^l}\right)^p\right)^{1/p} \\ & \lesssim \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{0\leqslant l \leqslant k} 2^{l-k}\widetilde{a}_{2^l}\right)^p\right)^{1/p} + \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{l\geqslant k + 1} \widetilde{a}_{2^l}\right)^p\right)^{1/p} =: I_1 + I_2. \end{align}\] Let’s estimate \(I_1\). By using Hardy’s inequality 7 \[\begin{align} I_1 & = \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{0\leqslant l \leqslant k} 2^{l-k}\widetilde{a}_{2^l}\right)^p\right)^{1/p} \leqslant\left(\sum_{k=0}^{\infty}\left(2^{\left(\frac{1}{p'}-1\right)k}\sum_{l=0}^k 2^l \widetilde{a}_{2^l}\right)^p\right)^{1/p}\\ & \lesssim \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\widetilde{a}_{2^k}\right)^p\right)^{1/p} \asymp \|a\|_{n_{p',p}}. \end{align}\] Let’s estimate \(I_2\). Since \(\{\widetilde{a}_{2^k}\}_{k=0}^{\infty}\) is a nonincreasing sequence, we obtain \[I_2 = \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{l\geqslant k+1} \widetilde{a}_{2^l}\right)^p\right)^{1/p} \leqslant\left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\widetilde{a}_{2^{k+1}}\right)^p\right)^{1/p} \lesssim \|a\|_{n_{p', p}}.\] Therefore, by using Theorem 3 \[I_p(f) \lesssim I_1 + I_2 \lesssim \|a\|_{n_{p',p}}\lesssim \|f\|_{L_p([-\pi,\pi])}.\] Applying Theorem 4, we get \[\label{ip} I_p(f) \asymp \|f\|_{L_p([-\pi,\pi])}.\tag{15}\]
Now, we show that for any \(1 < p < \infty\) and for arbitrary \(\{a_k\}_{k=-\infty}^{\infty}\), inequality \(J_p^*(f) \lesssim I_p(f)\) holds.
Repeating the same arguments as in 13 and 14 and using Lemma 4, we obtain \[\begin{align} J_p^*(f) & \asymp \|a\|_{l_{p'p}} \asymp \sup_{\|b\|_{l_{pp'}}=1}\left|\sum_{k=-\infty}^{\infty}a_k b_k\right| \lesssim \sup_{\|b\|_{l_{pp'}}=1}\|b\|_{n_{pp'}}I_p(f) \\ & \lesssim \sup_{\|b\|_{l_{pp'}}=1}\|b\|_{l_{pp'}}I_p(f) = I_p(f). \end{align}\]
On the other hand, since \(\{a_k\}_{k=-\infty}^{\infty} \in \textrm{GM}^*\), \[\begin{align} I_p(f) & = \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\Theta_{k}(f)\right)^p\right)^{1/p} \lesssim \left(\sum_{k=0}^{\infty}\left(2^{\frac{k}{p'}}\sup_{l \in \mathbb{N}_0} \min (1, 2^{l-k}) \widetilde{a}_{2^l}\right)^p\right)^{1/p} \\ & \lesssim \|a\|_{n_{p',p}} \lesssim \|a\|_{l_{p',p}} \asymp J_p^*(f). \end{align}\] Therefore, \[\label{jps} I_p(f) \asymp J_p^*(f).\tag{16}\] Combining relations 15 and 16 we prove Theorem 1. ◻
Remark 7. From the proof of Theorem 1, it follows that for function \(f(x)\) whose Fourier coefficients \(\{a_k\}_{k = -\infty}^{\infty}\) belong to the class \(\textrm{GM}^*\), the following equivalence holds: \[\|f\|_{L_p([-\pi,\pi])} \asymp I_p(f) = \left(\sum_{k=0}^\infty \left(2^{\frac{k}{p'}} \sum\limits_{[2^{k-1}] \leqslant|m| < 2^{k}}\left|\Delta a_{m}\right| \right)^p\right)^{1/p}, \quad 1 < p < \infty.\]
Proof of Theorem 2. For any \(1 < p < \infty\), we define \[A_p(a) := \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}} \sup_{r \in \mathbb{N}_0}\min(1,2^{r-k})\widehat{a}_{2^r}\right)^p\right)^{\frac{1}{p}}.\] Then, the following inequality holds: \[A_p(a) \lesssim \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}} \sup_{0 \leqslant r \leqslant k} 2^{r-k}\widehat{a}_{2^r}\right)^p\right)^{\frac{1}{p}} + \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}} \sup_{r \geqslant k + 1} \widehat{a}_{2^r}\right)^p\right)^{\frac{1}{p}} =: I_1 + I_2.\] Let’s estimate \(I_1\). For any \(k \in \mathbb{N}_0\), we have \[\begin{align} \sup_{0 \leqslant r \leqslant k} 2^{r-k}\widehat{a}_{2^r} & = \sup_{0 \leqslant r \leqslant k} 2^{r-k} \sup_{2^{r} \leqslant|m| < 2^{r+1}} \frac{1}{|m|+1} \left| \sum_{s=0}^{m} a_s\right| \\ & \leqslant\sup_{0 \leqslant r \leqslant k} 2^{r-k} \frac{1}{2^{r}} \sum_{s=-(2^{r+1}-1)}^{2^{r+1}-1} |a_s| \leqslant 2^{-k} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} |a_s|. \end{align}\] Therefore, \[\begin{align} I_1 & \lesssim \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}} 2^{-k} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} |a_s|\right)^p\right)^{\frac{1}{p}} = \left(\sum_{k=0}^{\infty} \left(2^{-\frac{k}{p}} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} |a_s|\right)^p\right)^{\frac{1}{p}}. \end{align}\]
Let \(-\frac{1}{p'} < \varepsilon< -\frac{1}{p'} + \frac{1}{p}\). Applying Hölder’s inequality, we get \[\label{2461} \begin{align} I_1 & \lesssim\left(\sum_{k=0}^{\infty} \left(2^{-\frac{k}{p}} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} |a_s|\right)^p\right)^{\frac{1}{p}}\\ & \leqslant \left(\sum_{k=0}^{+\infty}\left(2^{-\frac{k}{p}} \left(\sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} \left(|a_s|(|s|+1)^{-\varepsilon}\right)^p\right)^{\frac{1}{p}} \left(\sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} (|s|+1)^{\varepsilon p'}\right)^{\frac{1}{p'}} \right)^p\right)^{\frac{1}{p}}\\ & \asymp \left(\sum_{k=0}^{+\infty}2^{-k} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} \left(|a_s|(|s|+1)^{-\varepsilon}\right)^p \left(2^{k(\varepsilon p' + 1)}\right)^{\frac{p}{p'}} \right)^{\frac{1}{p}}\\ & = \left(\sum_{k=0}^{+\infty} 2^{k(\varepsilon p + p-2)} \sum_{s=-(2^{k+1}-1)}^{2^{k+1}-1} \left(|a_s|(|s|+1)^{-\varepsilon}\right)^p\right)^{\frac{1}{p}}\\ & \asymp \left(\sum_{s=-\infty}^{+\infty} \left(|a_s|(|s|+1)^{-\varepsilon}\right)^p \sum_{k = \log_2 (|s|+1)}^{+\infty} 2^{k(\varepsilon p + p-2)} \right)^{\frac{1}{p}} \\ & \asymp \left(\sum_{s=-\infty}^{+\infty} \left(|a_s|(|s|+1)^{-\varepsilon}\right)^p (|s|+1)^{\varepsilon p + p-2} \right)^{\frac{1}{p}} = J_p(f). \end{align}\tag{17}\] Now we estimate \(I_2\). For any \(r \geqslant k\), we have \[\begin{align} \widehat{a}_{2^r} \leqslant\sum_{t = k}^{\infty} \widehat{a}_{2^t}. \end{align}\] Therefore, by using Hardy’s inequality 6 , we obtain \[\label{24611} \begin{align} I_2 &\leqslant\left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}} \sup_{r \geqslant k} \widehat{a}_{2^r}\right)^p\right)^{\frac{1}{p}} \leqslant\left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\sum_{t = k}^{\infty} \widehat{a}_{2^t}\right)^p\right)^{\frac{1}{p}} \lesssim \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\widehat{a}_{2^{k}}\right)^p\right)^{\frac{1}{p}}\\ & = \left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\sup_{2^{k} \leqslant|m| < 2^{k+1}}\frac{1}{|m|+1}\left|\sum_{s=0}^{m} a_s\right|\right)^p\right)^{\frac{1}{p}} \leqslant\left(\sum_{k=0}^{\infty} \left(2^{\frac{k}{p'}}\frac{1}{2^{k}}\sum_{s=-2^{k+1}}^{2^{k+1}} |a_s|\right)^p\right)^{\frac{1}{p}}\\ &\lesssim J_p(f). \end{align}\tag{18}\]
Combining inequalities 17 and 18 , we have \[\label{246111} A_p(a) \lesssim J_p(f), \quad 1 < p < \infty.\tag{19}\]
For further arguments we consider two cases.
Let \(1 < p \leqslant 2\). By Hardy-Littlewood’s inequality [32] and by Theorem 4, we get \[\label{2462} \begin{align} J_p(f) \leqslant\|f\|_{L_p} \lesssim I_p(f). \end{align}\tag{20}\] Inequalities 19 and 20 imply \[\begin{align} A_p(a) \lesssim J_p(f) \leqslant\|f\|_{L_p} \lesssim I_p(f). \end{align}\] Since \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\), \[A_p(a) \lesssim J_p(f) \leqslant\|f\|_{L_p} \lesssim I_p(f) \lesssim A_p(a) .\] Therefore, \[\|f\|_{L_p} \asymp J_p(f).\]
Let \(2 \leqslant p <\infty\). For any \(m \in \mathbb{Z}\), we consider \(t \in \mathbb{N}_0\) such that \([2^{t-1}] \leqslant|m| < 2^{t}\). Then, since \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\), \[\begin{align} |a_m| \leqslant\sum_{|s|\geqslant|m|}|\Delta a_s| \leqslant\sum_{k=t}^{\infty}\sum_{[2^{k-1}] \leqslant|s| < 2^{k}}|\Delta a_s| \leqslant\sum_{k=t}^{\infty} \sup_{l \in \mathbb{N}_0} \min(1,2^{l-k})\widehat{a}_{2^l}. \end{align}\] Therefore, \[\begin{align} J_p(f) & = \left(\sum_{m=-\infty}^{\infty} |a_m|^p (|m|+1)^{\frac{p}{p'}-1}\right)^{\frac{1}{p}} \leqslant \left(\sum_{t=0}^{\infty} \sum_{[2^{t-1}] \leqslant|m| < 2^{t}}|a_m|^p(|m|+1)^{\frac{p}{p'}-1}\right)^{\frac{1}{p}}\\ & \lesssim \left(\sum_{t=0}^{\infty} \left(\sum_{k=t}^{\infty}\sum_{[2^{k-1}] \leqslant|s| < 2^k}|\Delta a_s|\right)^p\sum_{[2^{t-1}] \leqslant|m| < 2^{t}} (|m|+1)^{\frac{p}{p'}-1}\right)^{\frac{1}{p}}\\ & \asymp \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}}\sum_{k=t}^{\infty}\sum_{[2^{k-1}] \leqslant|s| < 2^k}|\Delta a_s|\right)^p\right)^{\frac{1}{p}}\\ &\lesssim \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}}\sum_{k=t}^{\infty}\sup_{l \in \mathbb{N}_0} \min(1,2^{l-k})\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}}. \end{align}\] By using Hardy’s inequality 6 , we get \[\begin{align} & \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}}\sum_{k=t}^{\infty}\sup_{l \in \mathbb{N}_0} \min(1,2^{l-k})\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} \lesssim \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}}\sup_{l \in \mathbb{N}_0} \min(1,2^{l-t})\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}}\\ &\lesssim \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{0 \leqslant l \leqslant t} 2^{l-t}\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} + \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{l \geqslant t+1} \widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} =: L_1 + L_2. \end{align}\] We estimate \(L_1\). \[\begin{align} L_1 &= \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{0 \leqslant l \leqslant t} 2^{l-t}\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} = \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}-t} \sup_{0 \leqslant l \leqslant t} 2^{l}\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}}\\ &\leqslant \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}-t} \sum_{l=0}^t 2^{l}\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}}. \end{align}\] By using Hardy’s inequality 7 , we derive \[\begin{align} L_1 \lesssim \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}-t} \sum_{l=0}^t 2^{l}\widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} \lesssim \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \widehat{a}_{2^t}\right)^p\right)^{\frac{1}{p}} \leqslant\|a\|_{n_{p',p}}. \end{align}\] Now we estimate \(L_2\). \[\begin{align} L_2 &\leqslant\left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{l \geqslant t} \widehat{a}_{2^l}\right)^p\right)^{\frac{1}{p}} = \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{l \geqslant t} \sup_{2^l \leqslant|m| < 2^{l+1}} \frac{1}{|m|+1}\left|\sum_{s=0}^{m} a_s\right|\right)^p\right)^{\frac{1}{p}}\\ &\leqslant \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{l \geqslant t} \sup_{|m| \geqslant 2^{l}} \frac{1}{|m|+1}\left|\sum_{s=0}^{m} a_s\right|\right)^p\right)^{\frac{1}{p}}\\ & \leqslant \left(\sum_{t=0}^{\infty} \left(2^{\frac{t}{p'}} \sup_{|m| \geqslant 2^{t}} \frac{1}{|m|+1}\left|\sum_{s=0}^{m} a_s\right|\right)^p\right)^{\frac{1}{p}} \leqslant\|a\|_{n_{p'p}}. \end{align}\] Finally, since \(2 \leqslant p < \infty\), by Hardy-Littlewood’s inequality [32], we have \[\begin{align} J_p(f) \lesssim \|a\|_{n_{p',p}} \lesssim \|f\|_{L_{p}} \lesssim J_{p}(f). \end{align}\] Therefore, \(J_p(f) \asymp \|f\|_{L_p}\). ◻
In this Section, at first, we compare classes \(\textrm{GM}_{\mathbb{R}}\) and \(\overline{\textrm{GM}}\), where \[\textrm{GM}_{\mathbb{R}} = \left\{\{a_k\}_{k=1}^{\infty} \in \textrm{GM}: \;\;a_k \in \mathbb{R}, \;\;k \geqslant 1\right\}.\] For this reason, we need a technique considered in [33] by M. Dyachenko and S. Tikhonov.
Without loss of generality, we may assume in the definition of the class \(\textrm{GM}\) that \(\lambda = 2^{\nu}\), where \(\nu\) is a natural number.
Let \(\{a_k\}_{k=1}^{\infty}\in \textrm{GM}_{\mathbb{R}}\). Denote for any \(n \geqslant 2\nu\) \[A_n := \max_{2^n \leqslant k \leqslant 2^{n+1}} |a_k|, \quad B_n := \max_{2^{n-2\nu} \leqslant k \leqslant 2^{n + 2\nu}}|a_k|.\]
Definition 6. Let \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\). We say that a integer number \(n \geqslant 2\nu\) is good, if \(B_n \leqslant 2^{2\nu}A_n\). The rest of integer numbers \(n \geqslant 0\) consists of bad numbers.
Denote \[M_n := \left\{k \in [2^{n-\nu}, 2^{n+\nu}] : |a_k| > \frac{A_n}{8C2^{2\nu}} \right\},\] and \[M_n^+ := \{k \in M_n: a_k > 0\}, \quad M_n^- := M_n \setminus M_n^{+}.\]
Lemma 7 (). Let a vanishing sequence \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\). Denote \(N_0: = [\log_2 (C^32^{10\nu + 8})] +1\). Then, for any good \(n \geqslant N_0\), there exists an interval \([l_n, m_n]\subseteq [2^{n-\nu}, 2^{n+\nu}]\) such that at least one of the following condition holds:
for any \(k \in [l_n, m_n]\), we have \(a_k \geqslant 0\) and \[|M_n^+ \cap [l_n, m_n]| \geqslant\frac{2^n}{C^32^{15\nu+8}};\]
for any \(k \in [l_n, m_n]\), we have \(a_k \leqslant 0\) and \[|M_n^- \cap [l_n, m_n]| \geqslant\frac{2^n}{C^32^{15\nu+8}}.\]
Lemma 8 (). Let a vanishing sequence \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\). Then for any bad number \(n \geqslant 2\nu\) there exists a set of integer numbers \[n = \xi_0 > \xi_1 > \xi_2 > \ldots > \xi_{s-1} > \xi_{s} =: \xi_{n,s}\] or \[n =\xi_0 < \xi_1 < \xi_2 < \ldots < \xi_{s-1} < \xi_{s}=: \xi_{n,s}\] such that \(\xi_1, \ldots, \xi_{s-1}\) are bad, \(\xi_{s}\) is good, and \[A_n < 2^{-2\nu}A_{\xi_1} < 2^{-4\nu}A_{\xi_2}< \ldots < 2^{-2s\nu}A_{\xi_{s}},\] \[|\xi_i - \xi_{i-1}| \leqslant 2\nu \quad i = 1,\ldots, s.\]
Remark 8. These properties of sequences from \(\textrm{GM}_{\mathbb{R}}\) were stated in [33]. Due to these properties, some classical results of Fourier analysis and approximation theory have been extended to the \(\textrm{GM}_{\mathbb{R}}\) class, see [3], [4], [33], [34], and see also [9], [29], [35], [36].
Lemma 9. Let \(\{a_k\}_{k=-\infty}^{\infty}\) be a sequence of real numbers such that \(a_k = 0\), for any \(k \leqslant 0\), and \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\), \(\lim\limits_{k \to +\infty} a_k = 0\). Then, for any \(n \geqslant 0\), \[\label{gmgm-1} \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{a}_{2^k}.\tag{21}\]
Proof. Let \(N_0 = N_0(C, \nu)\) be a constant defined in Lemma 7. We prove 21 by considering five cases.
Let \(1 \leqslant n \leqslant N_0\). Then, for any \(2^n \leqslant k \leqslant 2^{n+1}\), we have \[\begin{align} |a_k| & = \left|\sum_{i = 0}^{k} a_i - \sum_{i = 0}^{k-1} a_i \right| \leqslant(k+1) \cdot \frac{1}{k+1}\left|\sum_{i = 0}^{k} a_i \right| + k\cdot \frac{1}{k}\left|\sum_{i = 0}^{k-1} a_i \right| \\ & \leqslant(2^{N_0 + 1}+1) (2\widehat{a}_{2^{n}} + \widehat{a}_{2^{n+1}}) \lesssim \sup_{t \geqslant n} \widehat{a}_{2^{t}} \leqslant\sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n})\widehat{a}_{2^{t}}. \end{align}\] Therefore, \[\begin{align} \sum_{k = 2^n}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n})\widehat{a}_{2^{t}}\sum_{k = 2^n}^{2^{n+1}} \frac{1}{k} \lesssim \sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n})\widehat{a}_{2^{t}}. \end{align}\]
Let \(n > N_0\) be a good number. By Lemma 7 there exist integer numbers \(l_{n}, m_{n}\) such that \([l_{n}, m_{n}] \subset [2^{n - \nu}, 2^{n + \nu}]\), \(m_{n} - l_{n} \asymp 2^{n}\) and \[\frac{1}{2^{n}}\left|\sum_{k = l_{n}}^{m_{n}} a_k\right| \gtrsim A_{n} \geqslant\frac{1}{2^{n}} \sum_{k=2^n}^{2^{n+1}} |a_k| \gtrsim \sum_{k=2^n}^{2^{n+1}} \frac{|a_k|}{k}.\]
Consider case, when \(\left|\sum\limits_{k=0}^{l_{n}-1} a_k\right| \geqslant\frac{1}{2}\left|\sum\limits_{k=l_{n}}^{m_{n}} a_k\right|\), then \[\frac{1}{l_{n}}\left|\sum_{k=0}^{l_{n}-1} a_k\right| \geqslant\frac{1}{2l_n}\left|\sum_{k=l_{n}}^{m_{n}} a_k\right| \gtrsim \frac{1}{2^{n}}\left|\sum_{k=l_{n}}^{m_{n}} a_k\right| \gtrsim \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k}.\]
Define \(\eta\): \(2^{\eta} \leqslant l_{n} < 2^{\eta+1}\). Then \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \frac{1}{l_{n}}\left|\sum_{k=0}^{l_{n}-1} a_k\right| \leqslant\widehat{a}_{2^{\eta}}.\] Note that, since \(2^{n - \nu} \leqslant l_{n} \leqslant 2^{n + \nu}\), then \(n - \nu \leqslant\eta \leqslant n + \nu - 1\).
Now consider case, when \(\left|\sum\limits_{k=0}^{l_{n}-1} a_k\right| < \frac{1}{2}\left|\sum\limits_{k=l_{n}}^{m_{n}} a_k\right|\), then \[\left|\sum_{k=0}^{m_{n}}a_k\right| \geqslant \left|\sum_{k=l_{n}}^{m_{n}}a_k\right| - \left|\sum_{k=0}^{l_{n}-1}a_k\right| \geqslant\frac{1}{2}\left|\sum_{k=l_{n}}^{m_{n}}a_k\right|.\] Therefore, \[\frac{1}{m_{n}+1}\left|\sum_{k=0}^{m_{n}}a_k\right| \gtrsim \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k}.\] Hence, \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \widehat{a}_{2^{\eta}},\] where \(\eta\) is defined by relation: \(2^{\eta} \leqslant m_{n} + 1 < 2^{\eta+1}\). As in previous case such \(\eta\) satisfies condition: \(n - \nu \leqslant\eta \leqslant n + \nu - 1\).
In both cases, we obtain \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \widehat{a}_{2^{\eta}},\] where \(n - \nu \leqslant\eta \leqslant n + \nu - 1\). Therefore, \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \widehat{a}_{2^{\eta}} \leqslant\sup_{k \geqslant n - \nu}\widehat{a}_{2^{k}} \lesssim \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{a}_{2^k}.\]
Now let \(n > N_0\) be a bad number. Then, there exists a good number \(\xi_s = \xi_{n,s}\) satisfying conditions from Lemma 8. In particular, \(A_{\xi_s} \geqslant 2^{2s\nu}A_n\).
Let \(\xi_s > n\). Then by Lemma 7 there exist integer numbers \(l_{\xi_s}, m_{\xi_s}\) such that \([l_{\xi_s}, m_{\xi_s}] \subset [2^{\xi_s - \nu}, 2^{\xi_s + \nu}]\), \(m_{\xi_s} - l_{\xi_s} \asymp 2^{\xi_s}\) and \[\frac{1}{2^{\xi_s}}\left|\sum_{k = l_{\xi_s}}^{m_{\xi_s}} a_k\right| \gtrsim A_{\xi_s}.\] From Lemma 8 we have \[\frac{1}{2^{\xi_s}}\left|\sum_{k = l_{\xi_s}}^{m_{\xi_s}} a_k\right| \gtrsim A_{\xi_s} \geqslant 2^{2s\nu} A_n \gtrsim 2^{2s\nu} \sum_{k=2^n}^{2^{n+1}}\frac{|a_k|}{k}.\]
Repeating the same arguments as in Case 2, we derive \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \frac{1}{2\nu s} \cdot \frac{1}{2^{\xi_s}}\left|\sum_{k = l_{\xi_s}}^{m_{\xi_s}} a_k\right| \lesssim \frac{1}{2^{2\nu s}} \widehat{a}_{2^{\eta}},\] where \(\xi_s - \nu \leqslant\eta \leqslant\xi_s + \nu - 1\). Since \(\xi_s > n\), we get \(\eta > n - \nu\), and \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \frac{1}{2^{2\nu s}} \widehat{a}_{2^{\eta}} \leqslant\sup_{t > n - \nu}\widehat{a}_{2^{t}} \lesssim \sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n}) \widehat{a}_{2^t}.\]
Let \(N_0 \leqslant\xi_s < n\), then by Lemma 8, \(\xi_s < n \leqslant\xi_s +2\nu s\). The same arguments as in the Case 3 imply that there exists integer number \(\eta\) such that \(\eta - \nu + 1 < n \leqslant\eta + \nu + 2\nu s\) and \[\begin{align} \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} &\lesssim \frac{1}{2^{2\nu s}} \widehat{a}_{2^{\eta}} = \frac{2^n 2^{\eta}}{2^{2\nu s}2^n2^{\eta}} \widehat{a}_{2^{\eta}} = \frac{2^n}{2^{2\nu s}2^{\eta}} \cdot \frac{2^{\eta}}{2^{n}} \widehat{a}_{2^{\eta}}\\ & \lesssim \frac{2^{\eta}}{2^n} \widehat{a}_{2^{\eta}} \leqslant\sup_{t < n + \nu - 1} \frac{2^t}{2^n} \widehat{a}_{2^t} \lesssim \sup_{t \in \mathbb{N}_0} \min(1,2^{t-n})\widehat{a}_{2^t}. \end{align}\]
Let \(\xi_s < N_0 < n\). Then by Lemma 8, \(\xi_s < n \leqslant\xi_s +2\nu s\), and \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim A_n \leqslant\frac{1}{2^{2\nu s}}A_{\xi_s}.\] The same arguments as in Case 1 imply \[A_{\xi_s} \lesssim (2\widehat{a}_{2^{\xi_s}} + \widehat{a}_{2^{\xi_s + 1}}).\] Therefore, \[\begin{align} \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} & \lesssim \frac{1}{2^{2\nu s}}A_{\xi_s} \lesssim \frac{1}{2^{2\nu s}} (2\widehat{a}_{2^{\xi_s}} + \widehat{a}_{2^{\xi_s + 1}}) = \frac{2\cdot 2^n 2^{\xi_s}}{2^{2\nu s}\cdot 2^n 2^{\xi_s}} \widehat{a}_{2^{\xi_s}} + \frac{2^n 2^{\xi_s + 1}}{2^{2\nu s}\cdot 2^n 2^{\xi_s + 1}} \widehat{a}_{2^{\xi_s+1}}\\ & = \frac{2^{n+1}}{2^{2\nu s}2^{\xi_s}} \cdot \frac{2^{\xi_s}}{2^n} \widehat{a}_{2^{\xi_s}} + \frac{2^n}{2^{2\nu s}2^{\xi_s + 1}} \cdot \frac{2^{\xi_s + 1}}{2^n} \widehat{a}_{2^{\xi_s+1}} \lesssim \sup_{t \leqslant n} 2^{t-n}\widehat{a}_{2^t} \leqslant\sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n})\widehat{a}_{2^t}. \end{align}\]
Thus, relation 21 was proved for all good and bad numbers \(n\). ◻
Lemma 10. Let \(\{a_k\}_{k=-\infty}^{\infty}\) be a sequence of real numbers such that \(a_k = 0\), for any \(k \leqslant 0\), and \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\), \(\lim\limits_{k \to +\infty} a_k = 0\). Then \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\). In this sense, we interpret the following inclusion: \[\textrm{GM}_{\mathbb{R}} \subset \overline{\textrm{GM}}.\]
Proof. Let \(n \in \mathbb{N}\). Then, since \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}\), we have \[\sum_{[2^{n-1}] \leqslant|m| < 2^n} |\Delta a_m| = \sum_{2^{n-1} \leqslant m < 2^n} |\Delta a_m| \lesssim \sum_{k = 2^{n-\nu -1}}^{2^{n+\nu - 1}} \frac{|a_k|}{k} \leqslant\sum_{s = n - \nu - 1}^{n+\nu - 2}\sum_{k = 2^s}^{2^{s+1}} \frac{|a_k|}{k}.\] By using inequality 21 \[\begin{align} \sum_{[2^{n-1}] \leqslant|m| < 2^n} |\Delta a_m| &\lesssim \sum_{s = n - \nu - 1}^{n+\nu - 2}\sum_{k = 2^s}^{2^{s+1}} \frac{|a_k|}{k} \lesssim \sum_{s = n - \nu - 1}^{n+\nu - 2} \sup_{k \in \mathbb{N}_0} \min(1,2^{k-s})\widehat{a}_{2^k} \\ &\leqslant 2^{\nu+1}\nu \sup_{k \in \mathbb{N}_0} \min(1,2^{k-n})\widehat{a}_{2^k}. \end{align}\] Therefore, for any \(n \geqslant 1\), we obtain \[\label{gmgm-2} \sum_{[2^{n-1}] \leqslant|m| < 2^n} |\Delta a_m| \lesssim \sup_{k \in \mathbb{N}_0} \min(1,2^{k-n})\widehat{a}_{2^k}.\tag{22}\] It easy to see, that inequality 22 holds for \(n = 0\). Hence, \(\{a_n\}_{n= - \infty}^{\infty} \in \overline{\textrm{GM}}\). ◻
According to the notations in [10], we consider sector of the complex plane: \[S_{\alpha, \beta} = \{z \in \mathbb{C} : |\arg z -\alpha| \leqslant\beta \}, \quad 0 \leqslant\alpha < 2\pi, \quad 0 \leqslant\beta < \frac{\pi}{2}.\] Now we compare the classes \(\textrm{GM}_{\alpha, \beta}\) and \(\overline{\textrm{GM}}\), where \[\textrm{GM}_{\alpha, \beta} = \left\{\{a_k\}_{k=1}^{\infty} \in \textrm{GM}: \;\;a_k \in S_{\alpha, \beta}, \;\;k \geqslant 1\right\}.\]
Lemma 11. Let \(0 \leqslant\alpha < 2\pi\), \(0 \leqslant\beta < \frac{\pi}{2}\), and let \(\{a_k\}_{k=-\infty}^{\infty}\) be a sequence of complex numbers such that \(a_k = 0\), for any \(k \leqslant 0\), and \(\{a_k\}_{k=1}^{\infty} \in \textrm{GM}_{\alpha, \beta}\). Then \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\). In this sense, we interpret the following inclusion: \[\textrm{GM}_{\alpha, \beta} \subset \overline{\textrm{GM}}.\]
Proof. It is sufficient to prove that, for any \(n \geqslant 0\), \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{a}_{2^k}\]
By [10], we have \[\sum_{k = 2^{n}}^{2^{n+1}} |a_k| \leqslant\frac{1}{|\cos \beta|} \left|\sum_{k = 2^{n}}^{2^{n+1}} a_k \right|.\] Therefore, \[\begin{align} \sum_{k = 2^{n}}^{2^{n+1}}\frac{|a_k|}{k} \lesssim \frac{1}{2^n} \sum_{k = 2^{n}}^{2^{n+1}} |a_k| \leqslant\frac{1}{2^n |\cos \beta|} \left|\sum_{k = 2^{n}}^{2^{n+1}} a_k \right|. \end{align}\] Consider case, when \(\left|\sum\limits_{k=0}^{2^{n}-1} a_k\right| \geqslant\frac{1}{2}\left|\sum\limits_{k=2^{n}}^{2^{n+1}} a_k\right|\), then \[\frac{1}{2^{n}}\left|\sum\limits_{k=0}^{2^{n}-1} a_k\right| \geqslant\frac{1}{2^{n+1}}\left|\sum\limits_{k=2^{n}}^{2^{n+1}} a_k\right| \gtrsim \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k}.\] Therefore, \[\widehat{a}_{2^{n}} \gtrsim \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k}.\] Hence, \[\begin{align} \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \widehat{a}_{2^{n}} \leqslant \sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n}) \widehat{a}_{2^t}. \end{align}\]
Now consider case, when \(\left|\sum\limits_{k=0}^{2^{n}-1} a_k\right| < \frac{1}{2}\left|\sum\limits_{k=2^{n}}^{2^{n+1}} a_k\right|\), then \[\left|\sum\limits_{k=0}^{2^{n+1}} a_k\right| \geqslant\left|\sum\limits_{k=2^n}^{2^{n+1}} a_k\right| -\left|\sum\limits_{k=0}^{2^{n}-1} a_k\right| > \frac{1}{2} \left|\sum\limits_{k=2^n}^{2^{n+1}} a_k\right|.\] Hence, \[\frac{1}{2^{n+1}+1}\left|\sum\limits_{k=0}^{2^{n+1}} a_k\right| \gtrsim \frac{1}{2^{n}}\left|\sum\limits_{k=2^{n}}^{2^{n+1}} a_k\right| \gtrsim \sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k}.\] Thus, \[\sum_{k=2^{n}}^{2^{n+1}} \frac{|a_k|}{k} \lesssim \widehat{a}_{2^{n+1}} \leqslant \sup_{t \in \mathbb{N}_0} \min(1, 2^{t-n}) \widehat{a}_{2^t}.\] ◻
Proposition 1. There exists a sequence \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\) such that \(a_k = 0\), for any \(k \leqslant 0\) and \(\{a_k\}_{k=1}^{\infty} \notin \textrm{GM}\). In this sense we interpret the following relation: \[\overline{\textrm{GM}} \setminus \textrm{GM} \neq \varnothing.\]
Proof. Consider a sequence \(\{a_k\}_{k=-\infty}^{\infty}\) defined as follows \[a_k = \\ \begin{cases}0, & k \leqslant 15, \\ \frac{1}{2^n}, & 2^n \leqslant k < 2^{n+1}, \;\; n \geqslant 4, \;\;k \;\text{is odd};\\ 0, & 2^n \leqslant k < 2^n + [\sqrt{n}], \;\;n \geqslant 4, \;\;k \;\text{is even};\\ \frac{1}{2^n}, & 2^n + [\sqrt{n}] \leqslant k < 2^{n+1}, \;\;n \geqslant 4, \;\;k \;\text{is even}; \end{cases}\] For any \(n \geqslant 4\), we have \[\begin{align} \sum_{k=2^n}^{2^{n+1}-1}|\Delta a_k| \asymp \sum_{k=2^n}^{2^n + [\sqrt{n}]}\frac{1}{2^n} \asymp \frac{\sqrt{n}}{2^n}. \end{align}\] On the other hand, for given \(n \geqslant 4\), \(\lambda > 1\), we choose \(m, l \in \mathbb{N}\) such that \(2^m \leqslant\frac{2^n}{\lambda} < 2^{m+1}\), \(2^{l-1} < \lambda 2^n \leqslant 2^l\). Hence,
\[\begin{align} & \frac{1}{2^n} \sum_{k = \frac{2^n}{\lambda}}^{\lambda 2^n}|a_k| \leqslant\frac{1}{2^n} \sum_{k = 2^m}^{2^l}|a_k| \leqslant\frac{1}{2^n} \left(\sum_{s = m}^{l-1}\sum_{k = 2^s}^{2^{s+1}-1}|a_k| + |a_{2^l}|\right) \leqslant\frac{1}{2^n} (l-m+1) \leqslant\frac{3+ 2\log_2 \lambda}{2^n}. \end{align}\] Therefore, \(\{a_k\}_{k=1}^{\infty} \notin \textrm{GM}\).
On the other hand, for sufficient large \(n\), we have \[\begin{align} \widehat{a}_{2^n} & \geqslant\frac{1}{2^n} \left|\sum_{k = 0}^{2^n-1} a_k\right| \geqslant \frac{1}{2^n} \left(\sum_{s = 4}^{n-1} \sum_{k = 2^s}^{2^{s+1}-1} a_k\right) \geqslant\frac{1}{2^n} \left(\sum_{s = 4}^{n-1} \frac{2^s - [\sqrt{s}]}{2^s}\right)\\ & \geqslant\frac{1}{2^n} \left(n - 4 - \sum_{s = 0}^{n-1}\frac{[\sqrt{s}]}{2^s}\right) \gtrsim \frac{n}{2^n},\\ \end{align}\] since series \(\sum\limits_{n=0}^{\infty}\frac{[\sqrt{n}]}{2^n}\) is converging.
Hence, \[\begin{align} & \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{a}_{2^k} \geqslant\widehat{a}_{2^n} \gtrsim \frac{n}{2^n}. \end{align}\] Therefore, \(\{a_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\). ◻
The following example demonstrates that previously established generalizations of the Hardy-Littlewood theorem do not apply to certain series of the form \(\sum\limits_{k=1}^{\infty} (a_k + i b_k) e^{ikx}\) even while Theorems 1 and 2 remain applicable.
Proposition 2. There exist \(\{a_k\}_{k=1}^{\infty} \notin \textrm{GM}_{\mathbb{R}}\), \(\{b_k\}_{k=1}^{\infty} \in \textrm{GM}_{\mathbb{R}}\), such that \(\{c_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\), where \[c_k = \begin{cases} a_k + ib_k, & k > 0,\\ 0, & k \leqslant 0. \end{cases}\]
Proof. Consider sequence \(\{a_k\}_{k=1}^{\infty}\) defined as follows \[a_k = (-1)^k \frac{1}{2^{\frac{7}{4}n}},\;\; \text{for} \;\;2^n \leqslant k < 2^{n+1}, \;n \in \mathbb{N}_0.\]
First, we consider \[\label{eq1} \begin{align} \sum_{k = 2^n}^{2^{n+1}}|\Delta a_k| & > \sum_{k = 2^n}^{2^{n+1}-1}\frac{2}{2^{\frac{7}{4}n}} = \frac{2}{2^{\frac{3}{4}n}}. \end{align}\tag{23}\] On the other hand, for given \(n \in \mathbb{N}\), \(\lambda > 1\), define \(m, l \in \mathbb{N}\) such that \(2^m \leqslant\frac{2^n}{\lambda} < 2^{m+1}\), \(2^{l-1} < \lambda 2^n \leqslant 2^l\). Hence, \[\label{eq2} \begin{align} \frac{1}{2^n}\sum_{k = \frac{2^n}{\lambda}}^{\lambda 2^n}|a_k| &\leqslant\frac{1}{2^n}\sum_{k = 2^m}^{2^l}|a_k| = \frac{1}{2^n}\left(\sum_{s=m}^{l-1}\sum_{k = 2^s}^{2^{s+1}-1}|a_k|+|a_{2^{l}}|\right) \\ & = \frac{1}{2^n}\left(\sum_{s=m}^{l-1} \frac{1}{2^{\frac{3}{4}s}}+ \frac{1}{2^{\frac{7}{4}l}}\right) < \frac{4}{2^n}. \end{align}\tag{24}\] From 23 and 24 follows that there is no \(C > 0\), \(\lambda > 1\) such that inequality \[\sum_{k = 2^n}^{2^{n+1}} |\Delta a_k| \leqslant \frac{C}{2^n}\sum_{k = \frac{2^n}{\lambda}}^{\lambda 2^n} |a_k|\] holds for any \(n \in \mathbb{N}\).
Now we consider sequence \(\{b_k\}_{k=1}^{\infty}\) defined as follows \[b_k = \left(\frac{2}{3}\right)^n \;\; \text{for} \;\;2^n \leqslant k < 2^{n+1}, \;\;n \in \mathbb{N}_0.\] The sequence \(\{b_k\}_{k=1}^{\infty}\) is nonincreasing sequence, hence, \(\{b_k\}_{k=1}^{\infty} \in \textrm{GM}\).
Finally, we consider sequence \(\{c_k\}_{k=-\infty}^{\infty}\). Let \(n \in \mathbb{N}\). Then \[\begin{align} \sum_{k=2^n}^{2^{n+1}-1}|\Delta c_k| & = \sum_{k=2^n}^{2^{n+1}-1}\sqrt{(\Delta a_k)^2 + (\Delta b_k)^2} \\ & = \sum_{k=2^n}^{2^{n+1}-2}\sqrt{\frac{4}{2^{\frac{7}{2}n}}} + \sqrt{\left(\frac{1}{2^{\frac{7}{4}n}} + \frac{1}{2^{\frac{7}{4}(n+1)}}\right)^2 + \left(\left(\frac{2}{3}\right)^n - \left(\frac{2}{3}\right)^{n+1}\right)^2}\\ & \leqslant\frac{2}{2^{\frac{3}{4}n}} + \sqrt{\frac{4}{2^{\frac{7}{2}n}} + \left(\frac{4}{9}\right)^n} \leqslant\frac{2}{2^{\frac{3}{4}n}} + 3\left(\frac{2}{3}\right)^n < 5\left(\frac{2}{3}\right)^n. \end{align}\] On the other hand, \[\begin{align} \widetilde{c}_{2^n} & \geqslant\frac{1}{2^n}\left|\sum_{j = 1}^{2^n-1} c_k\right| = \frac{1}{2^n} \sqrt{\left(\sum_{k=0}^{n-1} \sum_{j = 2^k}^{2^{k+1}-1} a_j\right)^2 + \left(\sum_{k=0}^{n-1} \sum_{j = 2^k}^{2^{k+1}-1} b_j\right)^2} \\ & = \frac{1}{2^n} \sqrt{\left(\sum_{k=0}^{n-1} \sum_{j = 2^k}^{2^{k+1}-1} (-1)^j \frac{1}{2^{\frac{7}{4}k}}\right)^2 + \left(\sum_{k=0}^{n-1} \sum_{j = 2^k}^{2^{k+1}-1} \left(\frac{2}{3}\right)^k\right)^2} \\ & = \frac{1}{2^n} \sqrt{1 + \left(\sum_{k=0}^{n-1} \left(\frac{4}{3}\right)^k\right)^2} \geqslant\frac{1}{2^n} \sqrt{1 + \left(\frac{4}{3}\right)^{2n-2}} \geqslant\frac{3}{4}\left(\frac{2}{3}\right)^n. \end{align}\] Therefore, for any \(n \in \mathbb{N}\), \[\sum_{k=2^n}^{2^{n+1}-1}|\Delta c_k| \leqslant\frac{20}{3} \widehat{c}_{2^n} \lesssim \sup_{k \in \mathbb{N}_0} \min (1, 2^{k-n}) \widehat{c}_{2^k}.\] It is easy to check that this inequality holds for \(n = 0\). Hence, \(\{c_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\). ◻
The following example demonstrates that previously established generalizations of the Hardy-Littlewood theorem do not apply to certain series of the form \(\sum\limits_{k=-\infty}^{\infty} c_k e^{ikx}\) even while Theorems 1 and 2 remain applicable.
Proposition 3. There exists \(\{c_k\}_{k = -\infty}^{\infty}\) such that \(\{c_k\}_{k=-\infty}^{0} \notin \overline{\textrm{GM}}\), \(\{c_k\}_{k = 1}^{\infty} \in \overline{\textrm{GM}}\), but \(\{c_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\).
Proof. For integer \(k \geqslant 0\), let’s take \[c_k = \begin{cases} \left(\frac{2}{3}\right)^n, & \text{for} \;\;2^{n} \leqslant k < 2^{n+1}, n \geqslant 0 \\ \left(\frac{2}{3}\right)^n, & \text{for} \;\;k = -2^n, n \geqslant 0 \\ 0, & \text{for} \;\;-2^{n+1} < k < -2^{n}, n \geqslant 0 \\ 0, & \text{for} \;\;k = 0\\ \end{cases}\] It is easy to see that, \(\{c_k\}_{k=0}^{\infty}\) is a monotone sequence, whereas \(\{c_k\}_{k=-\infty}^{0}\) is lacunary.
First consider the sequence \(\{c_k\}_{k = -\infty}^{0}\). For \(n \geqslant 1\), we have \[\sum_{i = -2^{n} + 1}^{-2^{n-1}} |\Delta c_i| = \sum_{i = -2^{n} + 1}^{-2^{n-1}} |c_i - c_{i-1}| = \left(\frac{2}{3}\right)^n.\] Let \(k \geqslant 1\) and \(-2^k < m \leqslant-2^{k-1}\), then \[\frac{1}{|m|+1} \left|\sum_{j = m}^{0} c_j\right| \leqslant \frac{1}{2^{k-1}} \left(\sum_{j = -2^k}^{0} c_j\right) \leqslant \frac{1}{2^{k-1}} \sum_{j = 0}^{k} \left(\frac{2}{3}\right)^j \lesssim \frac{1}{2^k}.\] Therefore, for any \(k \geqslant 1\), \[{\widehat{c}_{2^k}}^- := \sup_{-2^k < m \leqslant-2^{k-1}} \frac{1}{|m|+1} \left|\sum_{j = m}^{0} c_j\right| \lesssim \frac{1}{2^{k}}.\] It is easy to see, that \[\widehat{c}_{2^0}^- = \sup_{-2^0 < m \leqslant-[2^{-1}]} \frac{1}{|m|+1} \left|\sum_{j = m}^{0} c_j\right| = c_0 = 0.\] Hence, \[\begin{align} \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{c}_{2^k}^- &= \max\left\{\max_{0\leqslant k \leqslant n} 2^{k-n}\widehat{c}_{2^k}^-, \sup_{k > n}\widehat{c}_{2^k}^-\right\}\\ & \leqslant\max\left\{\max_{0\leqslant k \leqslant n} 2^{k-n}\frac{1}{2^k}, \sup_{k > n}\frac{1}{2^k}\right\} \leqslant\frac{1}{2^{n}}. \end{align}\] Therefore, \(\{c_k\}_{k = -\infty}^{0} \notin \overline{\textrm{GM}}\).
Now consider the sequence \(\{c_k\}_{k = 0}^{+\infty}\). Let \(n \geqslant 1\), one can see that \[\sum_{i = 2^{n-1}}^{2^{n}-1} |\Delta c_i| = \sum_{i = 2^{n-1}}^{2^{n} - 1} |c_i - c_{i+1}| \lesssim \left(\frac{2}{3}\right)^n.\]
On the other hand, we have \[\begin{align} & \sup_{k \in \mathbb{N}_0} \min(1, 2^{k-n}) \widehat{c}_{2^k}^+ \geqslant \widehat{c}_{2^{n+2}}^+ = \sup_{2^{n+1} \leqslant m < 2^{n+2}} \frac{1}{m+1} \left|\sum_{j=0}^{m} c_j\right| \geqslant\frac{1}{2^{n+2}} \left(\sum_{j=0}^{2^{n+1}} c_j\right)\\ & \geqslant\frac{1}{2^{n+2}} \left(\sum_{s=0}^{n} \sum_{j = 2^s}^{2^{s+1}-1} \left(\frac{2}{3}\right)^s\right) = \frac{1}{2^{n+2}} \left(\sum_{s=0}^{n} \left(\frac{4}{3}\right)^s\right)\\ & = \frac{3}{2^{n+2}} \cdot \left(\left(\frac{4}{3}\right)^{n+1}-1\right) \gtrsim \frac{3}{2^{n+2}} \cdot \left(\frac{4}{3}\right)^{n} = \frac{3}{4}\cdot\left(\frac{2}{3}\right)^n. \end{align}\] Therefore, \(\{c_k\}_{k=0}^{\infty} \in \overline{\textrm{GM}}\). And, moreover, \(\{c_k\}_{k=-\infty}^{\infty} \in \overline{\textrm{GM}}\). ◻
We consider slightly modified notion of idempotent Fourier multiplier.
Definition 7. We will say that a sequence of numbers \(\{\lambda_k\}_{k=-\infty}^{\infty}\) is an idempotent multiplier in \(L_p([-\pi,\pi])\), \(1 < p < \infty\) if:
\(\lambda_k \in \{-1,1\}\), for all \(k \in \mathbb{Z}\);
The sequence \(\{\lambda_k\}_{k=-\infty}^{\infty}\) is an \(L_p\)-multiplier, i.e., for any function \(f \in L_p([-\pi, \pi])\) with the Fourier series \(\sum\limits_{k=-\infty}^{\infty} c_k e^{ikx}\), there exists a function \(f_{\lambda} \in L_p([-\pi,\pi])\) with the Fourier series \(\sum\limits_{k=-\infty}^{\infty} \lambda_k c_k e^{ikx}\) such that \(\|f_{\lambda}\|_p \lesssim \|f\|_p\).
Example 1. One can see that, a sequence \(\lambda_k = (-1)^k\), \(k \in \mathbb{Z}\), is the idempotent \(L_p\)-multiplier. Indeed, for a function \(f(x) \sim \sum\limits_{k = -\infty}^{\infty}c_k e^{ikx}\), a series \(\sum\limits_{k = -\infty}^{\infty}(-1)^k c_k e^{ikx} = \sum\limits_{k = -\infty}^{\infty}c_k e^{ik(x+\pi)}\) is a Fourier series of the function \(f_{\lambda}(x) = f(x + \pi)\). Since, \(\|f(\cdot)\|_{L_p} = \|f(\cdot + \pi)\|_{L_p}\), we conclude that \(\{(-1)^k\}_{k=-\infty}^{\infty}\) is the idempotent \(L_p\)-multiplier.
Example 2. By Marcinkiewicz’s multiplier theorem [37], the sequence \(\{\lambda_k\}_{k=-\infty}^{\infty}\) defined by \[\lambda_k = \begin{cases} (-1)^n, & \text{for} \;\;2^{n-1} \leqslant|k| < 2^n, \; \; n \geqslant 1\\ 1, & \text{for} \;\;k = 0 \end{cases}\] is an idempotent \(L_p\)-multiplier.
For any idempotent \(L_p\)-multiplier \(\lambda = \{\lambda_k\}_{k=-\infty}^{\infty}\), since \(\lambda_k^2 = 1\) and \(|\lambda_k| = 1\), we establish:
\(J_p(f) = J_p(f_{\lambda})\);
\(\|f\|_{L_p} \asymp \|f_{\lambda}\|_{L_p}\).
These properties imply the following Corollary.
Corollary 1. Let \(1 < p < \infty\), \(\lambda = \{\lambda_k\}_{k = -\infty}^{\infty}\) be an idempotent \(L_p\)-multiplier, and \(f(x) \sim \sum\limits_{k=-\infty}^{\infty} c_k e^{ikx}\) be an integrable on \([-\pi, \pi]\) function. If \(\lambda c = \{\lambda_k c_k\}_{k = -\infty}^{\infty} \in \overline{\textrm{GM}}\), then \[\|f\|_{L_p} \asymp J_p(f).\]