# ON WAVELET POLYNOMIALS AND WEYL MULTIPLIERS

ANNA KAMONT AND GRIGORI A. KARAGULYAN

ABSTRACT. For the wavelet type orthonormal systems $\phi_n$, we establish a new bound

$$ \left\| \max_{1 \le m \le n} \left| \sum_{j \in G_m} \langle f, \phi_j \rangle \phi_j \right| \right\|_p \lesssim \sqrt{\log(n+1)} \cdot \|f\|_p, \quad 1 < p < \infty, $$

where $G_m \subset \mathbb{N}$ are arbitrary sets of indexes. Using this estimate, we prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping wavelet polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.

## 1. INTRODUCTION

1.1. **An historical overview.** Recall some definitions well-known in the theory of orthogonal series (see [11]).

**Definition 1.1.** Let $\Phi = \{\phi_n : n = 1, 2, \dots\} \subset L^2(0, 1)$ be an orthonormal system. A sequence of positive numbers $\omega(n) \nearrow \infty$ is said to be an a.e. convergence Weyl multiplier (C-multiplier) if every series

$$ \sum_{n=1}^{\infty} a_n \phi_n(x), $$

with coefficients satisfying the condition $\sum_{n=1}^{\infty} a_n^2 w(n) < \infty$ is a.e. convergent. If such series converges a.e. after any rearrangement of the terms, then we say $w(n)$ is an a.e unconditional convergence Weyl multiplier (UC-multiplier) for $\Phi$.

The Menshov-Rademacher classical theorem ([12], [18], see also [11] or [6]) states that the sequence $\log^2 n$ is a C-multiplier for any orthonormal system. The sharpness of $\log^2 n$ in this theorem was proved by Menshov in the same paper [12]. That is any sequence $w(n) = o(\log^2 n)$ fails to be C-multiplier for some orthonormal system. The following inequality is the key ingredient in the proof of the Menshov-Rademacher theorem.

<footnote n="2010">2010 Mathematics Subject Classification. 42C05, 42C10, 42C20, 42C40.</footnote>

<footnote>Key words and phrases. wavelet type systems, Franklin system, non-overlapping polynomials, Weyl multiplier, Menshov-Rademacher theorem.</footnote>

<footnote>The second author is supported by the Science Committee of Armenia (the grant number will be provided later).</footnote>
**Theorem A** (Menshov-Rademacher, [12], [18]). If {$\phi_k : k = 1, 2, \dots, n$} $\subset L^2(0, 1)$ is an orthogonal system, then

$$ \left\| \max_{1 \le m \le n} \left| \sum_{k=1}^{m} \phi_k \right| \right\|_{2} \le c \cdot \log(n+1) \left\| \sum_{k=1}^{n} \phi_k \right\|_{2}, \qquad (1.1) $$

where $c > 0$ is an absolute constant.

Similarly, the counterexample of Menshov is based on the following results.

**Theorem B** (Menshov, [12]). For any natural number $n \in \mathbb{N}$ there exists an orthonormal system {$\phi_k : k = 1, 2, \dots, n$} $\subset L^2(0, 1)$, such that

$$ \left| \left\{ x \in (0, 1) : \max_{1 \le m \le n} \left| \frac{1}{\sqrt{n}} \sum_{k=1}^{m} \phi_k(x) \right| \ge c \log(n+1) \right\} \right| \gtrsim 1, $$

for an absolute constant $c > 0$.

Theorem B shows that the factor $\log(n+1)$ is optimal in inequality (1.1) when the class of all orthogonal systems is considered. However, if we narrow the class of orthogonal systems, it is of interest to find the best constant in inequality (1.1) for the class under consideration.

For the further discussion, we need to introduce some notation. The relation $a \lesssim b$ ($a \gtrsim b$) will stand for the inequality $a \le c \cdot b$ ($a \ge c \cdot b$), where $c > 0$ is an absolute constant. Given two sequences of positive numbers $a_n, b_n > 0$, we write $a_n \sim b_n$ if we have $c_1 \cdot a_n \le b_n \le c_2 \cdot a_n$, $n = 1, 2, \dots$ for some constants $c_1, c_2 > 0$. Throughout the paper, the base of log is equal 2.

Let $\Phi = \{\phi_k(x), k = 1, 2, \dots\} \subset L^\infty(0, 1)$ be an infinite orthonormal system. Given $g \in L^1(0, 1)$, consider the Fourier coefficients

$$ a_n = \langle g, \phi_n \rangle = \int_0^1 g \phi_n. $$

Let $f$ be a $\Phi$-polynomial (i.e. a finite linear combination of functions of $\Phi$). Then we say $f$ satisfies the relation $f \prec g$ (with respect to the system $\Phi$), if $f = \sum_{n=1}^{m} \lambda_n a_n \phi_n$, where $|\lambda_n| \le 1$. Given orthonormal system $\Phi$ and $1 < p < \infty$ we consider the numerical sequence

$$ \mathcal{A}_n^p(\Phi) = \sup_{\|g\|_p \le 1} \sup_{g_k \prec g, k=1,\dots,n} \left\| \max_{1 \le k \le n} |g_k(x)| \right\|_p, \qquad (1.2) $$

where the second $\sup$ is taken over all sequences of polynomials $g_k$, $k = 1, 2, \dots, n$, satisfying $g_k \prec g$. One can consider in (1.2) only the monotonic sequences of polynomials

$$ g_k = \sum_{j \in G_k} \langle g, \phi_k \rangle \phi_j \prec g, \quad k = 1, 2, \dots, n, $$

where $G_1 \subset G_2 \subset \dots \subset G_n \subset \mathbb{N}$, $\#G_n < \infty$. Then we will get another sequence $\mathcal{A}_{n,\text{mon}}^p(\Phi)$. If we additionally suppose that each $G_{k+1} \setminus G_k$ consists of a single integer,
then we will have the sequence $A_{n,sng}^p(\Phi)$. Clearly we have

$$A_{n,sng}^p(\Phi) \le A_{n,mon}^p(\Phi) \le A_n^p(\Phi). \qquad (1.3)$$

The case of $p = 2$ is special for (1.2). Theorem A implies $A_{n,mon}^2(\Phi) \lesssim \log(n+1)$ for every orthonormal system $\Phi$. On the other hand, applying Theorem B, one can also construct an infinite orthonormal system with the lower bound $A_{n,sng}^2(\Phi) \gtrsim \log(n+1)$, $n = 1, 2, \dots$. Thus we conclude that for the general orthonormal systems the logarithmic upper bound of $A_{n,mon}^2(\Phi)$ is optimal. As it was remarked in [7] from some results of Nikishin-Ulyanov [15] and Olevskii [16] it follows that $A_{n,mon}^2(\Phi) \gtrsim \sqrt{\log(n+1)}$ for any complete orthonormal system $\Phi$.

The recent papers of one of the authors [7–9] highlight the relation of sequences (1.3) in the study of almost everywhere convergence of special orthogonal series. It was proved in [7] that

**Theorem C.** If $\Phi$ is a martingale difference, then $A_{n,mon}^2(\Phi) \lesssim \sqrt{\log(n+1)}$.

**Theorem D.** For any generalized Haar system $\mathcal{H}$ we have the relation

$$A_{n,sng}^2(\mathcal{H}) \sim A_{n,mon}^2(\mathcal{H}) \sim \sqrt{\log(n+1)}.$$

The paper [7] also provides corollaries of these results like those that will be considered below. In the case of trigonometric system [8, 9] it was proved the following.

**Theorem E.** If $\mathcal{T}$ is the trigonometric system, then we have

$$A_{n,sng}^2(\mathcal{T}) \sim A_{n,mon}^2(\mathcal{T}) \sim \log(n+1). \qquad (1.4)$$

Note that the upper bound $A_{n,mon}^2(\mathcal{T}) \lesssim \log(n+1)$ in (1.4) follows from the Menshov-Rademacher theorem. So the novelty here is the estimate $A_{n,sng}^2(\mathcal{T}) \gtrsim \log(n+1)$, which shows that the trigonometric system has no better estimate of the sequences $A_n^2$ than the general orthonormal systems have.

1.2. The statement of the new results. In this paper we will consider orthonormal systems of wavelet type $\Phi = \{\phi_k(x) : k = 1, 2, \dots\}$ defined on the interval $[0, 1]$. For the precise definition of such systems we use the function

$$\xi(x) = \frac{1}{(1 + |x|)^{1+\delta}}, \quad 0 < \delta < 1,$$

as well as the notations

$$t_1 = \frac{1}{2}, \quad t_k = \frac{2j-1}{2^{n+1}}, \quad k \ge 2, \\ \text{where } k = 2^n + j, 1 \le j \le 2^n, n = 0, 1, 2, \dots \qquad (1.5)$$
Hence we suppose

$$ \int_0^1 \phi_k(t) dt = 0, \quad k \ge 1, \qquad (1.6) $$

$$ |\phi_k(x)| \le c2^{n/2} \cdot \xi(2^n(x - t_k)), \qquad (1.7) $$

$$ |\phi_k(t) - \phi_k(t')| \le c2^{n/2}(2^n|t - t'|)^\alpha \cdot \xi(2^n(t - t_k)) \quad \text{if } |t - t'| \le 2^{-n}, \qquad (1.8) $$

where $0 < \alpha \le 1$. Recall that such wavelet type systems in an other context were considered in [14]. Note that the Fourier series of any function $f \in L^p$, $1 < p < \infty$, in a wavelet type system converges in $L^p$ unconditionally (see Remark 1.5). This fact will be used in the statements of some results below. Besides, in the definition of $A_n^p(\Phi)$ instead of a $\Phi$-polynomial $f$ we can equivalently consider $f \in L^p$. Examples of wavelet type systems are considered in Remark 1.4 below.

From this moment the constants occurring in the relations $\lesssim$ and $\sim$ may depend on the constants form (1.7), (1.8) and the parameter $1 < p < \infty$. The main result of the paper is the following sharp estimate.

**Theorem 1.1.** If an orthonormal system $\Phi$ satisfies (1.6)-(1.8), then for any $1 < p < \infty$ it holds the relation

$$ A_n^p(\Phi) \lesssim \sqrt{\log(n+1)}, \quad n = 1, 2, \dots \qquad (1.9) $$

Note that in view of (1.3), a likewise bound holds also for the sequences $A_{n,mon}^p(\Phi)$ and $A_{n,sgn}^p(\Phi)$. The following two corollaries show that for a wide class of wavelet type systems bound (1.9) is sharp. Namely,

**Corollary 1.1.** If an orthonormal system $\Phi$ satisfies (1.6)-(1.8) and is a basis in $L^2$, then for any $1 < p < \infty$ we have

$$ A_{n,mon}^p(\Phi) \sim A_n^p(\Phi) \sim \sqrt{\log(n+1)}. $$

**Corollary 1.2.** If $\mathcal{F}$ is either Haar or Franklin system, then for any $1 < p < \infty$ we have

$$ A_{n,sng}^p(\mathcal{F}) \sim A_{n,mon}^p(\mathcal{F}) \sim A_n^p(\mathcal{F}) \sim \sqrt{\log(n+1)}. \qquad (1.10) $$

The following bound is an interesting phenomenon of the wavelet type systems and immediately follows from (1.9).

**Corollary 1.3.** Let $\Phi = \{\phi_k\}$ be a wavelet type system and $G_k \subset \mathbb{N}$, $k = 1, 2, \dots, n$ be an arbitrary sets of indexes. Then for any function $f \in L^p$, $1 < p < \infty$, we have

$$ \left\| \max_{1 \le m \le n} \left| \sum_{j \in G_m} \langle f, \phi_j \rangle \phi_j \right| \right\|_p \lesssim \sqrt{\log(n+1)} \cdot \|f\|_p. $$

In case of $p = 2$ Theorem 1.1 implies the following results concerning the C or UC multipliers for non-overlapping $\Phi$-polynomials.
**Corollary 1.4.** If $\Phi = \{\phi_k\}$ is a wavelet type system, then the sequence $\log n$ is a C-multiplier for any system of $L^2$-normalized non-overlapping $\Phi$-polynomials

$$p_n(x) = \sum_{j \in G_n} c_j \phi_j(x), \quad n = 1, 2, \dots,$$

where $G_n \subset \mathbb{N}$ are finite and pairwise disjoint.

The following particular case of Corollary 1.4 is also new and interesting even for the classical Franklin system.

**Corollary 1.5.** The sequence $\log n$ is a C-multiplier for any rearrangement of a wavelet type system.

**Corollary 1.6.** Let $\Phi = \{\phi_k\}$ be a wavelet type system and $\{p_n\}$ be a sequence of $L^2$-normalized non-overlapping $\Phi$-polynomials. If $w(n)/\log(n+1)$ is increasing and

$$\sum_{n=1}^{\infty} \frac{1}{nw(n)} < \infty, \qquad (1.11)$$

then $w(n)$ is an UC-multiplier for $\{p_n\}$.

The only prior result in the context of Wail multipliers, concerning to non Haar wavelet type systems, is due to Gevorkyan [2], who proved

**Theorem F** (Gevorkyan, [2]). The sequence $w(n) \nearrow$ is an UC-multiplier for the Franklin system if and only if it satisfies (1.11).

### 1.3. Remarks.

*Remark 1.1.* In the case of the Franklin system the optimality of $\log n$ in Corollary 1.5 as well as condition (1.11) in Corollary 1.6 both follows just from the direct combination of this result of Gevorkyan with a result of Ul'yanov-Poleshchuk [17, 20].

*Remark 1.2.* Corollary 1.3 is a direct consequence of Theorem 1.1. Corollaries 1.4-1.6 follow from Theorem 1.1 in exactly the same way as Theorem C implies Corollaries 1.4-1.6 in [7], therefore we have decided to skip the details. The upper bounds in Corollaries 1.1 and 1.2 immediately follows from Theorem 1.1. The lower bounds easily follow from certain results well-known in the theory of general orthogonal series and their proofs will be shortly provided immediately after the proof of Theorem 1.1.

*Remark 1.3.* Similarly, as in the proofs of Theorem C and Theorem D the main argument in the proof of Theorem 1.1 is a good-$\lambda$ inequality for the Haar system due to Chang-Wilson-Wolff [1]. The needed link between Haar and wavelet type expansions is provided by Lemma 3.4. An application of the inequality of [1] in the study of maximal functions of Mikhlin-Hörmander multipliers was considered in [3] by Grafakos-Honzík-Seeger. Namely, given multipliers $m_k$, $k = 1, 2, \dots, N$, on $\mathbb{R}^n$ with uniform estimates, it was proved the optimal $\sqrt{\log N}$ bound in $L^p$, $1 < p < \infty$, for the maximal function $\max_k |\mathcal{F}^{-1}(m_k \hat{f})|$.
**Remark 1.4.** Examples of wavelet type systems are the classical wavelets, orthonormal spline systems on $[0, 1]$, where an interesting case is the Franklin system. All the results of the paper can be applied also to the periodic wavelet systems, since those can be split into a union of two wavelet type systems. With a slight change in the proofs one can also establish the results for the biorthogonal systems $(\phi_n, \psi_n)$, where both $\phi_n$ and $\psi_n$ satisfy the conditions (1.7) and (1.8).

Finally, observe that the Haar system doesn't satisfy (1.8). Nevertheless, the proof of estimate (1.10) for the Haar system is straightforward and we will not need Lemma 3.4. Indeed, let $f = \sum_{j=1}^{\infty} b_j h_j \in L^p$ and $p_k \ll f$, $k = 1, 2, \dots, n$. Instead of (4.2) one needs to consider the function

$$ \mathcal{P}(x) = \sup_{1 \le k \le n} S(p_k)(x) \le S(f)(x), $$

where $S$ is the Haar square function. So we have $\|\mathcal{P}\|_p \le c_p \|f\|_p$ (see [11], chap. 3). Then repeating the argument of the proof of Theorem 1.1 given in Section 4, we will get (1.10) for the Haar system too.

**Remark 1.5.** Observe that the wavelet type systems share the following property of unconditional bases of $L^p$: for any sequence $\lambda = \{\lambda_k\}$, $|\lambda_k| \le 1$, and a function $f \in L^p$, $1 < p < \infty$, the series $T_\lambda(f) = \sum_k \lambda_k \langle f, \phi_k \rangle \phi_k$ converges in $L^p$ and $\|T_\lambda(f)\|_p \lesssim \|f\|_p$. This property can be established by a well-known argument (see for example [5]), proving that $K_\lambda(x, t) = \sum_k \lambda_k \phi_k(x) \phi_k(t)$ is a Calderón-Zygmund kernel, namely

$$ |K_\lambda(x, t)| \lesssim \frac{1}{|x - t|}, \qquad (1.12) $$

$$ |K_\lambda(x, t) - K_\lambda(x, t')| \lesssim \frac{|t - t'|^\beta}{|x - t|^{1+\beta}}, \quad |x - t| > 2|t - t'|, \qquad (1.13) $$

where $0 < \beta < \min\{\alpha, \delta\}$. Concerning to unconditional wavelet bases in $L^p$, $1 < p < \infty$, we can also refer the readers to the papers [4, 13, 21, 22].

**Remark 1.6.** Suppose $T_k$, $k = 1, 2, \dots, n$, is a collection of Calderón-Zygmund operators with kernels, satisfying the standard conditions (1.12) and (1.13) uniformly. It was proved in [10] that

$$ \left\| \max_{1 \le k \le n} |T_k(f)| \right\|_p \lesssim \log(n+1) \cdot \|f\|_p, \quad 1 < p < \infty \qquad (1.14) $$

(see [10], Corollary 2.11). Corollary 1.3 above shows that for specific Calderón-Zygmund operators, namely

$$ T_k(f) = \sum_{j \in G_k} \langle f, \phi_j \rangle \phi_j, \quad k = 1, 2, \dots, n, $$

inequality (1.14) holds with the bound $\sqrt{\log(n+1)}$.
## 2. NOTATIONS AND AUXILIARY ESTIMATES

Recall the definition of $L^2$-normalized Haar system $h_k(x)$, $k = 1, 2, \dots$, on $[0, 1]$. That is $h_1(x) = 1$ and for $k \ge 2$ of the form (1.5) we have

$$h_k(x) = \begin{cases} 2^{n/2} & \text{if } x \in \left[ \frac{j-1}{2^n}, \frac{2j-1}{2^{n+1}} \right), \\ -2^{n/2} & \text{if } x \in \left[ \frac{2j-1}{2^{n+1}}, \frac{j}{2^n} \right), \\ 0 & \text{otherwise.} \end{cases}$$

Given $1 < q < \infty$ define the maximal function

$$M_q(f)(x) = \sup_{I: I \supset x} \left( \frac{1}{|I|} \int_I |f|^q \right)^{1/q}, \quad f \in L^1(\mathbb{T}),$$

where $\sup$ is taken over all the open intervals $I \subset [0, 1)$ containing the point $x$. The case of $q = 1$, which is the classical Hardy-Littlewood maximal function, will be simply denoted by $M$. Recall the Fefferman-Stein vector valued maximal inequality (see [19], chap. 2)

$$\left\| \left( \sum_{m=1}^{\infty} |M_q(g_k)|^2 \right)^{1/2} \right\|_p \lesssim \left\| \left( \sum_{m=1}^{\infty} |g_k|^2 \right)^{1/2} \right\|_p, \qquad (2.1)$$

where $1 < p < \infty$, $1 \le q < \min\{2, p\}$. In this section we will establish standard estimates to be used in the proof of the main lemma.

**Lemma 2.1.** If $\{\phi_n\}$ is a wavelet type system, then for any coefficients $a_k$ there hold the bounds

$$\sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k(x)| \lesssim \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right) (x), \qquad (2.2)$$

$$\mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right) (x) \lesssim \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right) (x). \qquad (2.3)$$

*Proof.* Using (1.7) and a standard argument, we have

$$\begin{align*} \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k(x)| & \lesssim 2^{m/2} \sum_{k=2^{m-1}+1}^{2^m} |a_k| \xi(2^m(x-t_k)) \\ & \lesssim 2^m \int_0^1 \sum_{k=2^{m-1}+1}^{2^m} |a_k||h_k(t)|\xi(2^m|x-t|)dt \\ & \lesssim \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right) (x) \end{align*}$$
that gives (2.2). To prove (2.3) we use (1.8) and the normality condition $||\phi_k||_2 = 1$. So one can fix a number $a > 0$, depending on the constant in (1.7), such that

$$ \int_{t_k - a/2^m}^{t_k + a/2^m} |\phi_k(t)|^2 dx \ge \frac{1}{2}, \quad 2^{m-1} < k < 2^m. \qquad (2.4) $$

Given $\varepsilon > 0$ consider the set

$$ E_k = \{x \in (t_k - a/2^m, t_k + a/2^m) : |\phi_k(x)| > \varepsilon 2^{m/2}\}. $$

From (2.4) and (1.7) we obtain

$$ \frac{1}{2} \le c^2 2^m |E_k| + \varepsilon^2 2^m (2a/2^m - |E_k|), $$

that implies $|E_k| \gtrsim 1/n$ for a small enough $\varepsilon$ independent on $k$. Thus one can easily get

$$ \begin{aligned} \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right) (x) &\lesssim \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} 2^{m/2} |a_k| \mathbf{1}_{E_k} \right) (x) \\ &\lesssim \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k| |\phi_k| \right) (x) \end{aligned} $$

and so (2.3).

<figure><img src="image_1.png" /></figure>

The following lemma is a version of Littlewood-Paley inequality.

**Lemma 2.2.** Let $\{\phi_n\}$ be a wavelet type system and $1 < p < \infty$. Then for any function $f \in L^p$ it holds the inequality

$$ \left\| \left( \sum_{m=1}^{\infty} \left( \sum_{k=2^{m-1}+1}^{2^m} |\langle f, \phi_k \rangle \phi_k| \right)^2 \right)^{1/2} \right\|_p \lesssim \|f\|_p. \qquad (2.5) $$

*Proof.* Denote $a_k = \langle f, \phi_k \rangle$ and let $r_k(x)$ be the Rademacher system. According to Remark 1.5 and Khintchin's inequality we have

$$ \left\| \left( \sum_{k=1}^{\infty} a_k^2 \phi_k^2 \right)^{1/2} \right\|_p^p \sim \int_0^1 \int_0^1 \left| \sum_k r_k(t) a_k \phi_k(x) \right|^p dx dt \lesssim \|f\|_p^p. \qquad (2.6) $$
which is the classical version of the Littlewood-Paley inequality. Then, by (2.3) in particular we have $|h_k(t)| \lesssim \mathcal{M}(\phi_k)(t)$. Thus, from (2.2), (2.1) with $q = 1$ and (2.6), we obtain

$$ \begin{aligned} \left\| \left[ \sum_{m=1}^{\infty} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right)^2 \right]^{1/2} \right\|_p & \lesssim \left\| \left[ \sum_{m=1}^{\infty} \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right)^2 \right]^{1/2} \right\|_p \\ & \lesssim \left\| \left( \sum_{k=2}^{\infty} a_k^2 h_k^2 \right)^{1/2} \right\|_p \lesssim \left\| \left[ \sum_{k=2}^{\infty} (\mathcal{M}(a_k \phi_k))^2 \right]^{1/2} \right\|_p \\ & \lesssim \left\| \left( \sum_{k=2}^{\infty} a_k^2 \phi_k^2 \right)^{1/2} \right\|_p \lesssim \|f\|_p. \end{aligned} $$

□

### 3. THE MAIN LEMMA

Let $\Phi = \{\phi_k(x) : k = 1, 2, \dots\}$ be our wavelet type system, i.e. it satisfies (1.6), (1.7) and (1.8). For $f \in L^1(0, 1)$, we consider the Fourier partial sums

$$ \Phi_n(f)(x) = \sum_{j=1}^{2^n} \langle f, \phi_j \rangle \phi_j(x), \quad n = 0, 1, 2, \dots, $$

$$ \Delta\Phi_n(f)(x) = \sum_{j=2^{n-1}+1}^{2^n} \langle f, \phi_j \rangle \phi_j(x), \quad n \ge 1. $$

The Haar system analogues of such sums will be denoted by

$$ H_n(f)(x) = \sum_{j=1}^{2^n} \langle f, h_j \rangle h_j(x), \quad \Delta H_n(f)(x) = \sum_{j=2^{n-1}+1}^{2^n} \langle f, h_j \rangle h_j(x). \quad (3.1) $$

We denote by $I_n(x)$ the dyadic interval of the form $[(j-1)/2^n, j/2^n)$, containing a point $x \in [0, 1)$. For the Haar partial sum (3.1) we have

$$ H_n(f)(x) = \frac{1}{|I_n(x)|} \int_{I_n(x)} f, \quad x \in [0, 1) $$

(see [11], chap. 3). The following standard bound

$$ \sum_{2^{n-1} < k \le 2^n} |\phi_k(x) \phi_k(t)| \lesssim 2^n \xi(2^n(x-t)) \quad (3.2) $$

one can find for example in [5] (see chap. 5, Lemma 3.12).
**Lemma 3.1.** For any interval $I = [a, b) \subset [0, 1)$ we have

$$|\Delta\Phi_m(\mathbf{1}_I)(x)| \lesssim \frac{1}{(1 + 2^m|x-a|)^\delta} + \frac{1}{(1 + 2^m|x-b|)^\delta}, \quad x \in [0, 1), \quad (3.3)$$

$$|\Delta\Phi_m(\mathbf{1}_I)(x)| \lesssim 2^m |I| \xi(2^m(x-a)), \quad x \in [0, 1] \setminus 2I. \quad (3.4)$$

*Proof.* To prove (3.3) first consider the case $x \in [0, 1) \setminus I$ and suppose that $0 \le x < a$. Using (3.2), we obtain

$$|\Delta\Phi_m(\mathbf{1}_I)(x)| \lesssim \int_a^b \sum_{2^{m-1} < k \le 2^m} |\phi_k(x)\phi_k(t)|dt \lesssim \int_a^b \frac{2^m}{(1 + 2^m|x-t|)^{1+\delta}}dt \\ \lesssim \sum_{k \ge 2^m|x-a|} \frac{1}{k^{1+\delta}} \lesssim \frac{1}{(1 + 2^m|x-a|)^\delta}.$$

In the case $x \ge b$ we will have the bound $\lesssim \frac{1}{(1+2^m|x-b|)^\delta}$. If $x \in I$, then by (1.6) we can write

$$\Delta\Phi_m(\mathbf{1}_I)(x) = \Delta\Phi_m(\mathbf{1}_{[0,1)\setminus I})(x),$$

and (3.3) can be obtained similarly. Then, under the condition $x \in [0, 1] \setminus 2I$ we have

$$|\Delta\Phi_m(\mathbf{1}_I)(x)| \lesssim \int_I \frac{2^m}{(1 + 2^m|x - t|)^{1+\delta}}dt \\ \le |I| \cdot \max_{t \in I} \frac{2^m}{(1 + 2^m|x - t|)^{1+\delta}} \lesssim 2^m |I| \xi(2^m(x - a))$$

that gives us (3.4).

<figure><img src="image_2.png" /></figure>

For a function $f(t)$ defined on a set $E$ we denote

$$\mathrm{OSC}_E(f) = \sup_{x,t \in E} |f(x) - f(t)|.$$

**Lemma 3.2.** If $g \in L^1(0, 1)$, then for any integers $n = m \ge 1$ it holds the inequality

$$|\Delta H_n(\Delta\Phi_m(g))(x)| \lesssim 2^{\alpha(m-n)} \cdot \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} |\langle g, \phi_k \rangle| |\phi_k| \right) (x), \quad x \in [0, 1).$$
*Proof.* The case of $n = m$ is trivial. So we suppose $n > m$ and denote $a_k = \langle g, \phi_k \rangle$. First using (1.8) then (2.3), we obtain

$$
\begin{aligned}
|\Delta H_n(\Delta\Phi_m(g))(x)| &= \left| \frac{1}{|I_{n-1}(x)|} \int_{I_{n-1}(x)} \Delta\Phi_m(g) - \frac{1}{|I_n(x)|} \int_{I_n(x)} \Delta\Phi_m(g) \right| \\
&\le \sum_{k=2^{m-1}+1}^{2^m} |a_k| \mathrm{OSC}_{I_{n-1}(x)}(\phi_k) \\
&\lesssim 2^{m/2} \cdot 2^{\alpha(m-n)} \sum_{k=2^{m-1}+1}^{2^m} |a_k| \xi(2^m(x - t_k)) \\
&\lesssim 2^{\alpha(m-n)} \int_0^1 \sum_{k=2^{m-1}+1}^{2^m} |a_k| |h_k(t)| \cdot 2^m \xi(2^m(x - t)) dt \\
&\lesssim 2^{\alpha(m-n)} \cdot \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} a_k h_k \right) (x) \\
&\lesssim 2^{\alpha(m-n)} \cdot \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k| |\phi_k| \right) (x),
\end{aligned}
$$

and lemma is proved.

<figure><img src="image_3.png" /></figure>

**Lemma 3.3.** If $f \in L^1(0,1)$ and $m \ge n \ge 1$, then for any $q > 1$ with $q' = q/(q-1) > \delta^{-1}$ we have

$$
|H_n(\Delta\Phi_m(f))(x)| \lesssim 2^{(n-m)/q'} \mathcal{M}_q(\Delta\Phi_m(f))(x), \quad x \in [0,1]. \quad (3.5)
$$

*Proof.* Let $x \in [0, 1)$. By the orthogonality of $\{\phi_n\}$ we have $\Delta\Phi_m(f) = \Delta\Phi_m(\Delta\Phi_m(f))$. Let $J_0 = I_n(x)$, $J_k = 2^k I_n(x) \cap [0, 1]$, $k = 1, 2, \dots$. Thus, using a duality argument, we can write

$$
\begin{aligned}
|H_n(\Delta\Phi_m(f))(x)| &= \frac{1}{|I_n(x)|} \left| \int_{I_n(x)} \Delta\Phi_m(f)(u)du \right| \\
&= \frac{1}{|I_n(x)|} \left| \int_{I_n(x)} \Delta\Phi_m(\Delta\Phi_m(f))(u)du \right| \\
&= 2^n \left| \int_0^1 \Delta\Phi_m(\mathbf{1}_{I_n(x)})(u)\Delta\Phi_m(f)(u)du \right| \\
&\le 2^n \left| \int_{J_0} \Delta\Phi_m(\mathbf{1}_{I_n(x)})(u)\Delta\Phi_m(f)(u)du \right| \\
&\quad + 2^n \sum_{k \ge 1} \left| \int_{J_k \setminus J_{k-1}} \Delta\Phi_m(\mathbf{1}_{I_n(x)})(u)\Delta\Phi_m(f)(u)du \right|.
\end{aligned}
\quad (3.6)
$$
Thus using (3.3) for the first integral, we obtain

$$ 
\begin{aligned}
& 2^n \left| \int_{J_0} \Delta \Phi_m(\mathbf{1}_{I_n(x)})(u) \Delta \Phi_m(f)(u) du \right| \\
& \lesssim 2^n \left( \int_{J_0} |\Delta \Phi_m(f)(u)|^q du \right)^{1/q} \cdot \left( \int_{\mathbb{R}} \frac{dt}{(1+2^m|t|)^{q'\delta}} \right)^{1/q'} \\
& \lesssim 2^{(n-m)/q'} \mathcal{M}_q(\Delta \Phi_m(f))(x).
\end{aligned} 
\quad (3.7) $$

From (3.4) it follows that

$$ |\Delta \Phi_m(\mathbf{1}_{I_n(x)})(u)| \lesssim 2^{m-n} \left( \frac{2^{n-m}}{2^k} \right)^{1+\delta} = \frac{2^{\delta(n-m)}}{2^{k(1+\delta)}}, \quad u \in J_k \setminus J_{k-1}, $$

and so

$$ 
\begin{aligned}
& 2^n \left| \int_{J_k \setminus J_{k-1}} \Delta \Phi_m(\mathbf{1}_{I_n(x)})(u) \Delta \Phi_m(f)(u) du \right| \\
& \lesssim 2^n \frac{2^{\delta(n-m)}}{2^{k(1+\delta)}} \left| \int_{J_k} |\Delta \Phi_m(f)(u)| du \right| \\
& \lesssim 2^n \frac{2^{\delta(n-m)}}{2^{k(1+\delta)}} |J_k| \mathcal{M}(\Delta \Phi_m(f))(x) \\
& \leq \frac{2^{\delta(n-m)}}{2^{k\delta}} \mathcal{M}(\Delta \Phi_m(f))(x).
\end{aligned} 
\quad (3.8) $$

Hence, combining (3.6), (3.7) and (3.8), we get (3.5).

<figure><img src="image_4.png" /></figure>

Next, we will need a well-known discrete convolution inequality

$$ \left( \sum_{n \in \mathbb{Z}} \left( \sum_{k \in \mathbb{Z}} |a_k b_{n-k}| \right)^2 \right)^{1/2} \le \left( \sum_{n \in \mathbb{Z}} a_k^2 \right)^{1/2} \cdot \left( \sum_{n \in \mathbb{Z}} |b_k| \right). \quad (3.9) $$

Also, recall the definition of the Haar square function

$$ S(f)(x) = \left( \sum_{k=1}^{\infty} |\langle f, h_k \rangle|^2 h_k^2(x) \right)^{1/2}. $$

**Lemma 3.4 (main).** If $f \in L^p$, $1 < p < \infty$, then

$$ \left\| \sup_{|\lambda_k| \le 1} S \left( \sum_k \lambda_k a_k \phi_k \right) \right\|_p \lesssim \|f\|_p, $$

where the sup is taken over all the sequences $\lambda = \{\lambda_k\}$ with $|\lambda_k| \le 1$.
Proof. Clearly, we can suppose that $a_1 = 0$ and $0 < \delta < 1$. For a given sequence $\lambda = \{\lambda_k : |\lambda_k| \le 1\}$, we denote

$$f_{\lambda} = \sum_{k \ge 1} \lambda_k a_k \phi_k = \sum_{k \ge 2} \lambda_k a_k \phi_k.$$

Chose a number $q = \min\{(p+1)/2, (1-\delta/2)^{-1}, 3/2\}$, which is depended only on $p$ and $\delta$. Combining Lemma 3.2, Lemma 3.3, and inequality (3.9), we can write

$$ \begin{aligned} \left| \sum_{n=1}^{\infty} \Delta H_n \left( \sum_{k \ge 1} \lambda_k a_k \phi_k \right) (x) \right|^2 & \le \sum_{n=1}^{\infty} \left[ \sum_{m=1}^{\infty} \left| \Delta H_n (\Delta \Phi_m (f_{\lambda}))(x) \right| \right]^2 \\ & \lesssim \sum_{n=1}^{\infty} \left[ \sum_{m=n+1}^{\infty} 2^{(n-m)/q'} \mathcal{M}_q (\Delta \Phi_m (f_{\lambda}))(x) \right]^2 \\ & \qquad + \sum_{n=1}^{\infty} \left[ \sum_{m=1}^{n} 2^{\alpha(m-n)} (\mathcal{M} (\Delta \Phi_m (f_{\lambda}))(x)) \right]^2 \\ & \lesssim \sum_{n=1}^{\infty} \left[ \sum_{m=n+1}^{\infty} 2^{(n-m)/q'} \mathcal{M}_q \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right) (x) \right]^2 \\ & \qquad + \sum_{n=1}^{\infty} \left[ \sum_{m=1}^{n} 2^{\alpha(m-n)} \mathcal{M} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right) (x) \right]^2 \\ & \lesssim \sum_{m=1}^{\infty} \left[ \mathcal{M}_q \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right) (x) \right]^2. \end{aligned} $$

Thus we get the pointwise estimate

$$ \sup_{|\lambda_k| \le 1} S \left( \sum_{k \ge 1} \lambda_k a_k \phi_k \right) (x) \lesssim \left[ \sum_{m=1}^{\infty} \left( \mathcal{M}_q \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right) (x) \right)^2 \right]^{1/2}. $$

Then, using (2.1) and (2.5), we obtain

$$ \left\| \sup_{|\lambda_k| \le 1} S \left( \sum_{k \ge 1} \lambda_k a_k \phi_k \right) \right\|_p \lesssim \left\| \left[ \sum_{m=1}^{\infty} \left( \sum_{k=2^{m-1}+1}^{2^m} |a_k \phi_k| \right)^2 \right]^{1/2} \right\|_p \lesssim \|f\|_p. $$

Lemma is proved.

□
4. PROOF OF THEOREM 1.1 AND COROLLARIES 1.1 AND 1.2

A key argument in the proof of Theorem 1.1 is the following good-$\lambda$ inequality due to Chang-Wilson-Wolff (see [1], Corollary 3.1):

$$ \begin{aligned} & |\{x \in [0,1) : \mathcal{M}^d(f)(x) > \lambda, Sf(x) < \varepsilon\lambda\}| \\ & \lesssim \exp\left(-\frac{c}{\varepsilon^2}\right) |\{\mathcal{M}^d(f)(x) > \lambda/2\}|, \quad \lambda > 0, 0 < \varepsilon < 1, \quad (4.1) \end{aligned} $$

where $\mathcal{M}^d$ denotes the dyadic maximal function

$$ \mathcal{M}^d(f)(x) = \sup_{n \ge 1} \frac{1}{|I_n(x)|} \int_{I_n(x)} |f|. $$

Let $f \in L^p(0,1)$, $1 < p < \infty$ and $a_k = \langle f, \phi_k \rangle$. Suppose the functions $p_k \in L^p(\mathbb{T})$, $k = 1, 2, \dots, n$ satisfy $p_k < f$ with respect to our wavelet type system $\Phi$. It is clear that

$$ \mathcal{P}(x) = \sup_{1 \le k \le n} S(p_k)(x) \le \sup_{|\lambda_k| \le 1} S \left( \sum_{k=0}^{\infty} \lambda_k a_k \phi_k \right) (x). \quad (4.2) $$

Thus, according to Lemma 3.4, we have

$$ \|\mathcal{P}\|_p \lesssim \|f\|_p. $$

On the other hand, $|g(x)| \le \mathcal{M}^d g(x)$ a.e. for any function $g \in L^1$, as well as $S(p_k)(x) \le \mathcal{P}(x)$, $k = 1, 2, \dots, n$. Thus, applying inequality (4.1) with $\varepsilon_n = (c/\ln n)^{1/2}$, we obtain

$$ \begin{aligned} & |\{ |p_k(x)| > \lambda, \mathcal{P}(x) \le \varepsilon_n \lambda \}| \\ & \lesssim \exp\left(-\frac{c}{\varepsilon_n^2}\right) |\{\mathcal{M}^d p_k(x) > \lambda/2\}|. \end{aligned} \quad (4.3) $$

For $p^*(x) = \max_{1 \le k \le n} |p_k(x)|$ we obviously have

$$ \begin{aligned} & \{p^*(x) > \lambda\} \subset \{p^*(x) > \lambda, \mathcal{P}(x) \le \varepsilon_n \lambda\} \\ & \cup \{\mathcal{P}(x) > \varepsilon_n \lambda\} = A(\lambda) \cup B(\lambda), \end{aligned} $$

and thus

$$ \|p^*\|_p^p \le p \int_0^\infty \lambda^{p-1} |A(\lambda)| d\lambda + p \int_0^\infty \lambda^{p-1} |B(\lambda)| d\lambda. $$
From (4.3) it follows that

$$
\begin{align*}
\int_0^\infty \lambda^{p-1} |A(\lambda)| d\lambda &\le \sum_{k=1}^n \int_0^\infty \lambda^{p-1} |\{|p_k| > \lambda, \mathcal{P} \le \varepsilon_n \lambda\}| d\lambda \\
&\le \exp\left(-\frac{c}{\varepsilon_n^2}\right) \sum_{k=1}^n \int_0^\infty \lambda^{p-1} |\{\mathcal{M}^d p_k > \lambda/2\}| d\lambda \\
&\lesssim \frac{1}{n} \sum_{k=1}^n \|\mathcal{M}^d p_k\|_p^p \\
&\lesssim \frac{1}{n} \sum_{k=1}^n \|p_k\|_p^p \\
&\le \|f\|_p^p.
\end{align*}
$$

Combining this and

$$
p \int_0^\infty \lambda^{p-1} |B(\lambda)| d\lambda = \varepsilon_n^{-p} \|\mathcal{P}\|_p^p \lesssim (\log(n+1))^{p/2} \cdot \|f\|_p^p,
$$

we get

$$
\|p^*\|_p = \left\| \max_{1 \le m \le n} |p_m(x)| \right\|_p \lesssim \sqrt{\log(n+1)} \cdot \|f\|_p
$$

and so (1.9).

*Proof of Corollary 1.1.* In view of (1.3) it remains to show

$$
\mathcal{A}_{n,\text{mon}}^{p}(\Phi) \gtrsim \sqrt{\log(n+1)}. \qquad (4.4)
$$

We remarked in [7] that from some results of Nikishin-Ulyanov [15] and Olevskii [16] it
follows that $\mathcal{A}_{n,\text{mon}}^2(\Phi) \gtrsim \sqrt{\log(n+1)}$ for any complete orthonormal system $\Phi$. Thus
(4.4) holds in the case $p=2$. So there is a function $f \in L^2$ and a sequence of integer
sets $G_1 \subset G_2 \subset \dots \subset G_n$ such that

$$
\left\| \max_{1 \le m \le n} \left| \sum_{j \in G_m} \langle f, \phi_j \rangle \phi_j \right| \right\|_2 \ge c \sqrt{\log(n+1)} \cdot \|f\|_2,
$$

Thus for the operator

$$
U(f) = \max_{1 \le m \le n} \left| \sum_{j \in G_m} \langle f, \phi_j \rangle \phi_j \right|
$$

we have ||U||<sub>2→2</sub> ≥ c√log(n + 1). On the other, according to Theorem 1.1, we have
||U||<sub>p→p</sub> ≤ c<sub>p</sub>√log(n + 1) for all 1 < p < ∞. Combining these two estimates with the
"full version" of Marcinkiewicz interpolation theorem (see [23], Theorem 4.6) one can
easily obtain ||U||<sub>p→p</sub> ≥ √log(n + 1) that completes the proof of Corollary 1.1. □
*Proof of Corollary 1.2.* The upper bounds for both systems follow from Theorem 1.1. The lower bound $A_{n,\text{sgn}}^p(\Phi) \gtrsim \sqrt{\log(n+1)}$ for the Franklin system if $p=2$ follows from the remark (1) given in the introduction. For the Haar system the case of $p=2$ of the same bound is known from [15]. To show the general case $1 < p < \infty$ we again use the argument of the Marcinkiewicz interpolation theorem. □

## REFERENCES

[1] S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, *Some weighted norm inequalities concerning the Schrödinger operators*, Comment. Math. Helv. **60** (1985), no. 2, 217–246, DOI 10.1007/BF02567411. MR**800004**

[2] G. G. Gevorkyan, *On Weyl factors for the unconditional convergence of series in the Franklin system*, Mat. Zametki **41** (1987), no. 6, 789–797, 889 (Russian). MR**904246**

[3] Loukas Grafakos, Petr Honzík, and Andreas Seeger, *On maximal functions for Mikhlin-Hörmander multipliers*, Adv. Math. **204** (2006), no. 2, 363–378, DOI 10.1016/j.aim.2005.05.010. MR**2249617**

[4] Gustaf Gripenberg, *Wavelet bases in $L^p(\mathbb{R})$, Studia Math. **106** (1993), no. 2, 175–187, DOI 10.4064/sm-106-2-175-187. MR**1240312**

[5] Eugenio Hernández and Guido Weiss, *A first course on wavelets*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR**1408902**

[6] S. Kačmaž and G. Šteingauz, *Teoriya ortogonalnykh ryadov*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR**0094635**

[7] Grigori A. Karagulyan, *On systems of non-overlapping Haar polynomials*, Ark. Math. **58** (2020), no. 1, 121–131, DOI 10.4310/arkiv.2020.v58.n1.a8.

[8] __________, *On Weyl multipliers of the rearranged trigonometric system*, Sbornik Mathematics **211** (2020), no. 12, 1704–1736, DOI 10.1070/SM9422.

[9] __________, *A sharp estimate for the majorant norm of a rearranged trigonometric system*, Russian Math. Surveys **75** (2020), no. 3, 569–571, DOI 10.1070/RM9946.

[10] Grigori A. Karagulyan and Michael T. Lacey, *On logarithmic bounds of maximal sparse operators*, Math. Z. **294** (2020), no. 3-4, 1271–1281, DOI 10.1007/s00209-019-02314-9. MR**4074040**

[11] B. S. Kashin and A. A. Saakyan, *Orthogonal series*, Translations of Mathematical Monographs, vol. 75 American Mathematical Society, Providence, RI, 1989. Translated from the Russian by Ralph P. Boas; Translation edited by Ben Silver. MR**1007141**

[12] D. E. Menshov, *Sur les series de fonctions orthogonales I*, Fund. Math. **4** (1923), 82–105 (Russian).

[13] Yves Meyer, *Ondelettes et opérateurs. II*, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators]. MR**1085488**

[14] Paul F. X. Müller, *Isomorphisms between $H^1$ spaces*, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 66, Birkhäuser Verlag, Basel, 2005. MR**2157745**

[15] E. M. Nikišin and P. L. Ul'janov, *On absolute and unconditional convergence*, Uspehi Mat. Nauk **22** (1967), no. 3 (135), 240–242 (Russian). MR**0212488**

[16] A. M. Olevskii, *Divergent Fourier series*, Izv. Akad. Nauk SSSR Ser. Mat. **27** (1963), 343–366 (Russian). MR**0147834**

[17] S. N. Poleščuk, *On the unconditional convergence of orthogonal series*, Anal. Math. **7** (1981), no. 4, 265–275, DOI 10.1007/BF01908218 (English, with Russian summary). MR**648491**
[18] Hans Rademacher, *Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen*, Math. Ann. **87** (1922), no. 1-2, 112-138, DOI 10.1007/BF01458040 (German). MR1512104

[19] Elias M. Stein, *Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192

[20] P. L. Ul'janov, *Weyl multipliers for the unconditional convergence of orthogonal series*, Dokl. Akad. Nauk SSSR **235** (1977), no. 5, 1038-1041 (Russian). MR0450886

[21] P. Wojtaszczyk, *Wavelets as unconditional bases in $L_p(\mathbb{R})$, J. Fourier Anal. Appl. **5** (1999), no. 1, 73-85, DOI 10.1007/BF01274190. MR1682254

[22] B. Wolnik, *The wavelet type systems*, Banach center publications **72** (2006), 397-406.

[23] Antoni Zygmund, *Trigonometric series. 2nd ed. Vol. 2*, Cambridge University Press, New York, 1959. MR0107776

INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES, UL. ABRAHAMA 18, 80-825 SOPOT, POLAND

Email address: anna.kamont@impan.pl

FACULTY OF MATHEMATICS AND MECHANICS, YEREVAN STATE UNIVERSITY, ALEX MANOOGIAN, 1, 0025, YEREVAN, ARMENIA

Email address: g.karagulyan@ysu.am