Some new Fourier inequalities for unbounded orthogonal systems in Lorentz–Zygmund spaces

In this paper we prove some essential complements of the paper (J. Inequal. Appl. 2019:171, 2019) on the same theme. We prove some new Fourier inequalities in the case of the Lorentz–Zygmund function spaces Lq,r(logL)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{q,r}(\log L)^{\alpha }$\end{document} involved and in the case with an unbounded orthonormal system. More exactly, in this paper we prove and discuss some new Fourier inequalities of this type for the limit case L2,r(logL)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{2,r}(\log L)^{\alpha }$\end{document}, which could not be proved with the techniques used in the paper (J. Inequal. Appl. 2019:171, 2019).

In particular, Flett [5] generalized this to the case of Lorentz spaces and Maslov [13] derived generalizations of both Theorem 1.1 and Theorem 1.2 to the case of Orlicz spaces.
Some inequalities related to the summability of the Fourier coefficients in bounded orthonormal systems with functions from some Lorentz spaces were investigated e.g. by Stein [23], Bochkarev [3], Kopezhanova and Persson [9] and Kopezhanova [8].
Moreover, as a further generalization of a result of Kolyada [7] Kirillov proved an essential generalization of Theorem 1 [6].
holds, where μ ν n are defined by (2) and a k (f ) denote the Fourier coefficients of f with respect to an orthogonal system satisfying (1).
The methods of proofs of Theorems 1.3 and 1.4 presented in [2] are not sufficient to cover the case q = 2 in both cases. In this paper we fill in this gap by proving complements of Theorem 1 in [6] (see Theorem 2.1) and Theorem 1.2 (see Theorem 3.1) for this case. In Sect. 4 we include some concluding remarks with comparisons of other recent research of this type (see [8,15] and [17]) and suggestions of further possibilities for development of this area.

A complement of Theorem 1.3. The case q = 2
Our main result in this section reads as follows.
For the proof of this theorem we need the following well-known results of Kolyada [7].
By now taking the limit l → ∞ in (14) we get Finally, in this inequality we put m = 1 and use the norm property to conclude that The proof is complete.

A complement of Theorem 1.4. The case q = 2
Our main result in this section reads as follows.