Some new Fourier inequalities for unbounded orthogonal systems in Lorentz\u2013Zygmund spaces

G Akishev,D Lukkassen,L E Persson

doi:10.1186/s13660-020-02344-6

Abstract

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 L_{q,r}(log L)^{alpha } 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 L_{2,r}(log L)^{alpha }, which could not be proved with the techniques used in the paper (J. Inequal. Appl. 2019:171, 2019).

Highlights

Let q ∈ (1, +∞), r ∈ (0, +∞) and α ∈ R
If α = 0, the Lorentz–Zygmund space coincides with the Lorentz space Lq,r(log L)α =
4 Concluding remarks We say that a function f on (0, 1) or (0, ∞) is quasi-increasing if, for all x ≤ y and some C > 0, f (x) ≤ Cf (y) (f (y) ≤ Cf (x))

Summary

Introduction

Let q ∈ (1, +∞), r ∈ (0, +∞) and α ∈ R. If α = 0, the Lorentz–Zygmund space coincides with the Lorentz space Lq,r(log L)α =. Theorem 1.1 Assume that the orthogonal system {φn} satisfies the condition (1) and 2 ≤ p < s. There are several generalizations of Theorems 1.1 and 1.2 for different function spaces and systems 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]. < +∞, where ρn and μn are defined by (2), the series n=1 with respect to an orthogonal system {φn}∞ n=1, which satisfies the condition (1), converges to some function f ∈ Lq,r(log L)α and f q,r,α ≤ CΩq,r,α.

If f

Choose the number r such that and note that s