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

Abstract

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

Keywords

• Inequalities
• Fourier series
• Fourier coefficients
• Unbounded orthogonal systems
• Lorentz–Zygmund spaces