Аннотация
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 $L_{q,r}(\log L)^{\alpha }$ Lq,r(logL)α 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 $L_{2,r}(\log L)^{\alpha }$ L2,r(logL)α , which could not be proved with the techniques used in the paper (J. Inequal. Appl. 2019:171, 2019).