## Some new Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces.(English)Zbl 07460852

Summary: In this paper we prove some essential complements of the paper [the first author et al., ibid. 2019, Paper No. 171, 18 p. (2019; Zbl 07459199)] 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 [loc. cit.].

### MSC:

 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42B05 Fourier series and coefficients in several variables 26D15 Inequalities for sums, series and integrals 26D20 Other analytical inequalities 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Zbl 07459199
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### References:

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