On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces

We denote by $\Lambda_\beta(\lambda),$ $\beta>0,$ the Lorentz space equipped with the (quasi) norm
$$ \|f\|_{\Lambda_\beta(\lambda)}:=\left(\int_0^1\left(f^*(t)t\lambda\left(\frac1t\right)\right)^\beta\frac{dt}{t}\right)^{\frac1\beta} $$
for a function $f$ on [0,1] and with $\lambda$ positive and equipped with some additional growth properties. Some estimates of this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system.