Аннотация:Let $\Lambda_\beta,$ $\beta>0,$ denote the Lorentz space equipped with the (quasi) norm
$$
\|f\|_{\Lambda_\beta}:=\left(\int_0^1\left(f^*(t)t\lambda\left(\frac1t\right)\right)^\beta\frac{dt}{t}\right)^{\frac1\beta}
$$
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 general orthonormal regular systems. Under certain circumstances even two sided estimates are obtained.