Nikol'skii's Inequality for Different Metrics and Properties of the Sequence of Norms of the Fourier Sums of a Function in the Lorentz Space

Let $(X,Y)$ be a pair of normed spaces such that $X\subset Y\subset L_1[0,1]^n$ and $\{e_k\}_k$ be an expanding sequence of finite sets in $\mathbb Z^n$ with respect to a scalar or vector parameter $k$, $k\in \mathbb N$ or $k\in \mathbb N^n$. The properties of the sequence of norms $\{\|S_{e_k}(f)\|_X\}_k$ of the Fourier sums of a fixed function $f\in Y$ are studied. As the spaces $X$ and $Y$, the Lebesgue spaces $L_p[0,1]$, the Lorentz spaces $L_{p,q}[0,1]$, $L_{p,q}[0,1]^n$, and the anisotropic Lorentz spaces $L_{\mathbf p,\mathbf q^\star }[0,1]^n$ are considered. In the one-dimensional case, the sequence $\{e_k\}_k$ consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in $\mathbb Z^n$. For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces $L_{p,q}[0,1]^n$ and $L_{\mathbf p,\mathbf q^\star }[0,1]^n$ are obtained.