An inequality of different metrics in the generalized Lorentz space

The main goal of the paper is to prove the Jackson-Nikol'skii inequality for multiple trigonometric polynomials in the generalized Lorentz space $L_{\psi,\theta}(\mathbb{T}^{m})$. In the first section we give definitions of a symmetric space of functions, a fundamental function, and the Boyd index of a space. In particular, we define the generalized Lorentz and Lorentz-Zygmund spaces. In addition, definitions of a weakly varying function and of the Lorentz-Karamata space are given. In the second section we prove an analog of the inequality of different metrics for multiple trigonometric polynomials in generalized Lorentz spaces $L_{\psi,\theta}(\mathbb{T}^{m})$ with identical Boyd indices but different fundamental functions. In the Lorentz-Karamata space, the order-exact Jackson-Nikol'skii inequality for multiple trigonometric polynomials is obtained.