On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space

G. A. Akishev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 9-20

Résumé

We consider the generalized Lorentz space $L_{\psi,\tau}(\mathbb{T}^m)$ defined by some continuous concave function $\psi$ such that $\psi (0)=0$. For two spaces $L_{\psi_1,\tau_1}(\mathbb{T}^m)$ and $L_{\psi_2,\tau_2}(\mathbb{T}^{m})$ such that $\alpha_{\psi_{1}}={\underline\lim}_{t\rightarrow 0}\psi_{1}(2t)/\psi_{1}(t) = \beta_{\psi_{2}} = \overline{\lim}_{t\rightarrow 0}\psi_{2}(2t)/\psi_{2}(t)$, we prove an order-exact inequality of different metrics for multiple trigonometric polynomials. We also prove an auxiliary statement for functions of one variable with monotonically decreasing Fourier coefficients in a trigonometric system. In this statement we establish a two-sided estimate for the norm of the function $f\in L_{\psi, \tau}(\mathbb{T})$ in terms of the series composed of the Fourier coefficients of this function.

Détail

Keywords: generalized Lorentz space, Jackson–Nikol'skii inequality, trigonometric polynomial.