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

Abstract

We consider the generalized Lorentz space Lψ,τ (Tm) defined by some continuous concave function ψ such that ψ(0) = 0. For two spaces Lψ1,τ1 (Tm) and Lψ2,τ2 (Tm) such that αψ1 = limt→0ψ1(2t)/ψ1(t) = βψ2 = limt→0ψ2(2t)/ψ2(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 ∈ Lψ,τ (T) in terms of the series composed of the Fourier coefficients of this function.