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. © 2019 Mofid University - Center for Human Rights Studies. All rights reserved.