Order estimates for Lebesgue constants of Fourier sums in Orlicz spaces

Abstract

We consider the problem of order estimates for partial sums of trigonometric Fourier series as operators from Orlicz spaces Lϕ2π to the space of 2π-periodic continuous functions C2π. It is established that an arbitrary function ϕ generating an Orlicz class satisfies the estimate

Sn(f)

C2π ≤ Cϕ−1(n) ln(n + 1)

f

Lϕ 2π, (∗) where f ∈ Lϕ2π, n ∈ N, Sn(f) is the nth partial sum of the trigonometric Fourier series of f, and the constant C > 0 is independent of f and n. In addition, it is shown that if the function ϕ satisfies the ∆2-condition, then the estimate can be improved. More exactly,

Sn(f)

C2π ≤ Cϕ−1(n)

f

Lϕ 2π, f ∈ Lϕ2π, n ∈ N, C = C(ϕ). (∗∗) Counterexamples are constructed, which show that if ϕ satisfies the ∆2-condition, then estimate (∗∗) is unimprovable in order on the space Lϕ2π and, if ϕ satisfies the ∆2-condition, then estimate (∗) is unimprovable in order on the space Lϕ2π © 2021 The authors.