Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

where 1 ≤ 푞 < 푝 ≤ ∞ and ‖∗‖ is the usual norm on the Lebesgue spaces . For 푝 = ∞, this estimate has been proved by Jackson [3]. e proofs in [1–3] are based on Bernstein’s inequality. Inequalities between different (quasi-)norms of the same function are known as Nikol’skii–type (or Jackson–Nikol’skii– type) inequalities. ey play a crucial role in many areas of mathematics, e.g., theory of approximation, theory of functions of several variables, and functional analysis (embedding theorems for Besov spaces). Nessel and Wilmes [4] extended inequality (1) for 0 < 푞 < 푝 ≤ ∞. ey also observed that in inequality (1) one may in fact take into account the spectrum of the function involved for 푞 ≤ 2 or certain gaps in the spectrum for 푞 > 2. Sherstneva [5] extended inequality (1) in the Lorentz spaces 푝,푏 and showed that they are exact relative to the order . Moreover, she investigated the limiting case when Lorentz spaces on both sides have the same value of the main parameter. She proved that if 0 < 푝 < ∞ and 0 < 푏 < 푐 ≤ ∞, then Some Nikol’skii–type inequalities for the Lorentz–Zygmund spaces are considered in [6–8] and for the generalized Lorentz space in [9]. In other investigations, different sets of functions, domains, and measures were explored. For further information about these results and applications, we refer to [1–19] and references therein. In [19], the results of [5] were improved in two directions. First, the functions of the form ∑푛푘=1푐푘휑푘 were considered, where { } is an orthonormal system in 퐿2(M) uniformly bounded in 퐿∞(M) (with 휇(M) < ∞). No assumptions about smoothness of were made. Secondly, the inequalities (1) and (2) were extended to the Lorentz– Zygmund spaces. However, in [19], only the case 0 < 푞 ≤ 2 is obtained. The principal aim of this paper is to extend the results of [19] for the case 2 ≤ 푞 < ∞ for the trigonometric polynomials. The technique we apply relies on the observation that the power of a trigonometric polynomial is also a trigonometric polynomial [4, 10, 17, 18]. This paper is organized as follows. Section 2 contains necessary notations and definitions. Main results of this contribution are Theorems 4, 6, 8, and 9. They are formulated and proved in Section 3. Note that Theorems 8, and 9 deal with the limiting case. In Section 4, we reformulate Theorems 4, 6, 8, and 9 for trigonometric polynomials of degree at most . In Section 5, we give some applications to embeddings between approximation spaces in Lorentz–Zygmund spaces and between Besov spaces. (1) 儩儩儩儩푇푛儩儩儩儩푝≺ 푛((1/푞)−(1/푝))儩儩儩儩푇푛儩儩儩儩푞,

Inequalities between different (quasi-)norms of the same function are known as Nikol'skii-type (or Jackson-Nikol'skiitype) inequalities. ey play a crucial role in many areas of mathematics, e.g., theory of approximation, theory of functions of several variables, and functional analysis (embedding theorems for Besov spaces).
The principal aim of this paper is to extend the results of [19] for the case 2 ≤ 푞 < ∞ for the trigonometric polynomials. The technique we apply relies on the observation that the power of a trigonometric polynomial is also a trigonometric polynomial [4,10,17,18]. This paper is organized as follows. Section 2 contains necessary notations and definitions. Main results of this contribution are Theorems 4, 6, 8, and 9. They are formulated and proved in Section 3. Note that Theorems 8, and 9 deal with the limiting case. In Section 4, we reformulate Theorems 4, 6, 8, and 9 for trigonometric polynomials of degree at most . In Section 5, we give some applications to embeddings between approximation spaces in Lorentz-Zygmund spaces and between Besov spaces.

Preliminaries
We write 푋 ⊂ 푌 for two quasi-normed spaces and to indicate that is continuously embedded in . e notation 푋 ≅ 푌 means that 푋 ⊂ 푌 and 푌 ⊂ 푋. If and are positive functions, we write 푓 ≺ 푔 if 푓 ≤ 퐶푔, where the constant is independent on all significant quantities. We put 푓 ≈ 푔 if 푓 ≺ 푔 and 푔 ≺ 푓.

Main Results
Let Λ ⊂ be a finite set of lattice points. We denote the number of elements of the set Λ by #Λ. Everywhere below Our goal is to obtain Nikol'skii-type inequalities of the form where the constant does not depend on Λ. Or, writing it shorter Obviously, only the situations 퐿 푞,푐;훽 ⊂ /퐿 푝,푏;훼 are of interest, otherwise 퐺 = 1.
Both eorems 4 and 6 deal with the case < . e next two theorems examine the limiting case = .

Corollaries for Trigonometric Polynomials of Degree at Most
Let T be the set of all trigonometric polynomials of degree at most 푛 (푛 = 1, 2, . . .), i. e.

Applications
Let T be the set of all trigonometric polynomials of degree at most described above. e sequence T (푛 = 1, 2, . . .) allows construction of an approximation family in all Lorentz-Zygmund spaces, which produces approximation spaces. In Section 5.1 we start with some necessary definitions and auxiliary results dealing with approximation spaces. In Section 5.2 we present some corollaries of the statements from Section 4 dealing with embeddings of the approximation spaces 퐿 푝,푏;훼 (휎,훾) 푢 into Lorentz-Zygmund spaces and between these approximation spaces. In Sections 5.3 and 5.4 these corollaries will be reformulated in terms of Besov spaces.
en, for 0<u≤∞ and 훾 > −1/푢, we have based on and has classical smoothness and additional logarithmic smoothness with exponent . It is formed by all those ∈ such that with an obvious modification when 푢 = ∞. Here 휔 푓, 푡 (푘 > 휎, 푘 ∈ ) is the modulus of smoothness of order with respect to the quasi-norm on . It makes sense to consider the spaces 0,훾 푝,푢 with zero classical smoothness only for 훾 ≥ −1/푢. All we need for our application is the characterization of the Besov spaces by approximation. An important result in approximation theory states that (72) include a detailed proof below. " e author could also not find a precise reference and has included, therefore, the formulation and the proof of Lemma 2. In addition, the second referee have pointed out of the paper [26] to me. e author would also like to thank Doctor Dimitri Bulatov for their help during the preparation of the manuscript.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there is no conflict of interest regarding the publication of this paper.