tailieunhanh - Mixed modulus of continuity in the Lebesgue spaces with Muckenhoupt weights and their properties

Main properties of the mixed modulus of continuity in the Lebesgue spaces with Muckenhoupt weights are investigated. We use the mixed modulus of continuity to obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. | Turk J Math (2016) 40: 1169 – 1192 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Mixed modulus of continuity in the Lebesgue spaces with Muckenhoupt weights and their properties ¨ ∗ Ramazan AKGUN Department of Mathematics, Faculty of Arts and Sciences, Balıkesir University, C ¸ a˘ gı¸s Yerle¸skesi, Balıkesir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: Main properties of the mixed modulus of continuity in the Lebesgue spaces with Muckenhoupt weights are investigated. We use the mixed modulus of continuity to obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. We prove an equivalence between the mixed modulus of continuity and K -functional and realization functional. Key words: Direct theorem, inverse theorem, Muckenhoupt weights, modulus of continuity 1. Introduction and the main results In this paper we consider the properties of the mixed modulus of continuity Ω (f, δ, ξ)p,ω in the Lebesgue spaces ( ) ( ) ( ) Lpω T2 := Lp T2 , ω (x, y) with weights ω (x, y) belonging to the Muckenhoupt class Ap T2 , J where J is the set of rectangles in T2 := T × T , T := [0, 2π) with sides parallel to coordinate axes. In the particular case ω (x, y) ≡ 1 on T2 , classical mixed modulus of continuity was used to prove some results on trigonometric ( ) approximation by an angle for functions in the classical Lebesgue spaces Lp Td . For example, Potapov obtained a direct theorem [9, 12] and inverse estimate [13] on trigonometric approximation by an angle for ( ) functions in spaces Lp Td . Hardy–Littlewood, Marcinkiewicz, Littlewood–Paley, and embedding theorems were proved in [10, 11]. Transformed Fourier series and mixed modulus of continuity were investigated by Potapov et al. in [15] and [17]. Embeddings of the Besov–Nikolskii and Weyl–Nikolskii classes were .

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