1 - 10 of 15 articles
We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿 𝑝(·), λ (·) (Ω) over a bounded open set Ω ⊂ ℝ 𝑛 and a Sobolev type 𝐿 𝑝(·), λ (·) → 𝐿 𝑞(·), λ (·) -theorem for potential operators 𝐼 α (·) , also of variable order. In the case of constant α , the...
The essential norm of the Hilbert transform acting in weighted Lebesgue spaces with variable exponent is estimated from below.
The concept of 𝐴-level sets of real functions 𝑢(𝑥, 𝑦) (i.e., the solutions of 𝑢(𝑥, 𝑦) = 𝐴 = const ) in a given domain admits numerous interpretations in applied sciences: level sets are potential lines, streamlines in hydrodynamics, meteorology and electromagnetics, isobars in gas-dynamics,...
An analytic proof of Wiener's theorem on factorization of positive definite matrix-functions is proposed.
Using the generalized shift operator (GSO) generated by the Gegenbauer differential operator we introduce the notion of a Lebesgue–Gegenbauer (L-G)-point of a summable function 𝑓 on the interval 1,∞) and prove that almost all points of this interval are (L-G)-points of 𝑓. Furthermore, we give an...
We use the recently introduced concept of growth envelopes to characterize weighted spaces of type , where 𝑤 belongs to some Muckenhoupt 𝐴 𝑝 class, and discuss some applications.
The aim of this paper is to characterize variable 𝐿 𝑝 spaces 𝐿 𝑝(·) ( ) using wavelets with proper smoothness and decay. We obtain conditions for the wavelet characterizations of 𝐿 𝑝(·) ( ) with respect to the norm estimates and modular inequalities.
Some Hardy type inequalities for decreasing functions are characterized by one condition (Sinnamon), while others are described by two independent conditions (Sawyer). In this paper we make a new approach to deriving such results and prove a theorem, which covers both the Sinnamon result and the...
We study the Hardy–Littlewood maximal operator 𝑀 on 𝐿 𝑝(·) (Ω), where Ω ∈ is an open bounded domain with a Lipschitz boundary. Under the assumption that the exponent 𝑝 satisfies 1 < inf 𝑝(𝑥) ≤ sup 𝑝(𝑥) < ∞ and 𝑝 ∈ 𝑉𝑀𝑂 1/|log| (Ω), we prove that 𝑀 is bounded on 𝐿 𝑝(·) (Ω).
We consider some properties of sets and functions which are measurable (or nonmeasurable) with respect to certain classes of measures. In this context, the notion of an absolutely nonmeasurable set (function) is examined. Sierpiński-Zygmund type functions are constructed having additional...
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