Volume 15, Issue 2

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 limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

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, isotherms in thermodynamics, etc. In fact, the level sets of 𝑢 considered for all values 𝐴 make the “map” of this function and their interpretations in different sciences make the “maps” of the corresponding processes. In this paper we study the geometry of these maps for broad classes of functions and arbitrary values 𝐴. In particular, we study how much twisted or, speaking in general, how turbulent these maps are. The concepts and results admit some immediate interpretations and can be stated in terms of flow rotation and turbulence. The study gives a new, in fact a geometric description of these applied phenomena.

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 exact (by order) estimation of the approximation of locally summable functions by singular integrals generated by GSO (Gegenbauer singular integrals).

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 Sawyer result for the case where one weight is increasing. In all cases we point out that the characterizing condition is not unique and can even be chosen among some (infinite) scales of conditions.

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 properties closely connected with the above-mentioned notion. Also, some small subsets of uncountable commutative groups are discussed whose algebraic sum turns out to be an absolutely nonmeasurable set.

The paper investigates the questions of continuity and boundedness of a fractional integral on the curves of finite length in Lebesgue spaces depending on the geometrical properties of an integration curve.

The main aim of the paper is to study relations between the location of the essential spectrum and exponential decay estimates of the eigenfunctions of systems of elliptic partial differential equations. To solve this problem, we apply the limit operators method. Applications are given to the problem of exponential decay behavior of eigenfunctions of Schrödinger and Dirac operators.

We study multi-commutators on the Morrey spaces generated by BMO functions and singular integral operators or by BMO functions and fractional integral operators. We place ourselves in the setting of coming with a Radon measure μ which satisfies a certain growth condition. The Morrey-boundedness of commutators is established by M. Yan and D. Yang. However, the corresponding assertion of Morrey-compactness is still missing. The aim of this paper is to prove that the multi-commutators are compact if one of the BMO functions can be approximated with compactly supported smooth functions.

Let (𝑋,𝑑, μ ) be a normal space of homogeneous type, 𝑋 + be the upper half-space equipped with a Carleson measure β , and let Φ be an N-function and φ a suitable function satisfying the doubling property. We prove that the generalized Hardy–Littlewood maximal operator 𝑀 is bounded from the Orlicz–Campanato space 𝐿 Φ, φ (𝑋, μ ) to 𝐿 Φ, φ (𝑋 + , β ).

The paper deals with unconditional intrinsic wavelet bases in Lorentz spaces 𝐿 𝑝,𝑞 (Ω) and Zygmund spaces 𝐿 𝑝 (log 𝐿) 𝑎 (Ω) where Ω is an arbitrary domain in (with |Ω| < ∞ in the case of Zygmund spaces).