We investigate the problem of approximating function f in power-type weighted variable exponent Sobolev space Wr,p(.) ?(.) (0,1), (r = 1, 2, ...), by Hardy averaging operator A (f) (x) 1/x ?x0 f(t)dt. If lies ?(.)(0, 1), it is shown that A||(f)?f|| p(.),?(.)?rp(.) ? C ||f(r) p(.),?(.) , where a positive constant. Moreover, we consider boundedness grand Lebesgue spaces Lp(.),? ?(.)(0,1). The suf...

Singular integral operators of two types are considered on a piecewise smooth contour in weighted Lebesgue spaces with generalized Cauchy kernels, related to the parametrix elliptic systems first order plane. The type linear over field complex numbers and represented as usual combination singular operator multiplication by continuous matrix functions. second act space real vector functions and,...

In [3] we introduced a “monotone-type” (that is, Choquet-type) integral for realvalued functions, with respect to finitely additive positive set functions, with values in a Dedekind complete Riesz space. A “Lebesgue-type” integral for such kind of functions was investigated in [7]. In [4] we gave some comparison results for these types of integrals. In [10], a Choquet-type integral for real-val...

The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann-Liouville fractional integration operator has been obtained.

Let D be a domain in C and φ a holomorphic automorphism of D. Let C be the measure class of the Lebesgue measure in D, i.e., the set of all positive regular Borel measures on D whose null sets coincide with the Lebesgue null sets. Let φ∗ be the automorphism of C given by (φ∗μ)(B) = μ(φ−1(B)) (μ ∈ C , B ∈ B(D)), where B(D) denotes the Borel σ-algebra of D. Adopting the terminology introduced in ...

We find conditions for the embedding of grand Morrey-type spaces into so-called local Lebesgue spaces.

Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measur...

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group Γ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum Λ ⊆ Γ , the norm of the Hilbert transform and th...