Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, commutators integrals in grand variable exponent Lebesgue spaces are derived. The defined on quasi-metric measure doubling condition (spaces homogeneous type). proof the result regarding Hardy–Littlewood operator is based appropriate sharp weighted norm estimates we...

The �-dimensional Lebesgue space is the measurable space (E���(E�))— where E = [0 � 1) or E = R—endowed with the Lebesgue measure, and the “calculus of functions” on Lebesgue space is just “real and harmonic analysis.” The �-dimensional Gauss space is the same measure space (R���(R�)) as in the previous paragraph, but now we endow that space with the Gauss measure P� in place of the Lebesgue me...

In this paper the approximation of functions by linear means Fourier series in weighted variable exponent Lebesgue spaces was studied. This result applied to Faber Smirnov classes with defined on simply connected domain complex plane.

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...

In generalized Lebesgue spaces $L^{p(.)}$ with variable exponent $p(.)$ defined on the real axis, we obtain several inequalities of approximation by integral functions finite degree. Approximation properties Bernstein singular integrals in these are obtained. Estimates simultaneous degree proved.

We study, via -convergence, the asymptotic behavior of several classes of power–law functionals acting on fields belonging to variable exponent Lebesgue spaces and which are subject to constant rank differential constraints. Applications of the -convergence results to the derivation and analysis of several models related to polycrystal plasticity arising as limiting cases of more flexible power...