Operators of harmonic analysis in weighted spaces with non-standard growth

Abstract

Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vectorvalued analogues.

Approximation of functions by de la Vallée-Poussin sums in weighted Orlicz spaces

“…The problems of approximation theory in weighted and nonweighted Lebesgue spaces, weighted and nonweighted Orlicz spaces have been investigated by several authors (see, e.g., [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][30][31][32]36,37,45,46]). …”

Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

“…In particular, some direct and inverse theorems in weighted and nonweighted Lebesgue spaces with variable exponent have been obtained in [9,10,16,21,[28][29][30]34].…”

Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces

“…Note that in the cited papers the integral inequalities were studied as r → 0. Such inequalities are also of interest when they allow to impose different conditions as r → 0 and r → ∞; such a case was dealt with in [35], [23].…”