  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 vector-valued analogues.
