Approximating functions in the weighted variable exponent Sobolev space by the Hardy averaging operator

Abstract

We investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space W_{α(.)}^{r,p(.)}(0,1), (r=1,2,...,) by the Hardy averaging operator A(f)(x)=(1/x)∫₀^{x}f(t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space W_{α(.)}^{r,p(.)}(0,1), it is shown that ‖A(f)-f‖_{p(.),α(.)-rp(.)}≤C‖f^{(r)}‖_{p(.),α(.)}, where C is a positive constant . Moreover, we consider the problem of boundedness of Hardy averaging operator A in power-type weighted variable exponent grand Lebesgue spaces L_{α(.)}^{p(.),θ}(0,1). The sufficient criterion established on the power-type weight function α(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.