Approximating Functions in the Power-Type Weighted Variable Exponent Sobolev Space by the Hardy Averaging Operator

We investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space (Formula presented), by the Hardy averaging operator (Formula presented). If α(.) x 0 the function f lies in the power-type weighted variable exponent Sobolev space (Formula presented) (0, 1), it is shown that (Formula presented), 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 (Formula presented) (0, 1). The sufficient α(.) criterion established on the power-type weight function α(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.