Approximating Functions in the Power-Type 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-alpha(.)(r,p(.)) (0, 1), (r = 1, 2,...), by the Hardy averaging operator A (f) (x) = 1/x integral(x)(0) f (t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space W-alpha(.)(r,p(.)) (0, 1), it is shown that parallel to A(f) - f parallel to(p(.),alpha(.)-rp(.)) <= C parallel to f((r))parallel to(p(.),alpha(.)') 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 Lp(alpha(.))(p(.),theta) (0, 1). The sufficient criterion established on the power-type weight function alpha(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.