A Necessary and Sufficient Condition for Hardy's Operator in the Variable Lebesgue Space

The variable exponent Hardy inequality parallel to x(beta(x)-1) integral(x)(0) f(t)dt parallel to(LP(.)(0,l)) <= C parallel to x(beta(x)) f parallel to(LP(.)(0,l)), f >= 0 is proved assuming that the exponents p : (0,l) -> (1, infinity), beta : (0, l) -> R not rapidly oscilate near origin and 1/p'(0) - beta > 0. The main result is a necessary and sufficient condition on p, beta generalizing known results on this inequality.