On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces

We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space L (p(.))(R-n ). We prove equivalent conditions for L-p(.) -> L-q(.) boundness of the Hardy operator in the case of so called mixed exponents: q(0) >= p(0), q(infinity) < p (infinity) or q(0) < p(0), q(infinity) >= p(infinity). We show that a necessary and sufficient condition for such an inequality to hold coincides with conditions for the validity of two weight Hardy inequalities with constant exponents, provided that the exponents are regular at zero and at infinity.