On a weighted inequality of Hardy type in spaces Lp(⋅)

The boundedness of Hardy type operator is studied in weighted variable exponent Lebesgue spaces Lp(⋅). The necessary and sufficient criterion established on the weight functions v(x), ω(x) and exponents p(x), q(x) for the Hardy operator to be bounded from Lp(⋅)(ω) to Lq(⋅)(v). The exponents satisfy a modified logarithmic condition near zero and at infinity: ∃δ>0, ∃f∞, ∃f(0)∈R ; ∃N>1 supx∈Rn∖B(0,N)|f(x)−f∞|lnW(x)<∞, where .