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

The boundedness of Hardy type operator Hf (x) = integral((t is an element of R)n(:) (vertical bar t vertical bar <=vertical bar x vertical bar)) f(t)dt is studied in weighted variable exponent Lebesgue spaces L-p(.). The necessary and sufficient criterion established on the weight functions v(x), omega(x) and exponents p(x). q(x) for the Hardy operator to be bounded from L-p(.)(omega) to L-q(.)(v). The exponents satisfy a modified logarithmic condition near zero and at infinity: there exists delta > 0, there exists integral(infinity), there exists f(0) is an element of R sup(x is an element of B(o,delta)) vertical bar f(x) - f (0)vertical bar In I/W(x) < infinity: there exists N > 1 sup(x is an element of R)n(\B(0,N)) vertical bar f(x) - f(infinity)vertical bar In W(x) < infinity, where W(x) = integral({t is an element of R)n(:) (vertical bar t vertical bar <=vertical bar x vertical bar})omega(-1/(p(t)-1))(t)dt. (C) 2008 Elsevier Inc. All rights reserved.