Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition

Some functional inequalities in variable exponent lebesgue spaces are presented. the bi-weighted modular inequality with variable exponent p(.) for the hardy operator restricted to non-increasing function which is [mathematical formula] is studied. we show that the exponent p(.) for which these modular inequalities hold must have constant oscillation. also we study the boundedness of integral operator tf(x) = ʃ k(x; y)f(x)dy on lp^(.) when the variable exponent p(.) satises some uniform continuity condition that is named β-controller condition and so multiple interesting results which can be seen as a generalization of the same classical results in the constant exponent case, derived.