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 integral(infinity)(0) (1-x integral(x)(0) f(t)dt)(p(x)) v(x)dx <= C integral(infinity)(0) f (x)(p(x))u(x)dx, 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 T f (x) = f K (x, y) f (x)dy on L-p(.) when the variable exponent p(.) satisfies some uniform continuity condition that is named beta-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.