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

Some functional inequalitiesý ýin variable exponent Lebesgue spaces are presentedý. ýThe bi-weighted modular inequality with variable exponent p(.) for the Hardy operator restricted to noný- ýincreasing function which isý
ýýinti0nfty(frac1xintx0f(t)dt)p(x)v(x)dxleqýýCinti0nftyf(x)p(x)u(x)dxý,ý
ýýis studiedý. ýWe show that the exponent p(.) for which these modular inequalities hold must have constant oscillationý. ýAlso we study the boundedness of integral operator Tf(x)=intK(x,y)f(x)dy on Lp(.) when the variable exponent p(.) satisfies someý ýuniform continuity condition that is named beta -controller condition and so multiple interesting results which can beý ýseen as a generalization of the same classical results in the constant exponent caseý, ýderivedý.