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 is‎‎‎‎int_۰^infty (frac{۱}{x}int_۰^x f(t)dt)^{p(x)}v(x)dxleq‎‎Cint_۰^infty 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 Tf(x)=int 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.

Keywords:

Hardy type inequality , Variable exponent Lebesgue space , Modular type inequality.‎