Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 7 2 2016 12 20 Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition 29 38 439 10.22075/ijnaa.2016.439 EN Somayeh Saiedinezhad Assistant professor of Iran University of Science and technology Journal Article 2015 12 05 ‎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<br />‎$$‎<br />‎int_0^infty (frac{1}{x}int_0^x f(t)dt)^{p(x)}v(x)dxleq‎<br />‎Cint_0^infty f(x)^{p(x)}u(x)dx‎,<br />‎$$‎<br />‎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. https://ijnaa.semnan.ac.ir/article_439_eed5af6add95a9a6f1252739b1ad8c24.pdf