Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68227220161220Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition293843910.22075/ijnaa.2016.439ENSomayehSaiedinezhadAssistant professor of Iran University of Science and technologyJournal Article20151205Some 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_a9ff1b7775e024c726cd0418c812bd7b.pdf