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

Document Type: Research Paper

Author

Assistant professor of Iran University of Science and technology

Abstract

‎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
‎$$‎ ‎int_0^infty (frac{1}{x}int_0^x f(t)dt)^{p(x)}v(x)dxleq‎ ‎Cint_0^infty f(x)^{p(x)}u(x)dx‎, ‎$$‎
‎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.

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