Document Type : Research Paper

Author

• Somayeh Saiedinezhad

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
‎$$‎‎in{t}_0^{i}nfty(frac1xin{t}_0^{x}f(t)dt{)}^{p(x)}v(x)dxleq‎‎Cin{t}_0^{i}nftyf(x{)}^{p(x)}u(x)dx‎,‎$$‎
‎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 $Tf(x)=intK(x,y)f(x)dy$ 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.‎