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.
Keywords