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

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 Z 1 0 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) = R K(x;y)f(x)dy on L p(:) when the variable exponent p(:) satises some uniform continuity condition that is named -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.