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.
Saiedinezhad,S . (2016). Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition. International Journal of Nonlinear Analysis and Applications, 7(2), 29-38. doi: 10.22075/ijnaa.2016.439
MLA
Saiedinezhad,S . "Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition", International Journal of Nonlinear Analysis and Applications, 7, 2, 2016, 29-38. doi: 10.22075/ijnaa.2016.439
HARVARD
Saiedinezhad S. (2016). 'Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition', International Journal of Nonlinear Analysis and Applications, 7(2), pp. 29-38. doi: 10.22075/ijnaa.2016.439
CHICAGO
S Saiedinezhad, "Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition," International Journal of Nonlinear Analysis and Applications, 7 2 (2016): 29-38, doi: 10.22075/ijnaa.2016.439
VANCOUVER
Saiedinezhad S. Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition. IJNAA. 2016;7(2):29-38. doi: 10.22075/ijnaa.2016.439
International Journal of Nonlinear Analysis and Applications