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Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 2، دوره 7، شماره 2، اسفند 2016، صفحه 29-38 اصل مقاله (375.71 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2016.439 | ||
| نویسنده | ||
| Somayeh Saiedinezhad | ||
| Assistant professor of Iran University of Science and technology | ||
| تاریخ دریافت: 14 آذر 1394، تاریخ بازنگری: 04 خرداد 1395، تاریخ پذیرش: 14 خرداد 1395 | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Hardy type inequality؛ Variable exponent Lebesgue space؛ Modular type inequality. | ||
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