## Abstract and Applied Analysis

### A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space

#### Abstract

The variable exponent Hardy inequality ${∥{x}^{\beta (x)-1}{\int }_{0}^{x}\mathrm{‍}f(t)dt∥}_{{L}^{p(.)}(0,l)}\le C{∥{x}^{\beta (x)}f∥}_{{L}^{p(.)}(0,l)}$, $f\ge 0$ is proved assuming that the exponents $p:(0,l)\to (1,\infty )$, $\beta :(0,l)\to {\Bbb R}$ not rapidly oscilate near origin and $1/{p}^{\prime }(0)-\beta >0$. The main result is a necessary and sufficient condition on $p$, $\beta$ generalizing known results on this inequality.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 342910, 7 pages.

Dates
First available in Project Euclid: 7 October 2014

https://projecteuclid.org/euclid.aaa/1412687017

Digital Object Identifier
doi:10.1155/2014/342910

Mathematical Reviews number (MathSciNet)
MR3208529

Zentralblatt MATH identifier
07022194

#### Citation

Mamedov, Farman; Zeren, Yusuf. A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 342910, 7 pages. doi:10.1155/2014/342910. https://projecteuclid.org/euclid.aaa/1412687017

#### References

• H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing, 1978.
• A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy Inequality: About Its History and Some Related Results, Vydavatelský servis, 2007.
• J. L. Lions and E. Magenes, Non Homogeneous Boundary Prob-lem and Applications, vol. 2, Springer, New York, NY, USA, 1972.
• Y. Brundyi and N. Krugljak, Interpolation Functors and Interpolation Spaces, North-Holland Publisher, 1991.
• M. Ruzicka, Electrorheological Fluids Modeling and Mathematical the Ory, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
• V. V. Zhikov, “On some variational problems,” Russian Journal of Mathematical Physics, vol. 5, no. 1, pp. 105–116, 1997.
• P. Marcellini, “Regularity for elliptic equations with general growth conditions,” Journal of Differential Equations, vol. 105, no. 2, pp. 296–333, 1993.
• E. Acerbi and G. Mingione, “Regularity results for a class of fun-ctionals with non-standard growth,” Archive for Rational Mechanics and Analysis, vol. 156, no. 2, pp. 121–140, 2001.
• L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, Germany, 2011.
• D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Fon-dation and Harmonic Analysis, Birkhäuser, 2013.
• S. Samko, “Hardy inequality in the generalized Lebesgue spaces,” Fractional Calculus & Applied Analysis, vol. 6, no. 4, pp. 355–362, 2003.
• S. G. Samko, “Convolution type operators in ${L}^{p(x)}$,” Integral Tra-nsforms and Special Functions, vol. 7, no. 1-2, pp. 123–144, 1998.
• V. Kokilashvili and S. Samko, “Maximal and fractional operators in weighted ${L}^{p(x)}$ spaces,” Revista Matemática Iberoamericana, vol. 20, no. 2, pp. 493–515, 2004.
• P. Harjulehto, P. Hästö, and M. Koskenoja, “Hardy's inequality in a variable exponent Sobolev space,” Georgian Mathematical Journal, vol. 12, no. 3, pp. 431–442, 2005.
• D. E. Edmunds, V. Kokilashvili, and A. Meskhi, “On the boun-dedness and compactness of weighted Hardy operators in spaces ${L}^{p(x)}$,” Georgian Mathematical Journal, vol. 12, no. 1, pp. 27–44, 2005.
• H. Rafeiro and S. Samko, “Hardy type inequality in variable Lebesgue spaces,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 34, no. 1, pp. 279–289, 2009.
• L. Diening and S. Samko, “Hardy inequality in variable exponent Lebesgue spaces,” Fractional Calculus & Applied Analysis, vol. 10, no. 1, pp. 1–17, 2007.
• R. A. Mashiyev, B. Çekiç, F. I. Mamedov, and S. Ogras, “Hardy's inequality in power-type weighted ${L}^{p(\cdot\,\!)}(0,\infty )$ spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 289–298, 2007.
• F. I. Mamedov and A. Harman, “On a weighted inequality of Hardy type in spaces ${L}^{p(\cdot\,\!)}$,” Journal of Mathematical Analysis and Applications, vol. 353, no. 2, pp. 521–530, 2009.
• F. I. Mamedov and A. Harman, “On a Hardy type general wei-ghted inequality in spaces ${L}^{p(\cdot\,\!)}$,” Integral Equations and Operator Theory, vol. 66, no. 4, pp. 565–592, 2010.
• D. Cruz-Uribe and F. I. Mamedov, “On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces,” Revista Matemática Complutense, vol. 25, no. 2, pp. 335–367, 2012.
• F. I. Mamedov and Y. Zeren, “On equivalent conditions for the general weighted Hardy type inequality in space ${L}^{p(\cdot\,\!)}$,” Zeits-chrift für Analysis und ihre Anwendungen, vol. 31, no. 1, pp. 55–74, 2012.
• F. I. Mamedov, “On Hardy type inequality in variable exponent Lebesgue space ${L}^{p(\cdot\,\!)}(0,l)$,” Azerbaijan Journal of Mathematics, vol. 2, no. 1, pp. 90–99, 2012.
• A. Harman and F. I. Mamedov, “On boundedness of weighted Hardy operator in ${L}^{p(\cdot\,\!)}$ and regularity condition,” Journal of Inequalities and Applications, vol. 2010, Article ID 837951, 14 pages, 2010.
• F. I. Mamedov and F. M. Mammadova, “A necessary and suffic-ient condition for Hardy's operator in ${L}^{p(\cdot\,\!)}$,” Mathematische Nachrichten, vol. 287, no. 5-6, pp. 666–676, 2013.
• X. Fan and D. Zhao, “On the spaces ${L}^{p(x)}(\Omega )$ and ${W}^{m,p(x)}(\Omega )$,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001.
• O. Kováčik and J. Rákosník, “On spaces ${L}^{p(x)}$($\Omega$) and ${W}^{1,p(x)}$,” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991.
• N. K. Bari and S. B. Stečkin, “Best approximations and differen-tial properties of two conjugate functions,” Proceedings of Moscow Mathematical Society, vol. 5, pp. 483–522, 1956. \endinput