Compactness for the weighted Hardy operator in variable exponent spaces

Farman Mamedov; Sayali Mammadli

doi:10.1016/j.crma.2016.12.010

Harmonic analysis

Compactness for the weighted Hardy operator in variable exponent spaces

Presented by: the Editorial Board

Farman Mamedov ^1,²; Sayali Mammadli ¹

¹ Mathematics and Mechanic Institute of National Academy of Science, Baku 1141, B. Vahabzade, 9, Azerbaijan
² OilGasScientificResearchProject Inst., SOCAR, Baku 1012, H. Zardabi, 88A, Azerbaijan

Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 325-335.

Abstract (Original language)
Résumé

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator

H_{υ, ω} f (x) = υ (x) \int_{0}^{x} f (t) ω (t) d t

to be compactly acting from

L^{p (\cdot)} (0, \infty)

to

L^{q (\cdot)} (0, \infty)

.

Dans cette Note, nous prouvons une condition nécessaire et suffisante pour que l'opérateur de Hardy pondéré

H_{υ, ω} f (x) = υ (x) \int_{0}^{x} f (t) ω (t) d t

agisse de façon compacte de

L^{p (\cdot)} (0, \infty)

dans

L^{q (\cdot)} (0, \infty)

.

Received: 2016-07-14
Accepted: 2017-01-10
Published online: 2017-02-10

DOI: 10.1016/j.crma.2016.12.010

Author's affiliations:

Farman Mamedov ^{1, 2}; Sayali Mammadli ¹

¹ Mathematics and Mechanic Institute of National Academy of Science, Baku 1141, B. Vahabzade, 9, Azerbaijan
² OilGasScientificResearchProject Inst., SOCAR, Baku 1012, H. Zardabi, 88A, Azerbaijan

@article{CRMATH_2017__355_3_325_0,
     author = {Farman Mamedov and Sayali Mammadli},
     title = {Compactness for the weighted {Hardy} operator in variable exponent spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {325--335},
     publisher = {Elsevier},
     volume = {355},
     number = {3},
     year = {2017},
     doi = {10.1016/j.crma.2016.12.010},
     language = {en},
}

TY  - JOUR
AU  - Farman Mamedov
AU  - Sayali Mammadli
TI  - Compactness for the weighted Hardy operator in variable exponent spaces
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 325
EP  - 335
VL  - 355
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2016.12.010
LA  - en
ID  - CRMATH_2017__355_3_325_0
ER  -

%0 Journal Article
%A Farman Mamedov
%A Sayali Mammadli
%T Compactness for the weighted Hardy operator in variable exponent spaces
%J Comptes Rendus. Mathématique
%D 2017
%P 325-335
%V 355
%N 3
%I Elsevier
%R 10.1016/j.crma.2016.12.010
%G en
%F CRMATH_2017__355_3_325_0

Farman Mamedov; Sayali Mammadli. Compactness for the weighted Hardy operator in variable exponent spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 325-335. doi : 10.1016/j.crma.2016.12.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.010/

References
Cited by

[1] E. Acerbi; G. Mingione Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., Volume 156 (2011), pp. 121-140

[2] S. Boza; J. Soria Weighted Hardy modular inequalities in variable $L^{p (\cdot)}$ spaces for decreasing functions, J. Math. Anal. Appl., Volume 348 (2008) no. 1, pp. 383-388

[3] S. Boza; J. Soria Weighted weak modular and norm inequalities for the Hardy operator in variable $L^{p (\cdot)}$ space of monotone functions equalities, Rev. Mat. Complut., Volume 25 (2012) no. 2, pp. 459-474

[4] L. Chen Hardy Type Inequality in weighted variable exponent spaces and applications to $p (x)$ -Laplace type equations, 2007 www.paper.edu.cn/en_releasepaper/downPaper/200711-521

[5] M. Colombo; G. Mingione Regularity for double phase variational problems, Arch. Ration. Mech. Anal., Volume 215 (2015), pp. 443-496

[6] D. Cruz-Uribe; A. Florenzia Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkhauser, Basel, Switzerland, 2013

[7] D. Cruz-Uribe; F.I. Mamedov On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces, Rev. Mat. Complut., Volume 25 (2012) no. 2, pp. 335-367

[8] L. Diening; S. Samko Hardy inequality in variable exponent Lebesgue spaces, Fract. Calc. Appl. Anal., Volume 10 (2007) no. 1, pp. 1-17

[9] L. Diening; P. Harjulehto; P. Hasto; M. Ruziska Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math., vol. 2017, Springer, Heidelberg, Germany, 2011

[10] N. Dunford; J.T. Schwartz Linear Operators: General Theory, Part 1, Wiley, 1958

[11] D.E. Edmunds; P. Gurka; L. Pick Compactness of Hardy-type integral operators in weighted Banach function spaces, Stud. Math., Volume 109 (1994) no. 1, pp. 73-90

[12] D.E. Edmunds; V. Kokllashvili; A. Meshki Bounded and Compact Integral Operators, Mathematics and Its Applications, vol. 543, Kluwer Academic Publishers, Dordrecht, 2002

[13] D.E. Edmunds; V. Kokllashvili; A. Meshki On the boundedness and compactness of the weighted Hardy operators in spaces $L^{p (x)}$ , Georgian Math. J., Volume 12 (2005) no. 1, pp. 27-44

[14] A. Gogatishvili; A. Kufner; L.E. Persson; A. Wedestig An equivalence theorem for integral conditions related to Hardy's operator, Real Anal. Exch., Volume 29 (2003/2004) no. 2, pp. 867-880

[15] P. Harjulehto; P. Hasto; M. Koskinoja Hardy's inequality in variable exponent Sobolev spaces, Georgian Math. J., Volume 12 (2005) no. 3, pp. 431-442

[16] A. Harman; F.I. Mamedov On boundedness of weighted Hardy operator in $L^{p (\cdot)}$ and regularly condition, J. Inequal. Appl., Volume 2010 (2010) (14 pages)

[17] V. Kokilashvili; A. Meskhi; H. Rafeiro; S. Samko Integral Operators in Non-standard Function Spaces: Vol. I: Variable Exponent Lebesgue and Amalgam Spaces, Operator Theory: Advances and Applications, vol. 248, Birkhauser, 2016 (i–xx, 1–567 pp)

[18] V. Koklashvili; S. Samko Maximal and fractional operators in weighted $L^{p (x)}$ spaces, Rev. Mat. Iberoam., Volume 20 (2004) no. 2, pp. 493-515

[19] W.A.J. Luxemburg, Banach function spaces, Thesis, Delft, 1995.

[20] F.I. Mamedov On Hardy type inequality in variable exponent Lebesgue space $L^{p (\cdot)} (0, 1)$ , Azerb. J. Math., Volume 2 (2012) no. 1, pp. 90-99

[21] F.I. Mamedov; A. Harman On a weighted inequality of Hardy type in spaces $L^{p (\cdot)}$ , J. Math. Anal. Appl., Volume 353 (2009) no. 2, pp. 521-530

[22] F.I. Mamedov; A. Harman On a Hardy type general weighted inequality in spaces $L^{p (\cdot)}$ , Integral Equ. Oper. Theory, Volume 66 (2010) no. 4, pp. 565-592

[23] F.I. Mamedov; F. Mammadova A necessary and sufficient condition for Hardy's operator in $L^{p (\cdot)} (0, 1)$ , Math. Nachr., Volume 287 (2014) no. 5–6, pp. 666-676

[24] F.I. Mamedov; Y. Zeren On equivalent conditions for the general weighted Hardy type inequality in space $L^{p (\cdot)}$ , Z. Anal., Volume 31 (2012) no. 1, pp. 55-74

[25] F.I. Mamedov; Y. Zeren A necessary and sufficient condition for Hardy's operator in the variable Lebesgue space, Abstr. Appl. Anal., Volume 2014 (2014) (7 pages)

[26] F.I. Mamedov; F. Mammadova; M. Aliyev Boundedness criterions for the Hardy operator in weighted $L^{p (\cdot)} (0, l)$ space, J. Convex Anal., Volume 22 (2015) no. 2, pp. 553-568

[27] P. Marcellini Regularity for elliptic equations with general growth conditions, J. Differ. Equ., Volume 105 (1993), pp. 296-333

[28] R. Mashiyev; B. Cekic; F.I. Mamedov; S. Ogrash Hardy's inequality in power-type weighted $L^{p (\cdot)}$ spaces, J. Math. Anal. Appl., Volume 334 (2007) no. 1, pp. 289-298

[29] A. Meskhi; M.A. Zaighim Weighted kernel operators in $L^{p (x)}$ spaces, J. Math. Inequal., Volume 10 (2016) no. 3, pp. 623-639

[30] J. Muslelak Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983

[31] C.A. Okpotl; L.E. Persson; G. Sinnamon An equivalence theorem for some integral conditions with general measures related to Hardy's inequality, J. Math. Anal. Appl., Volume 326 (2007) no. 1, pp. 398-413

[32] V.D. Radulescu; D. Repovs Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, 2015

[33] H. Rafeiro; S.G. Samko Hardy inequality in variable Lebesgue spaces, Ann. Acad. Sci. Fenn., Volume 34 (2009) no. 1, pp. 279-289

[34] W. Rudin Functional Analysis, McGraw-Hill, New York, 1973

[35] M. Ruzicka Electrorheological Fluids Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000

[36] S. Samko Hardy inequality in the generalized Lebesgue spaces, Fract. Calc. Appl. Anal., Volume 6 (2003) no. 4, pp. 355-362

[37] S. Samko Hardy–Littlewood–Stein–Weiss inequality in the Lebesgue spaces with variable exponent, Fract. Calc. Appl. Anal., Volume 6 (2003) no. 4, pp. 421-440

[38] S. Samko On compactness of operators in variable exponent Lebesgue spaces, Oper. Theory, Adv. Appl., Volume 202 (2010), pp. 497-508

[39] V. Zhikov On some variational problems, J. Math. Phys., Volume 5 (1997), pp. 105-116 (Russian)

Cited by Sources:

Comments - Policy