New equivalent conditions for Hardy-type inequalities

Alois Kufner; Komil Kuliev; Gulchehra Kulieva; Mohlaroyim Eshimova

Mathematica Bohemica (2024)

  • Issue: 1, page 57-73
  • ISSN: 0862-7959

Abstract

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We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.

How to cite

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Kufner, Alois, et al. "New equivalent conditions for Hardy-type inequalities." Mathematica Bohemica (2024): 57-73. <http://eudml.org/doc/299222>.

@article{Kufner2024,
abstract = {We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.},
author = {Kufner, Alois, Kuliev, Komil, Kulieva, Gulchehra, Eshimova, Mohlaroyim},
journal = {Mathematica Bohemica},
keywords = {integral operator; norm; weight function; Lebesgue space; Hardy-type inequality; kernel},
language = {eng},
number = {1},
pages = {57-73},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New equivalent conditions for Hardy-type inequalities},
url = {http://eudml.org/doc/299222},
year = {2024},
}

TY - JOUR
AU - Kufner, Alois
AU - Kuliev, Komil
AU - Kulieva, Gulchehra
AU - Eshimova, Mohlaroyim
TI - New equivalent conditions for Hardy-type inequalities
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 57
EP - 73
AB - We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties.
LA - eng
KW - integral operator; norm; weight function; Lebesgue space; Hardy-type inequality; kernel
UR - http://eudml.org/doc/299222
ER -

References

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  1. Bloom, S., Kerman, R., 10.1090/S0002-9939-1991-1059623-6, Proc. Am. Math. Soc. 113 (1991), 135-141. (1991) Zbl0753.42010MR1059623DOI10.1090/S0002-9939-1991-1059623-6
  2. Breadley, J. S., 10.4153/CMB-1978-071-7, Can. Math. Bull. 21 (1978), 405-408. (1978) Zbl0402.26006MR0523580DOI10.4153/CMB-1978-071-7
  3. Drábek, P., Kuliev, K., Marletta, M., 10.1007/s00030-017-0433-2, NoDEA, Nonlinear Differ. Equ. Appl. 24 (2017), Article ID 11, 39 pages. (2017) Zbl1383.34044MR3608754DOI10.1007/s00030-017-0433-2
  4. Gogatishvili, A., Kufner, A., Persson, L.-E., 10.1007/s10474-009-8132-z, Acta Math. Hung. 123 (2009), 365-377. (2009) Zbl1199.26057MR2506756DOI10.1007/s10474-009-8132-z
  5. Gogatishvili, A., Kufner, A., Persson, L.-E., Wedestig, A., 10.14321/realanalexch.29.2.0867, Real Anal. Exch. 29 (2004), 867-880. (2004) Zbl1070.26015MR2083821DOI10.14321/realanalexch.29.2.0867
  6. Kalybay, A. A., Baiarystanov, A. O., Exact estimate of norm of integral operator with Oinarov condition, Mat. Zh. 21 (2021), 6-14. (2021) Zbl1488.26068
  7. Kokilashvili, V. M., On Hardy's inequalities in weighted spaces, Soobshch. Akad. Nauk Gruzin. SSR 96 (1979), 37-40 Russian. (1979) Zbl0434.26007MR0564755
  8. Kufner, A., Kuliev, K., Oinarov, R., 10.1186/1029-242X-2013-310, J. Inequal. Appl. 2013 (2013), Article ID 310, 15 pages. (2013) Zbl1290.26024MR3083528DOI10.1186/1029-242X-2013-310
  9. Kufner, A., Kuliev, K., Persson, L.-E., 10.1186/1029-242X-2012-69, J. Inequal. Appl. 2012 (2012), Article ID 69, 14 pages. (2012) Zbl1276.26048MR2931013DOI10.1186/1029-242X-2012-69
  10. Kufner, A., Maligranda, L., Persson, L.-E., The Hardy Inequality: About Its History and Some Related Results, Vydavatelský servis, Pilsen (2007). (2007) Zbl1213.42001MR2351524
  11. Kufner, A., Persson, L.-E., 10.1142/5129, World Scientific, Singapore (2003). (2003) Zbl1065.26018MR1982932DOI10.1142/5129
  12. Kufner, A., Persson, L.-E., Wedestig, A., 10.4064/bc64-0-11, Banach Center Publ. 64 (2004), 135-146. (2004) Zbl1052.26018MR2099465DOI10.4064/bc64-0-11
  13. Kuliev, K., Kulieva, G., Eshimova, M., 10.29229/uzmj.2021-4-10, Uzb. Math. J. 65 (2021), 117-127. (2021) Zbl07465549MR4378731DOI10.29229/uzmj.2021-4-10
  14. Martin-Reyes, F. J., Sawyer, E., 10.1090/S0002-9939-1989-0965246-8, Proc. Am. Math. Soc. 106 (1989), 727-733. (1989) Zbl0704.42018MR0965246DOI10.1090/S0002-9939-1989-0965246-8
  15. Maz'ja, V. G., 10.1007/978-3-662-09922-3, Springer, Berlin (1979). (1979) Zbl0692.46023MR0817985DOI10.1007/978-3-662-09922-3
  16. Muckenhoupt, B., 10.4064/sm-44-1-31-38, Stud. Math. 44 (1972), 31-38. (1972) Zbl0236.26015MR0311856DOI10.4064/sm-44-1-31-38
  17. Oinarov, R., Two-sided norm estimates for certain classes of integral operators, Proc. Steklov Inst. Math. 204 (1994), 205-214 translation from Tr. Mat. Inst. Steklova 204 1993 240-250. (1994) Zbl0883.47048MR1320028
  18. Opic, B., Kufner, A., Hardy-Type Inequalities, Pitman Research Notes in Mathematics 219. Longman Scientific & Technical, Harlow (1990). (1990) Zbl0698.26007MR1069756
  19. Persson, L.-E., Stepanov, V. D., Weighted integral inequalities with the geometric mean, Dokl. Math. 63 (2001), 201-202 translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 377 2001 439-440. (2001) Zbl1044.26014MR1862003
  20. Prokhorov, D. V., 10.1112/S002461079900856X, J. Lond. Math. Soc., II. Ser. 61 (2000), 617-628. (2000) Zbl0956.47019MR1760684DOI10.1112/S002461079900856X
  21. Prokhorov, D. V., Stepanov, V. D., Weighted estimates for the Riemann-Liouville operators and applications, Proc. Steklov Inst. Math. 243 (2003), 278-301 translation from Tr. Mat. Inst. Steklova 243 2003 289-312. (2003) Zbl1081.26004MR2054439
  22. Talenti, G., 10.1007/BF02924135, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185 Italian. (1969) Zbl0218.26011MR0280661DOI10.1007/BF02924135
  23. Tomaselli, G., A class of inequalities, Boll. Unione Mat. Ital., IV. Ser. 2 (1969), 622-631. (1969) Zbl0188.12103MR0255751
  24. Wedestig, A., Weighted Inequalities of Hardy-Type and their Limiting Inequalities: Doctoral Thesis, Luleå University of Technology, Luleå (2003). (2003) MR2044410

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