Discrete iterated Hardy-type inequalities with three weights
DOI:
https://doi.org/10.26577/JMMCS.2020.v105.i1.03Keywords:
inequalities, Hardy-type operator, weight, sequences, discrete Lebesgue spacesAbstract
Discrete, continuous Hardy-typ e inequalities are of great imp ortance and have numerous
applications in harmonic analysis, in the theory of integral, differential and difference op erators,
in the theory of emb eddings of function spaces and in other branches of mathematics. In recent
years, weighted estimates for multidimensional Hardy-typ e op erators have b een intensively studied,
which have an imp ortant application in the study of b oundedness prop erties of op erators from a
Leb esgue weighted space to a lo cal Morrey-typ e space, as well as in the study of bilinear op erators
in Leb esgue weighted spaces. The discrete case of Hardy typ e inequalities with three weights is an
op en problem. An inequality involving an iteration of the discrete Hardy op erator is traditionally
considered difficult to estimate b ecause it contains three indep endent weight sequences and three
parameters at their different ratios. In this pap er we prove some new discrete iterated Hardy-typ e
inequality involving three weights for the case 0 < p ≤ min { q , θ }.
References
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[16] Burenkov V.I. and Oinarov R., "Necessary and Sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesque space to a Morrey-type space" , Math. Inequal. Appl. vol. 16, no 1 (2013): 1-19.
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[18] Bernardis A.L. and Salvador P.O., "Some new iterated Hardy-type inequalities and applications" , J. Math. Ineq. vol. 11, no 2 (2017): 577-594.
[19] Canestro M.I.A., Salvador P.O., Torreblanca C.R., "Weighted bilinear Hardy inequalities" , J. Math. Anal. and Appl. 387 (2012): 320-334.
[20] Krepela M., "Iterating bilinear Hardy inequalities" , J. Math. Ineq. vol. 60, no 4 (2017): 955-971.
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[22] Temirhanova A.M., Omarbaeva B.K., "Vesovaja ocenka odnogo klassa kvazilinejnyh diskretnyh operatorov: sluchaj 0 <q < θ < p < ∞ , p > 1 [Weighted estimate of a class of quasilinear discrete operators: the case 0 < q < θ < p < ∞ , p > 1]" , Vestnik KazNPU im. Abaja vol. 67, no 3 (2019): 109-116.
[2] Kufner A., Maligranda L. and Persson L.-E., "The Prehistory of the Hardy Inequality" , Amer. Math. Monthly. vol. 113(8)(2006): 715–732.
[3] Kufner A., Persson L-E. and Samko N., "Weighted inequalities of Hardy type, Second Edition" , World Scientific Publishing Co. Pte.Ltd., New Jersey. (2017):459+XX.
[4] Goldman M. L., "Hardy type inequalities on the cone of quasimonotone functions" , Khabarovsk Computer Center Fart Eastern Brunch Russian Academy of Science, Research report no 98/31 (1998): 1-69.
[5] Oinarov R., Kalybay A.A., "Three-parameter weighted Hardy type inequalities" , Banach Journal Math. vol. 2, no 2 (2008): 85-93.
[6] Gogatishvili A., Mustafayev R., Persson L.-E., "Some new iterated Hardy-type inequalities" , J. Func. Spaces Appl. vol. 2012 (2012): 30.
[7] Gogatishvili A., Mustafayev R., Persson L.-E., "Some new iterated Hardy-type inequalities: the case θ = 1" , J. Inequal. Appl. vol. 2013, no 515 (2013): 29.
[8] Gogatishvili A., Mustafayev R., "Weighted iterated Hardy-type inequalities" , Math. Inequal. Appl. vol. 20, no 3 (2017): 683-728.
[9] Prohorov D.V., Stepanov V.D., "Vesovye ocenki klassa sublinejnyh operatorov [Weighted estimates for a class of sublinear operators]" , DAN vol. 453, no 5 (2013): 486-488.
[10] Prohorov D.V., Stepanov V.D., "O vesovyh neravenstvah Hardi v smeshannyh normah [On weighted Hardy inequalities in mixed norms]" , Tr. MIAN 283(2013): 155-170.
[11] Prohorov D.V., Stepanov V.D., "Ocenki odnogo klassa sublinejnyh integral’nyh operatorov [Estimates for a class of sublinear integral operators]" , DAN vol. 456, no 6 (2014): 645-649.
[12] Prohorov D.V., Stepanov V.D., "Vesovye neravenstva dlja kvazilinejnyh integral’nyh operatorov na poluosi i prilozhenija k prostranstvam Lorenca [Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces]" , Matem.sb. vol. 207, no 8 (2016): 135-162.
[13] Prohorov D. V., "Ob odnom klasse vesovyh neravenstv, soderzhashhih kvazilinejnye operatory [On a class of weighted inequalities containing quasilinear operators]" , Tr. MIAN 289(2016): 280-295.
[14] Krepela M., Pick L., "Weighted inequalities for iterated Copson Integral operators [electronic resource]" , (2019). URL: https://www.researchgate.net/publication/332962081 (date of the application: 21.10.2019).
[15] Stepanov V.D., Shambilova G.E., "On weighted iterated Hardy-type operators" , Analysis Math. vol. 44, no 2 ( 2018): 273–283.
[16] Burenkov V.I. and Oinarov R., "Necessary and Sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesque space to a Morrey-type space" , Math. Inequal. Appl. vol. 16, no 1 (2013): 1-19.
[17] Oinarov R. and Kalybay A., "On boundedness of the conjugate multidimensional Hardy operator from a Lebesque space to a local Morrey-type space" , Int. J. Math. Anal. vol. 8, no 11 (2014): 539-553.
[18] Bernardis A.L. and Salvador P.O., "Some new iterated Hardy-type inequalities and applications" , J. Math. Ineq. vol. 11, no 2 (2017): 577-594.
[19] Canestro M.I.A., Salvador P.O., Torreblanca C.R., "Weighted bilinear Hardy inequalities" , J. Math. Anal. and Appl. 387 (2012): 320-334.
[20] Krepela M., "Iterating bilinear Hardy inequalities" , J. Math. Ineq. vol. 60, no 4 (2017): 955-971.
[21] Gogatishvili A., Krepela M., Rastislav O., Pick L., "Weighted inequalities for discrete iterated Hardy operators [electronic resource]" , (2019). URL: https://arxiv.org/abs/1903.04313 (Submitted on 11 March 2019).
[22] Temirhanova A.M., Omarbaeva B.K., "Vesovaja ocenka odnogo klassa kvazilinejnyh diskretnyh operatorov: sluchaj 0 <q < θ < p < ∞ , p > 1 [Weighted estimate of a class of quasilinear discrete operators: the case 0 < q < θ < p < ∞ , p > 1]" , Vestnik KazNPU im. Abaja vol. 67, no 3 (2019): 109-116.
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Published
2020-04-05
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Section
Mathematics
