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- Abdikalikova, Zamira, et al.
(författare)
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Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1
- 2011
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Ingår i: Czechoslovak Mathematical Journal. - : Institute of Mathematics, Czech Academy of Sciences. - 0011-4642 .- 1572-9141. ; 61:1, s. 7-26
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Tidskriftsartikel (refereegranskat)abstract
- We consider a new Sobolev type function space called the space with multiweighted derivatives W-p(n),(alpha) over bar, where (alpha) over bar = (alpha(0), alpha(1), ......, alpha(n)), alpha(i) is an element of R, i = 0, 1,......,n, and parallel to f parallel to W-p(n),((alpha) over bar) = parallel to D((alpha) over bar)(n)f parallel to(p) + Sigma(n-1) (i=0) vertical bar D((alpha) over bar)(i)f(1)vertical bar, D((alpha) over bar)(0)f(t) = t(alpha 0) f(t), d((alpha) over bar)(i)f(t) = t(alpha i) d/dt D-(alpha) over bar(i-1) f(t), i = 1, 2, ....., n. We establish necessary and sufficient conditions for the boundedness and compactness of the embedding W-p,(alpha) over bar(n) -> W-q,(beta) over bar,(m) when 1 <= q < p < infinity, 0 <= m < n
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| 4. |
- Abylayeva, Akbota
(författare)
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Inequalities for some classes of Hardy type operators and compactness in weighted Lebesgue spaces
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis is devoted to investigate weighted differential Hardy inequalities and Hardy-type inequalities with the kernel when the kernel has an integrable singularity, and also the additivity of the estimate of a Hardy type operator with a kernel.The thesis consists of seven papers (Papers 1, 2, 3, 4, 5, 6, 7) and an introduction where a review on the subject of the thesis is given. In Paper 1 weighted differential Hardy type inequalities are investigated on the set of compactly supported smooth functions, where necessary and sufficient conditions on the weight functions are established for which this inequality and two-sided estimates for the best constant hold. In Papers 2, 3, 4 a more general class of -order fractional integrationoperators are considered including the well-known classical Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard operators. Here 0 < < 1. In Papers 2 and 3 the boundedness and compactness of two classes of such operators are investigated namely of Weyl and Riemann-Liouville type, respectively, in weighted Lebesgue spaces for 1 < p ≤ q < 1 and 0 < q < p < ∞. As applications some new results for the fractional integration operators of Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard are given and discussed.In Paper 4 the Riemann-Liouville type operator with variable upper limit is considered. The main results are proved by using a localization method equipped with the upper limit function and the kernel of the operator. In Papers 5 and 6 the Hardy operator with kernel is considered, where the kernel has a logarithmic singularity. The criteria of the boundedness and compactness of the operator in weighted Lebesgue spaces are given for 1 < p ≤ q < ∞ and 0 < q < p < ∞, respectively. In Paper 7 we investigated the weighted additive estimates for integral operators K+ and K¯ defined byK+ ƒ(x) := ∫ k(x,s) ƒ(s)ds, K¯ ƒ(x) := ∫ k(x,s)ƒ(s)ds.It is assumed that the kernel k of the operators K+and K- belongs to the general Oinarov class. We derived the criteria for the validity of these addittive estimates when 1 ≤ p≤ q < ∞
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| 5. |
- Abylayeva, Akbota M., et al.
(författare)
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Boundedness and compactness of a class of Hardy type operators
- 2016
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Ingår i: Journal of inequalities and applications. - : Springer Science and Business Media LLC. - 1025-5834 .- 1029-242X. ; :1
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Tidskriftsartikel (refereegranskat)abstract
- We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.
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| 7. |
- Arendarenko, Larissa, et al.
(författare)
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On the boundedness of some classes of integral operators in weighted Lebesgue spaces
- 2012
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Ingår i: Eurasian Mathematical Journal. - 2077-9879. ; 3:1, s. 5-17
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Tidskriftsartikel (refereegranskat)abstract
- Some new Hardy-type inequalities for Hardy-Volterra integral operators are proved and discussed. The case 1 < q < p < ∞ is considered and the involved kernels satisfy conditions, which are less restrictive than the usual Oinarov condition.
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