Lidé
doc. RNDr. Zdeněk Mihula, Ph.D.
Všechny publikace
A Weak-Type Expression of the Orlicz Modular
- Autoři: Mgr. Martin Křepela, Ph.D., doc. RNDr. Zdeněk Mihula, Ph.D., Soria, J.
- Publikace: MEDITERRANEAN JOURNAL OF MATHEMATICS. 2023, 20(3), 1-8. ISSN 1660-5446.
- Rok: 2023
- DOI: 10.1007/s00009-023-02315-3
- Odkaz: https://doi.org/10.1007/s00009-023-02315-3
- Pracoviště: Katedra matematiky
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Anotace:
An equivalent expression of Orlicz modulars in terms of measure of level sets of difference quotients is established. The result in a sense complements the famous Maz'ya-Shaposhnikova formula for the fractional Gagliardo-Slobodeckij seminorm and its recent extension to the setting of Orlicz functions.
Different degrees of non-compactness for optimal Sobolev embeddings
- Autoři: Lang, J., doc. RNDr. Zdeněk Mihula, Ph.D.,
- Publikace: Journal of Functional Analysis. 2023, 284(10), 1-22. ISSN 0022-1236.
- Rok: 2023
- DOI: 10.1016/j.jfa.2023.109880
- Odkaz: https://doi.org/10.1016/j.jfa.2023.109880
- Pracoviště: Katedra matematiky
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Anotace:
The structure of non-compactness of optimal Sobolev embed-dings of m-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bern-stein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular. (c) 2023 Elsevier Inc. All rights reserved.
Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
- Autoři: doc. RNDr. Zdeněk Mihula, Ph.D.,
- Publikace: MATHEMATISCHE NACHRICHTEN. 2023, 296(8), 3492-3538. ISSN 0025-584X.
- Rok: 2023
- DOI: 10.1002/mana.202200015
- Odkaz: https://doi.org/10.1002/mana.202200015
- Pracoviště: Katedra matematiky
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Anotace:
The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal rearrangement-invariant function spaces guaranteeing the boundedness of the operators from/to a given rearrangement-invariant function space are described. Second, the optimal rearrangement-invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions is addressed. Next, the relation between optimal rearrangement-invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy-type operators are also studied.
Compactness of Sobolev embeddings and decay of norms
- Autoři: Lang, J., doc. RNDr. Zdeněk Mihula, Ph.D., Pick, L.
- Publikace: Studia Mathematica. 2022, 265(1), 1-36. ISSN 0039-3223.
- Rok: 2022
- DOI: 10.4064/sm201119-29-9
- Odkaz: https://doi.org/10.4064/sm201119-29-9
- Pracoviště: Katedra matematiky
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Anotace:
We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with d-Ahlfors measures under certain restriction on the speed of their decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.
Compactness of Sobolev-type embeddings with measures
- Autoři: Cavaliere, P., doc. RNDr. Zdeněk Mihula, Ph.D.,
- Publikace: Communications in Contemporary Mathematics. 2022, 24(09), 1-41. ISSN 0219-1997.
- Rok: 2022
- DOI: 10.1142/S021919972150036X
- Odkaz: https://doi.org/10.1142/S021919972150036X
- Pracoviště: Katedra matematiky
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Anotace:
We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in R-n with respect to upper Ahlfors regular measures nu, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure nu and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz-Zygmund norms are also presented.
Discretization and antidiscretization of Lorentz norms with no restrictions on weights
- Autoři: Mgr. Martin Křepela, Ph.D., doc. RNDr. Zdeněk Mihula, Ph.D., Turcinova, H.
- Publikace: Revista Matemática Complutense. 2022, 35(2), 615-648. ISSN 1139-1138.
- Rok: 2022
- DOI: 10.1007/s13163-021-00399-7
- Odkaz: https://doi.org/10.1007/s13163-021-00399-7
- Pracoviště: Katedra matematiky
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Anotace:
We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality
Reduction principle for Gaussian K-inequality
- Autoři: Baena-Miret, S., Gogatishvili, A., doc. RNDr. Zdeněk Mihula, Ph.D., Pick, L.
- Publikace: Journal of Mathematical Analysis and Applications. 2022, 516(2), 1-23. ISSN 0022-247X.
- Rok: 2022
- DOI: 10.1016/j.jmaa.2022.126522
- Odkaz: https://doi.org/10.1016/j.jmaa.2022.126522
- Pracoviště: Katedra matematiky
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Anotace:
We study interpolation properties of operators (not necessarily linear) which satisfy a specific K-inequality corresponding to endpoints defined in terms of Orlicz-Karamata spaces modeled upon the example of the Gaussian-Sobolev embedding. We prove a reduction principle for a fairly wide class of such operators. (c) 2022 Elsevier Inc. All rights reserved.
Weighted Inequalities for a Superposition of the Copson Operator and the Hardy Operator
- Autoři: Gogatishvili, A., doc. RNDr. Zdeněk Mihula, Ph.D., Pick, L., Turcinova, H.
- Publikace: The Journal of Fourier Analysis and Application. 2022, 28(2), ISSN 1069-5869.
- Rok: 2022
- DOI: 10.1007/s00041-022-09918-6
- Odkaz: https://doi.org/10.1007/s00041-022-09918-6
- Pracoviště: Katedra matematiky
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Anotace:
We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely
Measure of noncompactness of Sobolev embeddings on strip-like domains
- Autoři: Edmunds, David E., Lang, J., doc. RNDr. Zdeněk Mihula, Ph.D.,
- Publikace: Journal of Approximation Theory. 2021, 269 1-13. ISSN 0021-9045.
- Rok: 2021
- DOI: 10.1016/j.jat.2021.105608
- Odkaz: https://doi.org/10.1016/j.jat.2021.105608
- Pracoviště: Katedra matematiky
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Anotace:
We compute the precise value of the measure of noncompactness of Sobolev embeddings W-0(1, p) (sic) L-p(D), p is an element of(1, infinity), on strip-like domains D of the form R-k x Pi(n-k)(j=1) (r(j), q(j)). We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict s-numbers of the embeddings in question coincide with their norms. We also prove that the maximal noncompactness of Sobolev embeddings on strip-like domains remains valid even when Sobolev-type spaces built upon general rearrangement-invariant spaces are considered. As a by-product we obtain the explicit form for the first eigenfunction of the pseudo- p-Laplacian on an n-dimensional rectangle. (C) 2021 Elsevier Inc. All rights reserved.