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Publications (10 of 11) Show all publications
Křepela, M. (2017). Bilinear weighted Hardy inequality for nonincreasing functions. Publications mathématiques (Bures-sur-Yvette), 61(1), 3-50
Open this publication in new window or tab >>Bilinear weighted Hardy inequality for nonincreasing functions
2017 (English)In: Publications mathématiques (Bures-sur-Yvette), ISSN 0073-8301, E-ISSN 1618-1913, Vol. 61, no 1, p. 3-50Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Barcelona: Universitat Autonoma de Barcelona, 2017
Keywords
Hardy operators, bilinear operators, weights, inequalities for monotone functions
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-35214 (URN)10.5565/PUBLMAT_61117_01 (DOI)000396538700001 ()
Available from: 2015-02-13 Created: 2015-02-13 Last updated: 2017-10-19Bibliographically approved
Křepela, M. (2017). Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$. Revista Matemática Complutense, 30(3), 547-587
Open this publication in new window or tab >>Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$
2017 (English)In: Revista Matemática Complutense, ISSN 1139-1138, E-ISSN 1988-2807, Vol. 30, no 3, p. 547-587Article in journal (Refereed) Published
Abstract [en]

Boundedness of a fundamental Hardy-type operator with a kernel is characterized between weighted Lebesgue spaces $L^p(v)$ and $L^q(w)$ for $0<q<1\le p<\infty$. The conditions are explicit and have a standard integral form.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
Hardy operators, Oinarov kernel, weighted Lebesgue spaces, weighted inequalities, integral operators
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41241 (URN)10.1007/s13163-017-0230-9 (DOI)000408650000007 ()
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2018-08-20Bibliographically approved
Křepela, M. (2017). Convolution inequalities in weighted Lorentz spaces: case 0<q<1. Mathematical Inequalities & Applications, 20(1), 191-201
Open this publication in new window or tab >>Convolution inequalities in weighted Lorentz spaces: case 0<q<1
2017 (English)In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 20, no 1, p. 191-201Article in journal (Refereed) Published
Abstract [en]

We characterize boundedness of a convolution operator between weighted Lorentz spaces $\Lambda^p(v)$and $\Gamma^q(w)$ in the case $0<q<1$.

Keywords
Convolution, Young inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41943 (URN)10.7153/mia-20-13 (DOI)000397414900012 ()
Available from: 2016-04-28 Created: 2016-04-28 Last updated: 2018-06-21Bibliographically approved
Gogatishvili, A., Křepela, M., Pick, L. & Soudsky, F. (2017). Embeddings of Lorentz-type spaces involving weighted integral means. Journal of Functional Analysis, 273(9), 2939-2980
Open this publication in new window or tab >>Embeddings of Lorentz-type spaces involving weighted integral means
2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 273, no 9, p. 2939-2980Article in journal (Refereed) Published
Abstract [en]

We solve the problem of characterizing weights on (0, infinity) for which the inequality involving two possibly different general inner weighted means (integral(infinity)(0)(integral(t)(0)f*(s)(m2)u(2)(s)ds)(p2/m2) w(2)(t)dt)(1/p2) <= C(integral(infinity)(0)(integral(t)(0)f*(s)(m2)u(1)(s)ds)(p1/m1) w(1)(t)dt)(1/p1) holds, where p(1), p(2), m(1), m(2) is an element of (0, infinity) and p(2) > m(2). The proof is based on a new approach combining duality techniques with sharp weighted estimates for iterated integral and supremum operators. (C) 2017 Elsevier Inc. All rights reserved.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kau:diva-65863 (URN)10.1016/j.jfa.2017.06.008 (DOI)000411422900005 ()
Available from: 2018-01-25 Created: 2018-01-25 Last updated: 2018-07-02Bibliographically approved
Křepela, M. (2017). Integral conditions for Hardy-type operators involving suprema. Collectanea Mathematica (Universitat de Barcelona), 68(1), 21-50
Open this publication in new window or tab >>Integral conditions for Hardy-type operators involving suprema
2017 (English)In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 68, no 1, p. 21-50Article in journal (Refereed) Published
Abstract [en]

The article contains characterizations of boundedness of an iterated supremal Hardy-type operator between weighted Lebesgue spaces, and an supremal Hardy operator restricted to positive decreasing functions between the same spaces. The found condtitions have an explicit integral/supremal form and cover all possible cases of positive exponents of the involved spaces.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
supremal operators, Hardy inequalities, weighted function spaces
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41239 (URN)10.1007/s13348-016-0170-6 (DOI)000392229000003 ()
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2017-09-07Bibliographically approved
Křepela, M. (2017). Iterating bilinear Hardy inequalities. Proceedings of the Edinburgh Mathematical Society
Open this publication in new window or tab >>Iterating bilinear Hardy inequalities
2017 (English)In: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839Article in journal (Refereed) Published
Abstract [en]

An iteration technique to characterize boundedness of certain types of multilinear operators is presented, reducing the problem into a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of a bilinear Hardy inequality in the weighted Lebesgue space setting. More equivalent characterizing conditions are presented. The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

Keywords
Hardy operators; bilinear operators; weights; operator inequalities
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kau:diva-41238 (URN)10.1017/S0013091516000602 (DOI)
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2017-12-06Bibliographically approved
Křepela, M. (2017). The Weighted Space Odyssey. (Doctoral dissertation). Karlstad: Karlstads universitet
Open this publication in new window or tab >>The Weighted Space Odyssey
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The common topic of this thesis is boundedness of integral and supremal operators between weighted function spaces.

The first type of results are characterizations of boundedness of a convolution-type operator between general weighted Lorentz spaces. Weighted Young-type convolution inequalities are obtained and an optimality property of involved domain spaces is proved. Additional provided information includes an overview of basic properties of some new function spaces appearing in the proven inequalities.

In the next part, product-based bilinear and multilinear Hardy-type operators are investigated. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.

Further objects of study are iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, thus completing the theory of the involved fundamental Hardy-type operators.

Abstract [en]

Operators acting on function spaces are classical subjects of study in functional analysis. This thesis contributes to the research on this topic, focusing particularly on integral and supremal operators and weighted function spaces.

Proving boundedness conditions of a convolution-type operator between weighted Lorentz spaces is the first type of a problem investigated here. The results have a form of weighted Young-type convolution inequalities, addressing also optimality properties of involved domain spaces. In addition to that, the outcome includes an overview of basic properties of some new function spaces appearing in the proven inequalities.

 Product-based bilinear and multilinear Hardy-type operators are another matter of focus. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.

 The last part of the presented work concerns iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, completing the theory of the involved fundamental Hardy-type operators.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2017. p. 57
Series
Karlstad University Studies, ISSN 1403-8099 ; 2017:1
Keywords
integral operators, supremal operators, weights, weighted function spaces, Lorentz spaces, Lebesgue spaces, convolution, Hardy inequality, multilinear operators, nonincreasing rearrangement
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41944 (URN)978-91-7063-734-6 (ISBN)978-91-7063-735-3 (ISBN)
Public defence
2017-02-10, 9C203, Karlstads universitet, Karlstad, 09:00 (English)
Opponent
Supervisors
Note

Artikel 9 publicerad i avhandlingen som manuskript med samma titel.

Available from: 2017-01-18 Created: 2016-04-28 Last updated: 2019-07-12Bibliographically approved
Křepela, M. (2016). Convolution in weighted Lorentz spaces of type Γ. Mathematica Scandinavica, 119(1), 113-132
Open this publication in new window or tab >>Convolution in weighted Lorentz spaces of type Γ
2016 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, p. 113-132Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Aarhus Universitetsforlag, 2016
Keywords
Convolution, Young inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-31752 (URN)10.7146/math.scand.a-24187 (DOI)000383815600007 ()
Note

This article was published as manuscript in Martin Křepelas licentiate thesis.

Available from: 2014-03-23 Created: 2014-03-23 Last updated: 2017-12-06Bibliographically approved
Křepela, M. (2014). Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function. Zeitschrift für Analysis und ihre Anwendungen, 33(4), 369-383
Open this publication in new window or tab >>Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function
2014 (English)In: Zeitschrift für Analysis und ihre Anwendungen, ISSN 0232-2064, E-ISSN 1661-4534, Vol. 33, no 4, p. 369-383Article in journal (Refereed) Published
Abstract [en]

We characterize boundedness of a convolution operator with a fixed kernel between the classes S p ( v), de fined in terms of oscillation, and weighted Lorentz spaces Gamma(q)(w), defined in terms of the maximal function, for 0 < p; q <= infinity. We prove corresponding weighted Young-type inequalities of the form parallel to f * g parallel to Gamma(q)(w) <= C parallel to f parallel to S-p(v)parallel to g parallel to Y and characterize the optimal rearrangement-invariant space Y for which these inequalities hold.

Keywords
Convolution, Young inequality, weighted Lorentz spaces, oscillation
National Category
Mathematics
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-41570 (URN)10.4171/ZAA/1517 (DOI)000347639000001 ()
Available from: 2016-04-22 Created: 2016-04-11 Last updated: 2017-11-30Bibliographically approved
Křepela, M. (2014). Convolution inequalities in weighted Lorentz spaces. Mathematical Inequalities & Applications, 17(4), 1201-1223
Open this publication in new window or tab >>Convolution inequalities in weighted Lorentz spaces
2014 (English)In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 17, no 4, p. 1201-1223Article in journal (Refereed) Published
Abstract [en]

We characterize boundedness of a convolution operator with a fixed kernel between the weighted Lorentz spaces Lambda(p)(v) and Gamma(q)(w) for 0 < p <= q <= infinity, 1 <= q < p < infinity and 0 < q <= p = infinity. We provide corresponding weighted Young-type inequalities and also study basic properties of some new involved r.i. spaces.

Place, publisher, year, edition, pages
Croatia: Element, 2014
Keywords
Convolution, Young inequality, O’Neil inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-31751 (URN)10.7153/mia-17-90 (DOI)000345462600001 ()
Available from: 2014-03-23 Created: 2014-03-23 Last updated: 2017-12-06Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-0234-1645

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