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  • 1.
    Gogatishvili, Amiran
    et al.
    Czech Acad Sci, Inst Math, Prague, Czech Republic.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Univ Freiburg, Inst Math.
    Pick, Lubos
    Charles Univ Prague, Czech republic.
    Soudsky, Filip
    Czech Tech Univ; Univ South Bohemia, Czech republic.
    Embeddings of Lorentz-type spaces involving weighted integral means2017In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 273, no 9, p. 2939-2980Article in journal (Refereed)
    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.

  • 2.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Bilinear weighted Hardy inequality for nonincreasing functions2017In: Publications mathématiques (Bures-sur-Yvette), ISSN 0073-8301, E-ISSN 1618-1913, Vol. 61, no 1, p. 3-50Article in journal (Refereed)
  • 3.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Charles University in Prague, Department of Mathematical Analysis.
    Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$2017In: Revista Matemática Complutense, ISSN 1139-1138, E-ISSN 1988-2807, Vol. 30, no 3, p. 547-587Article in journal (Refereed)
    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.

  • 4.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. Charles University in Prague, Department of Mathematical Analysis.
    Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function2014In: Zeitschrift für Analysis und ihre Anwendungen, ISSN 0232-2064, E-ISSN 1661-4534, Vol. 33, no 4, p. 369-383Article in journal (Refereed)
    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.

  • 5.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Convolution in weighted Lorentz spaces of type Γ2016In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, p. 113-132Article in journal (Refereed)
  • 6.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Convolution inequalities in weighted Lorentz spaces2014In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 17, no 4, p. 1201-1223Article in journal (Refereed)
    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.

  • 7.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Charles University in Prague, Department of Mathematical Analysis.
    Convolution inequalities in weighted Lorentz spaces: case 0<q<12017In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 20, no 1, p. 191-201Article in journal (Refereed)
    Abstract [en]

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

  • 8.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Forever Young: Convolution Inequalities in Weighted Lorentz-type Spaces2014Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is devoted to an investigation of boundedness of a general convolution operator between certain weighted Lorentz-type spaces with the aim of proving analogues of the Young convolution inequality for these spaces.

    Necessary and sufficient conditions on the kernel function are given, for which the convolution operator with the fixed kernel is bounded between a certain domain space and the weighted Lorentz space of type Gamma. The considered domain spaces are the weighted Lorentz-type spaces defined in terms of the nondecreasing rearrangement of a function, the maximal function or the difference of these two quantities.

    In each case of the domain space, the corresponding Young-type convolution inequality is proved and the optimality of involved rearrangement-invariant spaces in shown.

    Furthermore, covering of the previously existing results is also discussed and some properties of the new rearrangement-invariant function spaces obtained during the process are studied.

  • 9.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Integral conditions for Hardy-type operators involving suprema2017In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 68, no 1, p. 21-50Article in journal (Refereed)
    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.

  • 10.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. Charles University in Prague, Department of Mathematical Analysis.
    Iterating bilinear Hardy inequalities2017In: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839Article in journal (Refereed)
    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.

  • 11.
    Křepela, Martin
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    The Weighted Space Odyssey2017Doctoral 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.

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