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  • 1.
    Algervik, Robert
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On Fournier-Gagliardo mixed norm spaces2011In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 36, p. 493-508Article in journal (Refereed)
    Abstract [en]

    We study mixed norm spaces

    V (Rn)

    that arise in connection with embeddings of

    Sobolev spaces

    W

    1

    1

    (Rn). We prove embeddings of V (Rn)

    into Lorentz type spaces defined in terms

    of iterative rearrangements. Basing on these results, we introduce the scale of mixed norm spaces

    V

    p

    (Rn). We prove that V ½ V p

    and we discuss some questions related to this embedding.

  • 2.
    Barza, Sorina
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Soria, Javier
    Department of Applied Mathematics and Analysis, University of Barcelona, Spain.
    Sharp constants related to the triangle inequality in Lorentz spaces2009In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, no 10, p. 5555-5574Article in journal (Refereed)
  • 3.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity2005In: Russian Math. Surveys, ISSN 0036-0279, Vol. 60, no 6, p. 1147-1164Article in journal (Refereed)
  • 4.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Iterated rearrangements and Gagliardo-Sobolev type inequalities2012In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, Vol. 387, no 1, p. 335-348Article in journal (Refereed)
    Abstract [en]

    In this paper we consider Lorentz type spaces defined in terms of iterated rearrangements of functions of several variables (σ is a permutation of {1,…,n}). Further, we study Fournier–Gagliardo mixed norm spaces V(Rn) closely related to Sobolev spaces . We prove estimate of via ‖fV with the sharp constant. In particular, this gives a refinement of the known Sobolev type inequalities for the space .

  • 5.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Lp-moduli of continuity of sections of functions2010In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 22, no 1, p. 53-73Article in journal (Refereed)
  • 6.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Mixed norms and Sobolev type inequalities2006In: Banach Center Publications, ISSN 0137-6934, E-ISSN 1730-6299, Vol. 72, p. 141-160Article in journal (Refereed)
  • 7.
    Kolyada, Viktor
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    My First Meetings with Konstantin Oskolkov2013In: Recent Advances in Harmonic Analysis and Applications: In honor of Konstantin Oskolkov / [ed] Dmitriy Bilyk, Laura De Carli, Alexander Petukhov, Alexander M. Stokolos, Brett D. Wick, Springer, 2013, 1, Vol. 25, p. 27-29Chapter in book (Refereed)
    Abstract [en]

    This note tells about our first meetings with Konstatin Oskolkov. We discuss also optimal estimates of the rate of convergence of Fourier series obtained by Oskolkov in 1975.

  • 8.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On embedding theorems: survey paper2007In: Proceedings of NAFSA 2007, 2007, Vol. 8, p. 35-94Conference paper (Refereed)
  • 9.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On Fubini type property in Lorentz spaces2012In: Recent Advances in Harmonic Analysis and Applications / [ed] Bilyk, D., New York: Springer, 2012, p. 171-179Chapter in book (Refereed)
    Abstract [en]

    We study Fubini-type property for Lorentz spaces L p,r (R 2 ) . This problem is twofold. First we assume that all linear sections of a function f in directions of coordinate axes belong to L p,r (R) , and their one-dimensional Lp, r-norms belong to L p,r (R). We show that for pr it does not imply that f∈L p,r (R 2 ) (this complements one result by Cwikel). Conversely, we assume that f∈L p,r (R 2 ) , and we show that then for r < p almost all linear sections of f belong to L p,r (R) , but for p < r all linear sections may have infinite one-dimensional Lp, r-norms

  • 10.
    Kolyada, Viktor
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    On limiting relations for capacities2013In: Real Analysis Exchange, ISSN 0147-1937, Vol. 38, no 1, p. 211-240Article in journal (Refereed)
  • 11.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Optimal relationships between L-p-norms for the Hardy operator and its dual2014In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 2, p. 423-430Article in journal (Refereed)
    Abstract [en]

    We obtain sharp two-sided inequalities between -norms of functions and , where is the Hardy operator, is its dual, and is a nonnegative measurable function on In an equivalent form, it gives sharp constants in the two-sided relationships between -norms of functions and , where is a nonnegative nonincreasing function on with In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005-2013, 2008) for and by Boza and Soria (J Funct Anal 260:1020-1028, 2011) for all , and gives a sharp version of this result for 1 < p < 2.

  • 12.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Sections of Functions and Sobolev-Type Inequalities2014In: Proceedings of the Steklov Institute of Mathematics, ISSN 0081-5438, E-ISSN 1531-8605, Vol. 284, no 1, p. 192-203Article in journal (Refereed)
    Abstract [en]

    We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy the Lipschitz condition of order 0 < alpha a parts per thousand currency sign 1. We prove that if for a function f the Lip alpha-norms of these sections belong to the Lorentz space L (p,1)(a"e) (p = 1/alpha), then f can be modified on a set of measure zero so as to become bounded and uniformly continuous on a"e(2). For alpha = 1 this gives an extension of Sobolev's theorem on continuity of functions of the space W (1) (2,2) (a"e(2)). We show that the exterior L (p,1)-norm cannot be replaced by a weaker Lorentz L (p,q) -norm with q > 1.

  • 13.
    Kolyada, Viktor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Transplantation theorems for ultraspherical polynomials in Re H and BMO2005In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 22, no 2, p. 149-191Article in journal (Refereed)
  • 14.
    Kolyada, Viktor
    et al.
    Karlstad University, Division for Engineering Sciences, Physics and Mathematics.
    Lerner, A.K.
    On limiting embeddings of Besov spaces2005In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 171, p. 1-13Article in journal (Refereed)
  • 15.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Lind, Martin
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On functions of bounded p-variation2009In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 356, no 2, p. 582-604Article in journal (Refereed)
    Abstract [en]

    We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function fLp[0,1] in terms of its Lp-modulus of continuity ωp(f;δ). These estimates are sharp for any rate of the decay of ωp(f;δ). Moreover, the constant coefficients in them depend on parameters in an optimal way.

  • 16.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Lind, Martin
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On moduli of p-continuity2012In: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 137, no 3, p. 191-213Article in journal (Refereed)
    Abstract [en]

    Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove sharp estimates of the modulus of p-continuty in terms of the modulus of q-continuity (1<p<q<\infty).

  • 17.
    Kolyada, Viktor
    et al.
    Karlstad University, Division for Engineering Sciences, Physics and Mathematics.
    Marcellán, F.
    Kernels and best approximations related to the system of ultraspherical polynomials2005In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 133, p. 173-194Article in journal (Refereed)
  • 18.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Perez Lazaro, F. J.
    Univ La Rioja, Dept Matemat & Comp, Logrono 26004, Spain..
    On Gagliardo-Nirenberg Type Inequalities2014In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 20, no 3, p. 577-607Article in journal (Refereed)
    Abstract [en]

    We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of a function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev type embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities in some limiting cases. Our methods are based on estimates of rearrangements in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with p = 1.

  • 19.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Perez Lazaro, Francisco Javier
    Inequalities for partial moduli of continuity and partial derivatives2011In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 34, no 1, p. 23-59Article in journal (Refereed)
    Abstract [en]

    We obtain pointwise and integral type estimates of higher-order partial moduli of continuity in C via partial derivatives. Also, a Gagliardo–Nirenberg type inequality for partial derivatives in a fixed direction is proved. Our methods enable us to study the case when different partial derivatives belong to different spaces, including the space L

  • 20.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Pérez, F.J.
    Estimates of difference norms for functions in anisotropic Sobolev spaces2004In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 267, p. 46-64Article in journal (Refereed)
  • 21.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Roncal Gómez, L
    Jacobi Transplantations and Weyl integrals2010In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 61, no 2, p. 205-221Article in journal (Refereed)
  • 22.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Soria, Javier
    Hölder type inequalities in Lorentz spaces2010In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 189, no 3, p. 523-538Article in journal (Refereed)
  • 23.
    Kolyada, Viktor
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    Soria, Javier
    Univ Barcelona, Dept Appl Math & Anal, Gran Via 585, E-08007 Barcelona, Spain..
    Mixed Norms and Iterated Rearrangements2016In: Zeitschrift für Analysis und ihre Anwendungen, ISSN 0232-2064, E-ISSN 1661-4534, Vol. 35, no 2, p. 119-138Article in journal (Refereed)
    Abstract [en]

    We prove sharp estimates, and find the optimal range of indices, for the comparison of mixed norms for both functions and their iterated rearrangements.

1 - 23 of 23
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