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• 151.
University of Memphis, Department of Mathematical Sciences.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Convexity, concavity, type and cotype of Lorentz spaces1998In: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 9, no 3, p. 367-382Article in journal (Refereed)

The purpose of this article is to present necessary and sufficient conditions on convexity and concavity, lower and upper estimates and type and cotype of weighted Lorentz spaces Λp,w with 1 ≤ p < ∞ and a decreasing weight w.

• 152.
Department of Mathematical Sciences, University of Memphis.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Indices and regularizations of measurable functions2000In: Function spaces: the fifth conference : proceedings at the conference at Poznan, Poland, New York: Marcel Dekker Incorporated , 2000, p. 231-246Chapter in book (Other academic)
• 153.
Department of Mathematical Sciences, University of Memphis.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Indices, convexity and concavity of Calderón-Lozanovskii spaces2003In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 92, no 1, p. 141-160Article in journal (Refereed)

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space Eφ, generated by a quasi-Banach space E and an increasing Orlicz function φ. We give estimations of convexity and concavity indices of Eφ in terms of Matuszewska-Orlicz indices of φ as well as convexity and concavity indices of E. In the case when Eφ is a rearrangement invariant space we also provide some estimations of its Boyd indices. As corollaries we obtain some necessary and sufficient conditions for normability of Eφ, and conditions on its nontrivial type and cotype in the case when Eφ is a Banach space. We apply these results to Orlicz-Lorentz spaces receiving estimations, and in some cases the exact values of their convexity, concavity and Boyd indices

• 154.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Type, cotype and convexity properties of Orlicz spaces1997Report (Other academic)
• 155.
Kyushu Institute of Technology. Department of Mathematics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Okayama Prefactural University. Department of Systems Engineering.
Clarkson type inequalities and their relations to the concepts of type and cotype2000In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 51, no 3, p. 327-346Article in journal (Refereed)

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman's observation on Clarkson's inequalities and type. A similar investigation concerning the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several classical and some recent results of this type.

• 156.
Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
Persson, Lars-ErikLuleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Selected topics in mathematics: proceedings of the First Nordic Summer School for Female PhD Students of Mathematics1997Collection (editor) (Other academic)
• 157. Kokilashvili, Vakhtang
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Weighted norm inequalities for integral transforms with product kernels2009Book (Other academic)

The book may be considered as a systematic and detailed analysis of a wide class of integral transforms with product kernels from the two weighted boundedness point of view. The considered product kernels cover that case when factors of kernels have essential (less than one) singularities. The book intends to make a breakthrough in two directions: to cover multidimensional potentials, Hilbert transforms, strong maximal functions and at the same time, to present solutions of two weighted problems for them which are much more complicated than one weighted ones. In the given monograph two weighted boundedness criteria for multiple Hardy transforms is reflected in the case when the two dimensional weight on the right hand side of an appropriate inequality is a product of two weight functions of single variable. In this case we present simpler and transparent criteria than those of E Sawyer for general weights. Moreover, we prove some new multidimensional Hardy type inequalities with general kernels. Weighted integral in equalities for monotonic functions of several variables are also discussed. The main subjects of this book can be useful for applications both within various areas of the mathematical sciences (e.g. Fourier and Harmonic analysis, Fractional Calculus, BVP of PDE in Mathematical Physics, Stochastic Processes, Error Estimates in Numerical Analysis, etc.) as well as directly in some applied sciences.

• 158.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On summability of the Fourier coefficients for functions from some Lorentz type spaces2009Report (Other academic)
• 159.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. LN Gumilyov Eurasian Natl Univ, Fac Mech & Math, Astana, Kazakhstan.
RUDN Univ, Moscow, Russia. Lomonosov Moscow State Univ, Kazakhstan Branch, Astana, Kazakhstan. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Artic Univ Norway, UiT, Narvik, Norway.
A new generalization of Boas theorem for some Lorentz spaces lambda(q)(omega)2018In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 12, no 3, p. 619-633Article in journal (Refereed)

Let Lambda(q)(omega), q > 0, denote the Lorentz space equipped with the (quasi) norm parallel to f parallel to(Lambda q(omega)) := (integral(1)(0) (f*(t)omega(t))(q)dt/t)(1/q) for a function integral on [0,1] and with omega positive and equipped with some additional growth properties. A generalization of Boas theorem in the form of a two-sided inequality is obtained in the case of both general regular system Phi = {phi(k)}(k=1)(infinity) and generalized Lorentz Lambda(q) (omega) spaces.

• 160.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Faculty of Mechanics and Mathematics L. N. Gumilyov, Eurasian National University.
Kazakhstan Branch of Lomonosov, Moscow State University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new two-sided inequalities concerning the Fourier transform2017In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 20, no 3, p. 855-864Article in journal (Refereed)

The classical Hausdorff-Young and Hardy-Littlewood-Stein inequalities do not hold for p > 2. In this paper we prove that if we restrict to net spaces we can even derive a two-sided estimate for all p > 1. In particular, this result generalizes a recent result by Liflyand E. and Tikhonov S. [7] (MR 2464253).

• 161.
Kazakh Branch of Moscow State University, Astana.
Kazakh Branch of Moscow State University, Astana. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Relations between summability of the Fourier coefficients in regular systems and functions from some Lorentz type spaces2010In: Proceedings of A. Razmadze Mathematical Institute, ISSN 1512-0007, Vol. 152, p. 73-88Article in journal (Refereed)
• 162.
Faculty of Mechanics and Mathematics L. N. Gumilyov Eurasian National University, Astana, Kazakhstan.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces2010In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 1, no 2, p. 76-85Article in journal (Refereed)
• 163.
Eurasian National University, Astana.
Eurasian National University, Astana. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On inequalities for the Fourier transform of functions from Lorentz spaces2011In: Russian Academy of Sciences. Mathematical Notes, ISSN 1067-9073, Vol. 90, no 5-6, p. 767-770Article in journal (Refereed)
• 164.
Luleå University of Technology, Department of Engineering Sciences and Mathematics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics.
On Friedrichs-type estimates in domains with rapidly vanishing perforation along the boundary2011In: Book of Abstracts of the International Conference "Differential Equations and Related Topics'' dedicated to the Centenary Anniversary of Ivan G.Petrovskii: (XXIII Joint Session of Petrovskii Seminar and Moscow Mathematical Society) (May 29-June 4, 2011, Moscow, Russia), Moscow: Moscow State University Press, 2011, p. 65-66Conference paper (Refereed)
• 165.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On Friedrichs-type inequalities in domains rarely perforated along the boundary2011Report (Other academic)
• 166.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On Friedrichs-type inequalities in domains rarely perforated along the boundary2011In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2011Article in journal (Refereed)

This paper is devoted to the Friedrichs inequality, where the domain isperiodically perforated along the boundary. It is assumed that the functionssatisfy homogeneous Neumann boundary conditions on the outer boundary andthat they vanish on the perforation. In particular, it is proved that thebest constant in the inequality converges to the best constant in aFriedrichs-type inequality as the size of the perforation goes to zero muchfaster than the period of perforation. The limit Friedrichs-type inequalityis valid for functions in the Sobolev space $H^{1}$.

• 167.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On quantizer distortion and the upper bound for exponential entropy1991In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 37, no 4, p. 1168-1172Article in journal (Refereed)

A sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions. This result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources. It is known that for a large class of quantizers (both zero-memory and vector) the rth power distortion, as well as some other distortion criteria, are bounded below by a constant, depending on r, multiplied by a certain integral of the source's probability density. It is pointed out that this bound can be rewritten in terms of the exponential entropy. The exponential entropy measures the quantitative extent or range of the source distribution. This fact gives a physical interpretation of the indicated limits of quantizer performance, further elucidated by the main result

• 168.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some properties of generalized exponential entropies with applications to data compression1992In: Information Sciences, ISSN 0020-0255, E-ISSN 1872-6291, Vol. 62, no 1, p. 103-132Article in journal (Refereed)

I. Csiszár discussed generalized entropies in his lecture at the Sixth Prague Conference on Information Theory. The authors emphasize that Csiszár noted the link between certain lower bounds for the quantization error and Rényi's differential entropy of order $\alpha$. Another important reference is the paper by L. L. Campbell where the concept of an exponential entropy was introduced. The authors investigate "several consequences that are of interest in the theory of data (or signal) compression". They also "investigate especially the exponential families of distributions, in particular the Miller-Thomas (or generalized Gaussian) family of distributions". The paper is a detailed discussion of the aforementioned problems coupled with examples and details of the possible applications. Exponential entropy is calculated for the uniform distribution, the univariate Gaussian distribution, the Laplace distribution, the Miller-Thomas distribution, an infinite-dimensional Gaussian exponential family, the Gauss-Laplace mixture and the multivariate Gaussian distribution. The extent of a distribution is given for the shape parameter in the Miller-Thomas distribution. Campbell's representation for E$[\alpha, 1 ; f]$ and the connection between an entropy series and data compression are discussed. A lower bound for the entropy of a partition (as defined in the paper) is given. Examples and proofs are illustrated with outputs from Mathematica.

• 169. Koski, Timo
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, University of Stockholm.
ε-entropy, ε-rate, and interpolation spaces revisited with an application to linear communication channels1994In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 186, no 1, p. 265-276Article in journal (Refereed)

Some results concerning the ϵ-rate and the ϵ-entropy presented by one of the authors (cf. [Peetre, Ricerche Mat. 17 (1968), pp. 216-220]) are investigated, updated, and generalized in various ways. In addition we give some examples, as well as a discussion, related to the problems presented by Hajela and Honig of this interpolation theory technique in an abstract model of digital communication.

• 170. Kruglyak, Natan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A Carlson type inequality with blocks and interpolation1993In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 104, no 2, p. 161-180Article in journal (Refereed)
• 171. Kruglyak, Natan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On an elementary approach to the fractional Hardy inequality2000In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 128, no 3, p. 727-734Article in journal (Refereed)

Let be the usual Hardy operator, i.e., . We prove that the operator is bounded and has a bounded inverse on the weighted spaces for -1$" type="#_x005F_x0000_t75">and . Moreover, by using these inequalities we derive a somewhat generalized form of some well-known fractional Hardy type inequalities and also of a result due to Bennett-DeVore-Sharpley, where the usual Lorentz norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential. • 172. Kruglyak, Natan Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. The failure of the Hardy inequality and interpolation of intersections1999In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 37, no 2, p. 323-244Article in journal (Refereed) The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=−1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L p(w0), N∩Lp(w1)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L p(w0),N∩L p (w1)), which turns out to be essentially different from theK-functional for (L p(w0), Lp(w1)), even for the case whenN∩L p(wi) is dense inL p(wi) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result. • 173. Kruglyak, Natan Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Structure of the Hardy operator related to Laguerre polynomials and Euler differential equation2006In: Revista Matematica Complutense, ISSN 1139-1138, Vol. 19, no 2, p. 467-476Article in journal (Refereed) • 174. University of Zagreb. University of Zagreb. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Some new Hardy type inequalities with general kernels2009In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 12, no 3, p. 473-485Article in journal (Refereed) We state and prove some new weighted Hardy type inequalities with an integral operator A(k) defined by A(k)f(x) := 1/K(x) integral(Omega 2) k(x,y)f(y)d mu(2) (y), where k : Omega(1) x Omega(2) --> R is a general nonnegative kernel, (Omega(1), mu(1)) and (Omega(2), mu(2)) are measure spaces and K(x) := integral(Omega 2) k(x,y)d mu(2) (y), x is an element of Omega(1). In particular, the obtained results unify and generalize most of the results of this type (including the classical ones by Hardy, Hilbert and Godunova). • 175. Department of Mathematics, University of West Bohemia. Department of Mathematics, University of West Bohemia. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Some higher order Hardy inequalities2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed) We investigate the k-th order Hardy inequality (1-1) for functions satisfying rather general boundary conditions (1-2), show which of these conditions are admissible and derive sufficient, and necessary and sufficient, conditions (for 0 1) on u, v for (1-1) to hold. • 176. Mathematical Institute, Academy Sciences of the Czesh Republic. Department of Mathematics, University of West Bohemia. Department of Mathematics, University of Agriculture, Abeokuta. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Generalized weighted inequality with negative powers2007In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 1, no 2, p. 269-280Article in journal (Refereed) In this paper necessary and sufficient conditions for the validity of the generalized Hardy inequality for the case -∞ < q p < 0 and 0 < p q < 1 are derived. Furthermore, some special cases are considered • 177. Department of Mathematics, University of West Bohemia. Department of Mathematics, University of West Bohemia. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Some higher order Hardy inequalities2011Report (Other academic) • 178. Kufner, Alois Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. The Hardy inequality: about its history and some related results2007Book (Other academic) The Hardy inequality has a fascinating past and will have (hopefully) also a fascinating future. Here, the authors present some important steps of the development of the classical Hardy inequality , of its early weighted generalizations (i.e. furnished with general weights) and of its various modifications and extensions. Besides the continuous (i.e. integral) case, the (originally) discrete one (i.e. for sequences) is dealt with. Eighteen theorems are formulated, most of them with proof. Although the choice of material is a matter of personal taste and knowledge, the authors intended to include as many results as possible, using papers published up to 2005. • 179. Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. The prehistory of the Hardy inequality2006In: The American mathematical monthly, ISSN 0002-9890, E-ISSN 1930-0972, Vol. 113, no 8, p. 715-732Article in journal (Refereed) • 180. Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Hardy inequalities of fractional order via interpolation1994In: Inequalities and applications, Singapore: World Scientific Publishing Co Pte Ltd , 1994, p. 417-430Chapter in book (Other academic) • 181. Kufner, Alois Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Integral inequalities with weights2000Book (Other academic) • 182. Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Weighted inequalities of Hardy type2003Book (Other academic) • 183. Mathematical Institute, Czech Academy of Sciences, Department of Mathematics, University of West Bohemia. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Hardy type inequalities with kernels: The current status and some new results2017In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 290, no 1, p. 57-65Article in journal (Refereed) We consider the general Hardy type operator inline image where inline image is a positive and measurable kernel. To characterize the weights u and v so that inline image is still an open problem for any parameters p and q. However, for special cases the solution is known for some parameters p and q. In this paper the current status of this problem is described and discussed mainly for the case inline image In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised. • 184. Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Instituto Superior Tecnico, Research center CEAF. Some New Scales of Weight Characterizations of Hardy-type Inequalities2013In: Operator Theory, Pseudo-Differential Equations, and Mathematical Physics: The Vladimir Rabinovich Anniversary Volume, Basel: Encyclopedia of Global Archaeology/Springer Verlag, 2013, p. 261-274Chapter in book (Refereed) In this paper we present, discuss and illustrate some new scales of conditions to characterize modern forms of Hardy′s inequalities which can not be found in the newest books in this area. Moreover, some results of importance as motivation for these scales are presented and discussed in a historical perspective. • 185. Mathematical Institute, Czech Academy of Sciences. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Weighted inequalities of Hardy type2017 (ed. 2)Book (Other academic) In this second edition, all chapters in the first edition have been updated with new information. Moreover, a new chapter contains new and complementary information concerning: (a) a convexity approach to prove and explain Hardy-type inequalities; (b) sharp constants; (c) scales of inequalities to characterize Hardy-type inequalities; (d) Hardy-type inequalities in other function spaces; and (e) a number of new open questions. • 186. Academy of Sciences of the Czech Republic. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. A study of some constants characterizing the weighted Hardy inequality2004In: Orlicz centenary volume: proceedings of the conferences "The Wladyslaw Orlicz centenary conference" and "Function spaces VII2", Poznan, Poland, July 21--25, 2003 / [ed] Zbigniew Cieselski; Alesander Pelczynski; Leszek Skrzypczak, Institute of Mathematics, Polish Academy of Sciences , 2004, Vol. Volume I: Plenary lectures, p. 135-146Conference paper (Refereed) The modern form of Hardy's inequality means that we have a necessary and sufficient condition on the weights$u$and~$v$on$[0,b]$so that the mapping $$H:L^p(0,b;v)\rightarrow L^q(0,b;u)$$ is continuous, where$Hf(x)=\int_{0}^xf(t)\,dt$is the Hardy operator. We consider the case$1

• 187.
Luleå tekniska universitet.
Luleå tekniska universitet. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Parabolic variational inequalities with singularities2007Report (Other academic)
• 188.
University of West Bohemia.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
An extension of Rothe's method to noncylindrical domains2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 5, p. 365-389Article in journal (Refereed)

In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied

• 189.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Elliptic variational inequalities on weighted Sobolev spaces2006Report (Other academic)
• 190.
Department of Mathematics, Uppsala University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, University of Zagreb.
Multiplicative inequalities of Carlson type and interpolation2006Book (Other academic)
• 191.
Department of Mathematics, Uppsala University.
Faculty of Textile Technology. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
An extension of the Landau and Levin-Steckin inequalities2004In: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 70, no 1-2, p. 25-34Article in journal (Refereed)

The authors prove an inequality for sums, which generalizes both Landau's sharpening of Carlson's inequality and the corresponding complementary result by Levin and Ste\v{c}kin. The inequality is optimal, in the sense that necessary and sufficient conditions on the parameters for which the inequality holds are given. In some cases, sharp constants are obtained, also in situations not covered by the classical results.

• 192.
Department of Mathematics, Uppsala University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Carlson's inequality and interpolation2006In: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 55, no 3, p. 182-188Article in journal (Refereed)

In this survey, we explain and discuss some recent results concerning the close connection between Carlson type inequalities and interpolation theory. In particular, we point out that a fairly general Carlson type inequality can be used to extend the usefulness of the Gustavsson-Peetre $\langle·\rangle_\phi$ interpolation method

• 193.
Department of Mathematics, Uppsala University.
Institute of Mathematics and Informatics, Lajos Kossuth University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Carlson type inequalities for finite sums and integrals on bounded intervals2005In: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 71, no 2, p. 275-284Article in journal (Refereed)

We investigate Carlson type inequalities for finite sums, that is, inequalities of the form ∑mk=1ak < C (∑mk=1ka1akr+1) μ(∑mk=1ka2akr+1)λ, to hold for some constant C independent of the finite, non-zero set a1,⋯,am of non-negative numbers. We find constants C which are strictly smaller than the sharp constants in the corresponding infinite series case. Moreover, corresponding results for integrals over bounded intervals are given and a case with any finite number of factors on the right-hand side is proved

• 194.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A study of bending waves in infinite and anisotropic plates1997In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 42, no 3, p. 213-232Article in journal (Refereed)

In this paper we present a unified approach to obtain integral representation formulas for describing the propagation of bending waves in infinite plates. The general anisotropic case is included and both new and well-known formulas are obtained in special cases (e.g. the classical Boussinesq formula). The formulas we have derived have been compared with experimental data and the coincidence is very good in all cases

• 195.
Collège de France.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of monotone operators2000In: Comptes rendus de l'Académie des sciences. Série 1, Mathématique, ISSN 0764-4442, E-ISSN 1778-3577, Vol. 330, no 8, p. 675-680Article in journal (Refereed)

In this Note we study reiterated homogenization of nonlinear equations of the form −div(a(x/,x/2,Du))=f, where a is periodic in the first two arguments and monotone in the third. We state that u converges weakly in W1,p(Ω) (and even in some multiscale sense), as →0, to the solution u0 of a limit problem. Moreover, we give an explicit expression for the limit problem.

• 196.
Coolège de France.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of nonlinear monotone operators2001In: Chinese Annals of Mathematics. Series B, ISSN 0252-9599, E-ISSN 1860-6261, Vol. 22B, no 1, p. 1-12Article in journal (Refereed)

In this paper, the authors study reiterated homogenization of nonlinear equations of the form -div(a(x, x/ε, x/ε2,Duε)) = f, where a is periodic in the first two arguments and monotone in the third. It is proved that uε converges weakly in W1,p(Ω) (and even in some multiscale sense), as ε → 0 to the solution u0 of a limit problem. Moreover, an explicit expression for the limit problem is given. The main results were also stated in [15]. This article presents the complete proofs of these results

• 197. Lukkassen, Dag
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå tekniska universitet.
Computing effective conductivities by bounds and numerical methods1997In: ICCE 4: Fourth International Conference on Composites Engineering / [ed] David Hui, 1997, p. 199-200Conference paper (Refereed)
• 198.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Narvik University College, 8505 Narvik, Norway. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations2012In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 35, no 11, p. 1300-1311Article in journal (Refereed)

We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces L p,Ψ(ℝ n,w), defined by an almost increasing function Ψ(r) and radial type weights. We obtain sufficient conditions, in terms of numerical characteristics, that is, index numbers of the weight functions and the function Ψ. In relation with the wide usage of singular integral equations in applications, we show how the solvability of such equations in the generalized Morrey spaces depends on the main characteristics of the space, which allows to better control both the singularities and regularity of solutions

• 199.
Narvik University College, Narvik, Norway. Norut Narvik, Narvik, Norway.
Narvik University College, Narvik, Norway. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Nonlinear variational methods for estimating effective properties of multiscale materials2010In: Nonlinear analysis and variational problems: in honor of George Isac, Berlin: Encyclopedia of Global Archaeology/Springer Verlag, 2010, p. 367-414Chapter in book (Other academic)

We consider homogenization of sequences of integral functionals with natural growth conditions. Some means are analyzed and used to discuss some fairly new bounds for the homogenized integrand corresponding to integrands which are periodic in the spatial variable. These bounds, which are obtained by variational methods, are compared with the nonlinear bounds of Wiener and Hashin-Shtrikman type. We also point out conditions that make our bounds sharp. Several applications are presented. Moreover, we also discuss bounds for some linear reiterated two-phase problems with m different micro-levels in the spatial variable. In particular, the results imply that the homogenized integrand becomes optimal as m turns to infinity. Both the scalar case (the conductivity problem) and the vector-valued case (the elasticity problem) are considered. In addition, we discuss bounds for estimating the effective behavior described by homogenizing a problem which is a generalization of the Reynold equation.

• 200. Lukkassen, Dag
Centre of Mathematics, Lund University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On some iterated means arising in homogenization theory2004In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 49, no 4, p. 343-356Article in journal (Refereed)

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.

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