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Journal articles on the topic 'Hardy's inequalities'

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Published: 6 September 2023

Last updated: 7 September 2023

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1

Chen, Xu, Hong-Yi Su, Zhen-Peng Xu, Yu-Chun Wu, and Jing-Ling Chen. "Quantum nonlocality enhanced by homogenization." International Journal of Quantum Information 12, no. 06 (September 2014): 1450040. http://dx.doi.org/10.1142/s0219749914500403.

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Homogenization proposed in [Y.-C Wu and M. Żukowski, Phys. Rev. A 85 (2012) 022119] is a procedure to transform a tight Bell inequality with partial correlations into a full-correlation form that is also tight. In this paper, we check the homogenizations of two families of n-partite Bell inequalities: the Hardy inequality and the tight Bell inequality without quantum violation. For Hardy's inequalities, their homogenizations bear stronger quantum violation for the maximally entangled state; the tight Bell inequalities without quantum violation give the boundary of quantum and supra-quantum, but their homogenizations do not have the similar properties. We find their homogenization are violated by the maximally entangled state. Numerically computation shows the the domains of quantum violation of homogenized Hardy's inequalities for the generalized GHZ states are smaller than those of Hardy's inequalities.

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2

Lefèvre, Pascal. "Weighted discrete Hardy's inequalities." Ukrains’kyi Matematychnyi Zhurnal 75, no. 7 (July 25, 2023): 1009–12. http://dx.doi.org/10.37863/umzh.v75i7.7201.

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UDC 517.5 We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, Arch. Math. (Basel), <strong>114</strong>, № 2, 195–198 (2020)].

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3

Oguntuase, J., and B. Popoola. "Refinement of Hardy's Inequalities Involving Many Functions Via Superquadratic Functions." Annals of the Alexandru Ioan Cuza University - Mathematics 57, no. 2 (January 1, 2011): 271–83. http://dx.doi.org/10.2478/v10157-011-0026-z.

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Refinement of Hardy's Inequalities Involving Many Functions Via Superquadratic FunctionsSome new refined Hardy type integral inequalities involvingnfunctions (n∈ Z+) via superquadratic functions are established forp≥ 2 and their dual inequalities are also derived. In particular, the results obtained complement and improve some recent results of Oguntuase and Persson.

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4

Bicheng, Yang, and Lokenath Debnath. "Generalizations of Hardy's integral inequalities." International Journal of Mathematics and Mathematical Sciences 22, no. 3 (1999): 535–42. http://dx.doi.org/10.1155/s0161171299225355.

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This paper deals with some new generalizations of Hardy's integral inequalities. Some cases concerning whether the constant factors involved in these inequalities are best possible are discussed in some detail.

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5

SATAKE, Makoto. "Hardy's inequalities for Laguerre expansions." Journal of the Mathematical Society of Japan 52, no. 1 (January 2000): 17–24. http://dx.doi.org/10.2969/jmsj/05210017.

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6

Krnić, Mario, and Josip Pečarić. "General Hilbert's and Hardy's inequalities." Mathematical Inequalities & Applications, no. 1 (2005): 29–51. http://dx.doi.org/10.7153/mia-08-04.

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7

GALAKTIONOV, VICTOR A. "ON EXTENSIONS OF HARDY'S INEQUALITIES." Communications in Contemporary Mathematics 07, no. 01 (February 2005): 97–120. http://dx.doi.org/10.1142/s0219199705001659.

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8

Sababheh, Mohammad. "Hardy Inequalities on the Real Line." Canadian Mathematical Bulletin 54, no. 1 (March 1, 2011): 159–71. http://dx.doi.org/10.4153/cmb-2010-091-8.

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AbstractWe prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

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9

Ruzhansky, Michael, and Daulti Verma. "Hardy inequalities on metric measure spaces." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2223 (March 2019): 20180310. http://dx.doi.org/10.1098/rspa.2018.0310.

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In this note, we give several characterizations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on R n , on homogeneous groups, on hyperbolic spaces and on Cartan–Hadamard manifolds. We note that doubling conditions are not required for our analysis.

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10

C˘iz˘mes˘ija, Aleksandra, and Josip Pec˘arić. "Some new generalisations of inequalities of Hardy and Levin–Cochran–Lee." Bulletin of the Australian Mathematical Society 63, no. 1 (February 2001): 105–13. http://dx.doi.org/10.1017/s000497270001916x.

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In this paper finite versions of Hardy's discrete, Hardy's integral and the Levin–Cochran–Lee inequalities will be considered and some new generalisations of these inequalities will be derived. Moreover, it will be shown that the constant factors involved in the right-hand sides of the integral results obtained are the best possible.

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11

Pachpatte, B. G. "On a new class of Hardy type inequalities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 105, no. 1 (1987): 265–74. http://dx.doi.org/10.1017/s0308210500022095.

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SynopsisIn this paper we establish a new class of integral inequalities which originate from the well-known Hardy's inequality. The analysis used in the proofs is quite elementary and is based on the idea used by Levinson to obtain generalisations of Hardy's inequality.

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12

Mohapatra, R. N., and D. C. Russell. "Integral inequalities related to Hardy's inequality." Aequationes Mathematicae 28, no. 1 (December 1985): 199–207. http://dx.doi.org/10.1007/bf02189411.

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13

Love, E. R. "Generalizations of Hardy's Integral Inequality." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 100, no. 3-4 (1985): 237–62. http://dx.doi.org/10.1017/s0308210500013792.

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SynopsisExtensions of the integral version of Hardy's Inequality were given by Kadlec and Kufner (1967) and by Copson (1976). This paper provides several levels of further generalization of their results, obtained mostly by specializing four main inequalities. Most of the inequalities have the form ∥Kf∥ ≤ C ∥f∥, where K is an integral transform and ∥.∥ is a generalized Lp-norm; some have the inequality sign reversed. Best possible constants C are obtained in several cases, under mild extra hypotheses.

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14

Cižmešija, Aleksandra, and Josip Pecaric. "On Bicheng-Debnath's generalizations of Hardy's integral inequality." International Journal of Mathematics and Mathematical Sciences 27, no. 4 (2001): 237–50. http://dx.doi.org/10.1155/s0161171201005816.

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We consider Hardy's integral inequality and we obtain some new generalizations of Bicheng-Debnath's recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameterkfrom Hardy's original relation. Moreover, we prove the constant factors involved in the right-hand sides of some particular inequalities from both classes to be the best possible, that is, none of them can be replaced with a smaller constant.

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15

Zhongxue, Lü, and Xie Hongzheng. "On new generalizations of Hardy's integral inequalities." International Journal of Mathematics and Mathematical Sciences 30, no. 9 (2002): 515–20. http://dx.doi.org/10.1155/s0161171202111276.

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16

Mitrinović, Dragoslav S., and Josip E. Pečarić. "On inequalities of Hilbert and Widder." Proceedings of the Edinburgh Mathematical Society 34, no. 3 (October 1991): 411–14. http://dx.doi.org/10.1017/s0013091500005186.

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17

HWANG, DAH-YAN. "A MANY VARIABLE GENERALIZATION OF THE DISCRETE HARDY'S INEQUALITY." Tamkang Journal of Mathematics 27, no. 2 (June 1, 1996): 125–32. http://dx.doi.org/10.5556/j.tkjm.27.1996.4350.

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In the present note we establish the discrete Hardy's inequalities in many variables. The main tools used for deriving the inequalities are base on the Fubini's theorem and some application of the fundamental inequalities.

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18

Pečarić, J. E., and E. R. Love. "Still more generalizations of Hardy's inequality." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 59, no. 2 (October 1995): 214–24. http://dx.doi.org/10.1017/s1446788700038611.

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19

Kanjin, Yuichi. "Hardy's Inequalities for Hermite and Laguerre Expansions." Bulletin of the London Mathematical Society 29, no. 3 (May 1997): 331–37. http://dx.doi.org/10.1112/s0024609396002627.

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20

Colin, F., and Y. Hupperts. "Minimization problems related to generalized Hardy's inequalities." Nonlinear Analysis: Theory, Methods & Applications 52, no. 8 (March 2003): 1933–45. http://dx.doi.org/10.1016/s0362-546x(02)00287-0.

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21

Lam, Nguyen, Guozhen Lu, and Lu Zhang. "Geometric Hardy's inequalities with general distance functions." Journal of Functional Analysis 279, no. 8 (November 2020): 108673. http://dx.doi.org/10.1016/j.jfa.2020.108673.

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22

ng Liu, Xiaoj, Toshio Horiuchi, and Hiroshi Ando. "One dimensional weighted Hardy's inequalities and application." Journal of Mathematical Inequalities, no. 4 (2020): 1203–22. http://dx.doi.org/10.7153/jmi-2020-14-78.

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23

Wei, Shihshu Walter, and Ye Li. "Generalized sharp Hardy type and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds." Tamkang Journal of Mathematics 40, no. 4 (December 23, 2009): 401–13. http://dx.doi.org/10.5556/j.tkjm.40.2009.604.

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We prove generalized Hardy's type inequalities with sharp constants and Caffarelli-Kohn-Nirenberg inequalities with sharp constants on Riemannian manifolds $M$. When the manifold is Euclidean space we recapture the sharp Caffarelli-Kohn-Nirenberg inequality. By using a double limiting argument, we obtain an inequality that implies a sharp Hardy's inequality, for functions with compact support on the manifold $M $ (that is, not necessarily on a punctured manifold $ M \backslash \{ x_0 \} $ where $x_0$ is a fixed point in $M$). Some topological and geometric applications are discussed.

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24

Páles, Zsolt, and Lars-Erik Persson. "Hardy-type inequalities for means." Bulletin of the Australian Mathematical Society 70, no. 3 (December 2004): 521–28. http://dx.doi.org/10.1017/s0004972700034778.

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In this paper we consider inequalities of the form , Where M is a mean. The main results of the paper offer sufficient conditions on M so that the above inequality holds with a finite constant C. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.

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25

Yang, Gou-Sheng, and Dah-Yan Hwang. "Some refinements of Hardy's and Copson's inequality for convex function." Tamkang Journal of Mathematics 32, no. 1 (March 31, 2001): 33–37. http://dx.doi.org/10.5556/j.tkjm.32.2001.365.

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26

KANJIN, Yuichi. "Hardy's inequalities for Hermite and Laguerre expansions revisited." Journal of the Mathematical Society of Japan 63, no. 3 (July 2011): 753–67. http://dx.doi.org/10.2969/jmsj/06330753.

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27

Hanjš, Željko, E. R. Love, and Josip Pečarić. "On Some Inequalities Related to Hardy's Integral Inequality." Mathematical Inequalities & Applications, no. 3 (2001): 357–68. http://dx.doi.org/10.7153/mia-04-34.

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28

en Xiao, Jin, and Jian un He. "On Hardy's inequalities for the special Hermite expansions." Mathematical Inequalities & Applications, no. 2 (2017): 491–500. http://dx.doi.org/10.7153/mia-20-33.

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29

Pachpatte, B. G. "On some integral inequalities similar to Hardy's inequality." Journal of Mathematical Analysis and Applications 129, no. 2 (February 1988): 596–606. http://dx.doi.org/10.1016/0022-247x(88)90274-0.

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30

Hamza, A. E., M. A. Alghamdi, and S. A. Alasmi. "Quantum Lp-spaces and inequalities of Hardy's type." Journal of Mathematics and Computer Science 31, no. 03 (May 12, 2023): 274–86. http://dx.doi.org/10.22436/jmcs.031.03.04.

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31

Boggarapu, Pradeep, Luz Roncal, and Sundaram Thangavelu. "On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians." Communications on Pure & Applied Analysis 18, no. 5 (2019): 2575–605. http://dx.doi.org/10.3934/cpaa.2019116.

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32

Imoru, Christopher O. "On some extensions of Hardy’s inequality." International Journal of Mathematics and Mathematical Sciences 8, no. 1 (1985): 165–71. http://dx.doi.org/10.1155/s0161171285000151.

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33

Sayed, A. G., S. H. Saker, and A. M. Ahmed. "Some fractional dynamic inequalities on time scales of Hardy's type." Journal of Mathematics and Computer Science 23, no. 02 (October 15, 2020): 98–109. http://dx.doi.org/10.22436/jmcs.023.02.03.

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In this paper, we prove some new fractional dynamic inequalities on time scales of Hardy's type due to Yang and Hwang. The results will be proved by employing the chain rule, Hölder's inequality, and integration by parts on fractional time scales. Several well-known dynamic inequalities on time scales will be obtained as special cases from our results.

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34

Wang, Jianxiong. "L ^p Hardy's identities and inequalities for Dunkl operators." Advanced Nonlinear Studies 22, no. 1 (January 1, 2022): 416–35. http://dx.doi.org/10.1515/ans-2022-0020.

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Abstract The main purpose of this article is to establish the L p {L}^{p} Hardy’s identities and inequalities for Dunkl operator on any finite balls and the entire space R N {{\mathbb{R}}}^{N} . We also prove Hardy’s identities and inequalities on certain domains with distance function to the boundary ∂ Ω \partial \Omega . In particular, we use the notion of Bessel pairs introduced in Ghoussoub and Moradifam to extend Hardy’s identities for the classical gradients obtained by Lam et al., Duy et al., Flynn et al. to Dunkl gradients introduced by Dunkl. Our Hardy’s identities with explicit Bessel pairs significantly improve many existing Hardy’s inequalities for Dunkl operators.

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35

Pachpatte, B. G., and E. R. Love. "On some new inequalities related to Hardy's integral inequality." Journal of Mathematical Analysis and Applications 149, no. 1 (June 1990): 17–25. http://dx.doi.org/10.1016/0022-247x(90)90282-k.

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36

Oguntuase, James Adedayo, and Lars-Erik Persson. "Refinement of Hardy's inequalities via superquadratic and subquadratic functions." Journal of Mathematical Analysis and Applications 339, no. 2 (March 2008): 1305–12. http://dx.doi.org/10.1016/j.jmaa.2007.08.007.

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37

Ando, Hiroshi, and Toshio Horiuchi. "Weighted Hardy's inequalities and the variational problem with compact perturbations." Mathematical Journal of Ibaraki University 52 (2020): 15–26. http://dx.doi.org/10.5036/mjiu.52.15.

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38

Arcozzi, Nicola, Sorina Barza, J. L. Garcia-Domingo, and Javier Soria. "Hardy's inequalities for monotone functions on partly ordered measure spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 5 (October 2006): 909–19. http://dx.doi.org/10.1017/s0308210500004790.

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We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.

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39

Martín, Joaquim, and Javier Soria. "New Lorentz spaces for the restricted weak-type Hardy's inequalities." Journal of Mathematical Analysis and Applications 281, no. 1 (May 2003): 138–52. http://dx.doi.org/10.1016/s0022-247x(02)00584-x.

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40

Guseinov, R. V. "Hardy's inequalities in function spaces containing derivatives of noninteger order." Mathematical Notes 63, no. 5 (May 1998): 593–97. http://dx.doi.org/10.1007/bf02312839.

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41

Ruzhansky, Michael, and Durvudkhan Suragan. "Uncertainty relations on nilpotent Lie groups." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20170082. http://dx.doi.org/10.1098/rspa.2017.0082.

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We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic R n , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

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42

Saker, S. H., and I. Kubiaczyk. "Higher Summability and Discrete Weighted Muckenhoupt and Gehring Type Inequalities." Proceedings of the Edinburgh Mathematical Society 62, no. 4 (March 11, 2019): 949–73. http://dx.doi.org/10.1017/s0013091519000014.

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AbstractIn this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.

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43

Saker, S. H., M. M. A. El-sheikh, and A. M. Madian. "Some new generalized weighted dynamic inequalities of Hardy's type on time scales." Journal of Mathematics and Computer Science 23, no. 04 (November 22, 2020): 289–301. http://dx.doi.org/10.22436/jmcs.023.04.02.

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44

Johansson, Maria, Lars-Erik Persson, and Anna Wedestig. "A New Approach to the Sawyer and Sinnamon Characterizations of Hardy's Inequality for Decreasing Functions." gmj 15, no. 2 (June 2008): 295–306. http://dx.doi.org/10.1515/gmj.2008.295.

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Abstract Some Hardy type inequalities for decreasing functions are characterized by one condition (Sinnamon), while others are described by two independent conditions (Sawyer). In this paper we make a new approach to deriving such results and prove a theorem, which covers both the Sinnamon result and the Sawyer result for the case where one weight is increasing. In all cases we point out that the characterizing condition is not unique and can even be chosen among some (infinite) scales of conditions.

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45

Saker, Samir H., and Ravi P. Agarwal. "Discrete Hardy's type inequalities and structure of discrete class of weights satisfy reverse Hölder's inequality." Mathematical Inequalities & Applications, no. 2 (2021): 521–41. http://dx.doi.org/10.7153/mia-2021-24-36.

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46

Methot, A. A., and V. Scarani. "An anomaly of non-locality." Quantum Information and Computation 7, no. 1&2 (January 2007): 157–70. http://dx.doi.org/10.26421/qic7.1-2-10.

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Ever since the work of Bell, it has been known that entangled quantum states can produce non-local correlations between the outcomes of separate measurements. However, for almost forty years, it has been assumed that the most non-local states would be the maximally entangled ones. Surprisingly it is not the case: non-maximally entangled states are generally more non-local than maximally entangled states for all the measures of non-locality proposed to date: Bell inequalities, the Kullback-Leibler distance, entanglement simulation with communication or with non-local boxes, the detection loophole and efficiency of cryptography. In fact, one can even find simple examples in low dimensions, confirming that it is not an artefact of a specifically constructed Hilbert space or topology. This anomaly shows that entanglement and non-locality are not only different concepts, but also truly different resources. We review the present knowledge on this anomaly, point out that Hardy's theorem has the same feature, and discuss the perspectives opened by these discoveries.

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47

Benaissa, Bouharket. "More on reverses of Minkowski’s inequalities and Hardy’s integral inequalities." Asian-European Journal of Mathematics 13, no. 03 (December 13, 2018): 2050064. http://dx.doi.org/10.1142/s1793557120500643.

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In 2012, Sulaiman [Reverses of Minkowski’s, Hölder’s, and Hardy’s integral inequalities, Int. J. Mod. Math. Sci. 1(1) (2012) 14–24] proved integral inequalities concerning reverses of Minkowski’s and Hardy’s inequalities. In 2013, Banyat Sroysang obtained a generalization of the reverse Minkowski’s inequality [More on reverses of Minkowski’s integral inequality, Math. Aeterna 3(7) (2013) 597–600] and the reverse Hardy’s integral inequality [A generalization of some integral inequalities similar to Hardy’s inequality, Math. Aeterna 3(7) (2013) 593–596]. In this article, two results are given. First one is further improvement of the reverse Minkowski inequality and second is further generalization of the integral Hardy inequality.

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48

Benaissa, Bouharket, and Aissa Benguessoum. "Reverses Hardy-type inequalities via Jensen integral inequality." Mathematica Montisnigri 52 (2021): 43–51. http://dx.doi.org/10.20948/mathmontis-2021-52-5.

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The integral inequalities concerning the inverse Hardy inequalities have been studied by a large number of authors during this century, of these articles have appeared, the work of Sulaiman in 2012, followed by Banyat Sroysang who gave an extension to these inequalities in 2013. In 2020 B. Benaissa presented a generalization of inverse Hardy inequalities. In this article, we establish a new generalization of these inequalities by introducing a weight function and a second parameter. The results will be proved using the Hölder inequality and the Jensen integral inequality. Several the reverses weighted Hardy’s type inequalities and the reverses Hardy’s type inequalities were derived from the main results.

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49

El-Deeb, Ahmed A., Alaa A. El-Bary, Jan Awrejcewicz, and Kamsing Nonlaopon. "Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-Conformable Derivatives on Time Scales." Symmetry 14, no. 12 (December 17, 2022): 2674. http://dx.doi.org/10.3390/sym14122674.

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We established some new α-conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of α-conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions of α-conformable calculus. Our results are proved by using two-dimensional dynamic Jensen’s inequality and Fubini’s theorem on time scales. When α=1, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy’s inequality for T=R,T=Z and T=hZ. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

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50

Flynn, Joshua, Nguyen Lam, and Guozhen Lu. "Sharp Hardy Identities and Inequalities on Carnot Groups." Advanced Nonlinear Studies 21, no. 2 (March 12, 2021): 281–302. http://dx.doi.org/10.1515/ans-2021-2123.

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Abstract In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

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