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On the interpolation constant for Orlicz spaces

Authors: Alexei Yu. Karlovich and Lech Maligranda
Journal: Proc. Amer. Math. Soc. 129 (2001), 2727-2739
MSC (1991): Primary 46B70, 46E30; Secondary 26D07
Published electronically: April 17, 2001
MathSciNet review: 1838797
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Abstract | References | Similar Articles | Additional Information


In this paper we deal with the interpolation from Lebesgue spaces $L^p$ and $L^q$, into an Orlicz space $L^\varphi$, where $1\le p<q\le\infty$ and $\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})$for some concave function $\rho$, with special attention to the interpolation constant $C$. For a bounded linear operator $T$ in $L^p$ and $L^q$, we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

\begin{displaymath}\Vert T\Vert _{L^\varphi\to L^\varphi} \le C\max\Big\{ \Vert T\Vert _{L^p\to L^p}, \Vert T\Vert _{L^q\to L^q} \Big\}, \end{displaymath}

where the interpolation constant $C$ depends only on $p$ and $q$. We give estimates for $C$, which imply $C<4$. Moreover, if either $1< p<q\le 2$ or $2\le p<q<\infty$, then $C< 2$. If $q=\infty$, then $C\le 2^{1-1/p}$, and, in particular, for the case $p=1$ this gives the classical Orlicz interpolation theorem with the constant $C=1$.

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  • 1. C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, London 1988. MR 89e:46001
  • 2. J. Bergh, A generalization of Steffensen's inequality, J. Math. Anal. Appl. 41 (1973), 187-191. MR 47:3621
  • 3. J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin 1976. MR 58:2349
  • 4. D. W. Boyd, Spaces between a pair of reflexive Lebesgue spaces, Proc. Amer. Math. Soc. 18 (1967), 215-219. MR 35:3427
  • 5. Yu. A. Brudnyi, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, I, North-Holland, Amsterdam-New York-Tokyo 1991. MR 93b:46141
  • 6. A. Cianchi, An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces, J. Funct. Anal. 153 (1998), 357-381. MR 99d:46039
  • 7. I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Pitman Monographs and Surveys in Pure and Applied Math. 92, Addison Wesley Longman 1998. CMP 2001:03
  • 8. J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), 33 - 59. MR 55:11021
  • 9. M. A. Krasnoselskii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen 1961. MR 23:A4016
  • 10. S. G. Krein, Ju. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, AMS Trans. of Math. Monographs 54, Providence, R.I. 1982. MR 84j:46103
  • 11. N. Krugljak, L. Maligranda, L.-E. Persson, A Carlson type inequality with blocks and interpolation, Studia Math. 104 (1993), 161-180. MR 94k:46153
  • 12. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer Verlag, New York-Berlin 1979. MR 81c:46001
  • 13. G. G. Lorentz, T. Shimogaki, Interpolation theorems for the pairs of spaces $(L^p,L^\infty)$and $(L^1, L^q)$, Trans. Amer. Math. Soc. 159 (1971), 207-221. MR 52:1347
  • 14. L. Maligranda, Indices and interpolation, Dissert. Math. 234 (1985), 1-49. MR 87k:46059
  • 15. L. Maligranda, Some remarks on Orlicz's interpolation theorem, Studia Math. 95 (1989), 43-58. MR 90k:46160
  • 16. L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Univ. Estadual de Campinas, Campinas SP, Brazil 1989.
  • 17. V. I. Ovchinnikov, Interpolation theorems resulting from an inequality of Grothendieck, Funkts. Analiz i Prilozh., 10 (1976), 45-54 (in Russian); English transl.: Funct. Anal. Appl. 10 (1976), 287-294. MR 55:3818
  • 18. V. I. Ovchinnikov, The Method of Orbits in Interpolation Theory, Math. Rep. 1 (1984), No. 2, 349-515. MR 88d:46136
  • 19. J. Peetre, On interpolation functions, Acta Math. Sci. Szeged 27 (1966), 167-171. MR 34:6523
  • 20. J. Peetre, A new approach in interpolation spaces, Studia Math. 34 (1970), 23-42. MR 41:8985; Correction MR 58:23647
  • 21. I. B. Simonenko, Interpolation and extrapolation of linear operators in Orlicz spaces, Mat. Sb. (N.S.) 63(1964), 536-553 (in Russian). MR 33:7839
  • 22. G. Sparr, Interpolation of weighted $L_p$-spaces, Studia Math. 62 (1978), 229 - 271. MR 80d:46055

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Additional Information

Alexei Yu. Karlovich
Affiliation: Department of Mathematics and Physics, South Ukrainian State Pedagogical University, Staroportofrankovskaya 26, 65020 Odessa, Ukraine
Address at time of publication: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

Lech Maligranda
Affiliation: Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden

Keywords: Orlicz spaces, interpolation constant, interpolation of operators, $K$-functional, convex function, concave function
Received by editor(s): January 24, 2000
Published electronically: April 17, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society

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