On the interpolation constant for Orlicz spaces
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- by Alexei Yu. Karlovich and Lech Maligranda PDF
- Proc. Amer. Math. Soc. 129 (2001), 2727-2739 Request permission
Abstract:
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, \[ \|T\|_{L^\varphi \to L^\varphi } \le C\max \Big \{ \|T\|_{L^p\to L^p}, \|T\|_{L^q\to L^q} \Big \}, \] 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 $C=1$.References
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- J. Bergh, A generalization of Steffensen’s inequality, J. Math. Anal. Appl. 41 (1973), 187–191. MR 315072, DOI 10.1016/0022-247X(73)90193-5
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
- David W. Boyd, Spaces between a pair of reflexive Lebesgue spaces, Proc. Amer. Math. Soc. 18 (1967), 215–219. MR 212556, DOI 10.1090/S0002-9939-1967-0212556-3
- Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR 1107298
- Andrea Cianchi, An optimal interpolation theorem of Marcinkiewicz type in Orlicz spaces, J. Funct. Anal. 153 (1998), no. 2, 357–381. MR 1614590, DOI 10.1006/jfan.1997.3193
- 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.
- Jan Gustavsson and Jaak Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), no. 1, 33–59. MR 438102, DOI 10.4064/sm-60-1-33-59
- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
- S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
- Natan Ya. Kruglyak, Lech Maligranda, and Lars Erik Persson, A Carlson type inequality with blocks and interpolation, Studia Math. 104 (1993), no. 2, 161–180. MR 1211816, DOI 10.4064/sm-104-2-161-180
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- George G. Lorentz and Tetsuya 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 380447, DOI 10.1090/S0002-9947-1971-0380447-9
- Lech Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 49. MR 820076
- Lech Maligranda, Some remarks on Orlicz’s interpolation theorem, Studia Math. 95 (1989), no. 1, 43–58. MR 1024274, DOI 10.4064/sm-95-1-43-58
- L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Univ. Estadual de Campinas, Campinas SP, Brazil 1989.
- V. I. Ovčinnikov, Interpolation theorems that arise from Grothendieck’s inequality, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 45–54 (Russian). MR 0430813
- V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), no. 2, i–x and 349–515. MR 877877
- J. Peetre, On interpolation functions, Acta Sci. Math. (Szeged) 27 (1966), 167–171. MR 206706
- Walter Leighton and W. T. Scott, A general continued fraction expansion, Bull. Amer. Math. Soc. 45 (1939), 596–605. MR 41, DOI 10.1090/S0002-9904-1939-07046-8
- I. B. Simonenko, Interpolation and extrapolation of linear operators in Orlicz spaces, Mat. Sb. (N.S.) 63 (105) (1964), 536–553 (Russian). MR 0199696
- Gunnar Sparr, Interpolation of weighted $L_{p}$-spaces, Studia Math. 62 (1978), no. 3, 229–271. MR 506669, DOI 10.4064/sm-62-3-229-271
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
- MR Author ID: 606850
- Email: karlik@paco.net, akarlov@math.ist.utl.pt
- Lech Maligranda
- Affiliation: Department of Mathematics, LuleåUniversity of Technology, 971 87 Luleå, Sweden
- MR Author ID: 118770
- Email: lech@sm.luth.se
- Received by editor(s): January 24, 2000
- Published electronically: April 17, 2001
- Communicated by: Jonathan M. Borwein
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2727-2739
- MSC (1991): Primary 46B70, 46E30; Secondary 26D07
- DOI: https://doi.org/10.1090/S0002-9939-01-06162-7
- MathSciNet review: 1838797