Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T19:24:14.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Real Interpolation with Logarithmic Functors and Reiteration

Published online by Cambridge University Press:  20 November 2018

W. D. Evans  and
B. Opic
W. D. Evans
Affiliation:
School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff CF24 4YH, United Kingdom email: [email protected]
B. Opic
Affiliation:
Mathematical Institute of the Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present “reiteration theorems” with limiting values $\theta =0$ and $\theta =1$ for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in $[\text{D}]$.

Keywords

46B7026D1046E30Real interpolationbroken-logarithmic functorsreiterationweighted inequalities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 5 , 01 October 2000 , pp. 920 - 960
Copyright
Copyright © Canadian Mathematical Society 2000

References

[A] Adams, R. A., Sobolev spaces. Academic Press, New York, 1975.Google Scholar
[B] Bennet, C., Intermediate spaces and the class L log+ L. Ark. Mat. 11(1973), 215–228.Google Scholar
[BS] Bennet, C. and Sharpley, R., Interpolation of operators. Academic Press, New York, 1988.Google Scholar
[BK] Brudnyĭ, Yu. A. and Krugljak, N. Ya., Interpolation functors and interpolation spaces. NorthHolland, Amsterdam 1, 1991.Google Scholar
[BL] Bergh, J. and Löfström, J., Interpolation spaces. An Introduction, Springer, New York, 1976.Google Scholar
[D] Doktorskii, R. Ya., Reiteration relations of the real interpolation method. Soviet Math. Dokl. 44(1992), 665–669.Google Scholar
[DO] Dmitriev, V. I. and Ovchinnikov, V. I., On interpolation in real method spaces. Soviet. Math. Dokl. 20(1979), 538–542.Google Scholar
[EOP] Evans, W. D., Opic, B. and Pick, L., Real interpolation with logarithmic functors. to appear.Google Scholar
[ET] Edmunds, D. E. and Triebel, H., Function spaces, entropy numbers and differential operators. University Press, Cambridge, 1996.Google Scholar
[GM] Gomez, M. E. and Milman, M., Extrapolation spaces and almost-everywhere convergence of singular integrals. J. LondonMath. Soc. 34(1986), 305–316.Google Scholar
[G] Gustavsson, J., A function parameter in connection with interpolation of Banach spaces. Math. Scand. 42(1978), 289–305.Google Scholar
[He] Heinig, H. P., Interpolation of quasi-normed spaces involving weights. CMS Conference Proceedings 1(1981), 245–267.Google Scholar
[Ho] Holmstedt, T., Interpolation of quasi-normed spaces. Math. Scand. 26(1970), 177–199.Google Scholar
[Ka] Kalugina, T. F., Interpolation of Banach spaces with a functional parameter. The reiteration theorem. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 30(1975), 68–77.Google Scholar
[Kr] Krée, P., Interpolation d’espaces qui ne sont ni normés, ni complets. Applications, Seminaire Lions-Schwartz, semmestre 1964–1965, Secrétariat Mathématique 11, rue Pierre Curie, Paris 5e.Google Scholar
[L] Lai, S., Weighted norm inequalities for general operators on monotone functions. Trans. Amer.Math. Soc. 340(1993), 811–836.Google Scholar
[Me1] Merucci, C., Interpolation réelle avec function paramètre: réitération et applications aux espaces Λp(ϕ) (0 < p≤ +∞). C. R. Acad. Sci. Paris Sér. I. Math. 295(1982), 427–430.Google Scholar
[Me2] Merucci, C., Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. Lecture Notes in Math. 1070, Springer, 1984, 183–201.Google Scholar
[Mi] Milman, M., Extrapolation and optimal decompositions. LectureNotes in Math. 1580, Springer, 1994.Google Scholar
[N] Nilsson, P., Reiteration theorems for real interpolation and approximation spaces. Math. Pures Appl. 132(1982), 291–330.Google Scholar
[OP] Opic, B. and Pick, L., On generalized Lorentz-Zygmund spaces. Math. Inequal. & Appl. 2(1999), 391–467.Google Scholar
[Pe] Peetre, J., Espaces d’interpolation, généralisations, applications. Rend. Sem. Mat. Fis. Milano 34(1964), 133–164.Google Scholar
[Per] Persson, L. E., Interpolation with a parameter function. Math. Scand. 59(1986), 199–222.Google Scholar
[Pi] Pietsch, A., Eigenvalues and s-numbers. University Press, Cambridge, 1987.Google Scholar
[S] Sagher, Y., An application of interpolation theory to Fourier series. Studia Math. 41(1972), 169–181.Google Scholar