Real Interpolation with Logarithmic Functors and Reiteration

Evans, W. D. ; Opic, B.

Canadian journal of mathematics, Tome 52 (2000) no. 5, pp. 920-960

Résumé

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 $[\text{D}]$ .

Détail

DOI : 10.4153/CJM-2000-039-2

Mots-clés : 46B70, 26D10, 46E30, Real interpolation, broken-logarithmic functors, reiteration, weighted inequalities

Bibliographie

[A] [A] Adams, R. A., Sobolev spaces. Academic Press, New York, 1975. Google Scholar

[B] [B] Bennet, C., Intermediate spaces and the class L log+ L. Ark. Mat. 11(1973), 215–228. Google Scholar

[BS] [BS] Bennet, C. and Sharpley, R., Interpolation of operators. Academic Press, New York, 1988. Google Scholar

[BK] [BK] Brudnyĭ, Yu. A. and Krugljak, N. Ya., Interpolation functors and interpolation spaces. NorthHolland, Amsterdam 1, 1991. Google Scholar

[BL] [BL] Bergh, J. and Löfström, J., Interpolation spaces. An Introduction, Springer, New York, 1976. Google Scholar

[D] [D] Doktorskii, R. Ya., Reiteration relations of the real interpolation method. Soviet Math. Dokl. 44(1992), 665–669. Google Scholar

[DO] [DO] Dmitriev, V. I. and Ovchinnikov, V. I., On interpolation in real method spaces. Soviet. Math. Dokl. 20(1979), 538–542. Google Scholar

[EOP] [EOP] Evans, W. D., Opic, B. and Pick, L., Real interpolation with logarithmic functors. to appear. Google Scholar

[ET] [ET] Edmunds, D. E. and Triebel, H., Function spaces, entropy numbers and differential operators. University Press, Cambridge, 1996. Google Scholar

[GM] [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] [G] Gustavsson, J., A function parameter in connection with interpolation of Banach spaces. Math. Scand. 42(1978), 289–305. Google Scholar

[He] [He] Heinig, H. P., Interpolation of quasi-normed spaces involving weights. CMS Conference Proceedings 1(1981), 245–267. Google Scholar

[Ho] [Ho] Holmstedt, T., Interpolation of quasi-normed spaces. Math. Scand. 26(1970), 177–199. Google Scholar

[Ka] [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] [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] [L] Lai, S., Weighted norm inequalities for general operators on monotone functions. Trans. Amer.Math. Soc. 340(1993), 811–836. Google Scholar

[Me1] [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] [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] [Mi] Milman, M., Extrapolation and optimal decompositions. LectureNotes in Math. 1580, Springer, 1994. Google Scholar

[N] [N] Nilsson, P., Reiteration theorems for real interpolation and approximation spaces. Math. Pures Appl. 132(1982), 291–330. Google Scholar

[OP] [OP] Opic, B. and Pick, L., On generalized Lorentz-Zygmund spaces. Math. Inequal. & Appl. 2(1999), 391–467. Google Scholar

[Pe] [Pe] Peetre, J., Espaces d’interpolation, généralisations, applications. Rend. Sem. Mat. Fis. Milano 34(1964), 133–164. Google Scholar

[Per] [Per] Persson, L. E., Interpolation with a parameter function. Math. Scand. 59(1986), 199–222. Google Scholar

[Pi] [Pi] Pietsch, A., Eigenvalues and s-numbers. University Press, Cambridge, 1987. Google Scholar

[S] [S] Sagher, Y., An application of interpolation theory to Fourier series. Studia Math. 41(1972), 169–181. Google Scholar