Identification of some real interpolation spaces. (English) Zbl 1100.46013
The author considers the relations of the real interpolation spaces between the couples \((X, Y)\), \((X+Y, Y)\), \((X, X\cap Y)\) and \((X+Y, X\cap Y)\). He shows, among other things, the following identities between such real interpolation spaces for \(1\leq p \leq\infty\) and \(0\leq \theta \leq 1\):
\[
\begin{aligned} &(X+Y, X)_{\theta, p}\cap (X+Y, Y)_{\theta, p} = (X+Y, X\cap Y)_{\theta, p}, \\ &(X+Y, X)_{\theta, p}\cap (X+Y, Y)_{1-\theta, p} = (X, Y)_{\theta, p}, \\ &(X, X\cap Y)_{\theta, p } + (Y, X\cap Y)_{\theta, p} = (X+Y, X\cap Y)_{\theta, p}, \\ &(X, X\cap Y)_{\theta, p} + (Y, X\cap Y)_{1-\theta, p} = (X, Y)_{\theta, p}. \end{aligned}
\]
Some applications of the obtained results to the study of real interpolation spaces for sectorial operators are also given.
Reviewer: Shangquan Bu (Beijing)
MSC:
| 46B70 | Interpolation between normed linear spaces |
| 47A60 | Functional calculus for linear operators |
| 47D06 | One-parameter semigroups and linear evolution equations |
References:
| [1] | Pascal Auscher, Alan McIntosh, and Andrea Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), no. 2, 375 – 403. · Zbl 0903.47011 · doi:10.1512/iumj.1997.46.1180 |
| [2] | Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071 |
| [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. · Zbl 0743.46082 |
| [4] | Giovanni Dore, \?^{\infty } functional calculus in real interpolation spaces. II, Studia Math. 145 (2001), no. 1, 75 – 83. · Zbl 0985.47015 · doi:10.4064/sm145-1-5 |
| [5] | Pierre Grisvard, Interpolation non commutative, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 11 – 15 (French, with Italian summary). · Zbl 0247.46053 |
| [6] | Bernhard Haak, Markus Haase, and Peer Kunstmann. Perturbation, interpolation and maximal regularity. Adv. Differential Equations, 11(2):201-240, 2006. · Zbl 1110.47032 |
| [7] | Markus Haase. The Functional Calculus for Sectorial Operators. Book manuscript. To appear in: Operator Theory: Advances and Applications, Birkhäuser, Basel. Preliminary version available at http://www.mathematik.uni-ulm.de/m5/haase, 2005. |
| [8] | 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. |
| [9] | Alessandra Lunardi. Interpolation Theory. Appunti, Scuola Normale Superiore, Pisa., 1999. · Zbl 1165.41300 |
| [10] | Lech Maligranda, The \?-functional for symmetric spaces, Interpolation spaces and allied topics in analysis (Lund, 1983) Lecture Notes in Math., vol. 1070, Springer, Berlin, 1984, pp. 169 – 182. · Zbl 0543.46017 · doi:10.1007/BFb0099100 |
| [11] | Lech Maligranda, On commutativity of interpolation with intersection, Proceedings of the 13th winter school on abstract analysis (Srní, 1985), 1985, pp. 113 – 118 (1986). · Zbl 0616.41001 |
| [12] | Lech Maligranda, Interpolation between sum and intersection of Banach spaces, J. Approx. Theory 47 (1986), no. 1, 42 – 53. · Zbl 0636.46063 · doi:10.1016/0021-9045(86)90045-6 |
| [13] | Celso Martínez Carracedo and Miguel Sanz Alix, The theory of fractional powers of operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001. · Zbl 0971.47011 |
| [14] | Jaak Peetre, Über den Durchschnitt von Interpolationsräumen, Arch. Math. (Basel) 25 (1974), 511 – 513 (German). · Zbl 0293.46025 · doi:10.1007/BF01238718 |
| [15] | Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. · Zbl 0830.46028 |
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