Interpolation functions and the Lions–Peetre interpolation construction

V. I. Ovchinnikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 4, pp. 681-741

Résumé

The generalization of the Lions–Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces $\varphi(X_0,X_1)_{p_0,p_1}$ considered here have three parameters: two positive numerical parameters $p_0$ and $p_1$ of equal standing, and a function parameter $\varphi$. For $p_0\ne p_1$ these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces $\varphi(X_0,X_1)_{p_0,p_1}$, together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces $L_p$ and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter $\varphi$ can be an arbitrary natural functional parameter in the interpolation. Bibliography: 43 titles.

Détail

Keywords: interpolation functors with function parameters, orbits with respect to von Neumann–Schatten operators, optimal interpolation theorems, embedding theorems for Orlicz–Sobolev spaces.
Mots-clés : interpolation spaces, interpolation orbits

Bibliographie

[1] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 528 pp. ; North-Holland Math. Library, 18, North-Holland Publishing Co., Amsterdam–New York, 1978, 528 pp. | MR | MR | MR | Zbl | Zbl | Zbl

[2] S. G. Kreĭn, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, RI, 1982, xii+375 pp. | MR | MR | Zbl | Zbl

[3] J. Bergh, J. Löfström, Interpolation spaces. An introduction, Grundlehren Math. Wiss., 223, Springer-Verlag, Berlin–New York, 1978, x+207 pp. | MR | MR | Zbl

[4] V. I. Ovchinnikov, “The method of orbits in interpolation theory”, Math. Rep., 1:2 (1984), i–x + 349–515 | MR | Zbl

[5] C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Inc., Boston, MA, 1988, xiv+469 pp. | MR | Zbl

[6] Yu. A. Brudnyi, N. Ya. Krugliak, Interpolation functors and interpolation spaces, v. I, North-Holland Math. Library, 47, North-Holland Publishing Co., Amsterdam, 1991, xvi+718 pp. | MR | Zbl

[7] V. I. Ovchinnikov, “Interpolation orbits in couples of $L_p$ spaces”, C. R. Math. Acad. Sci. Paris, 334:10 (2002), 881–884 | DOI | MR | Zbl

[8] Ming Fan, “Interpolation methods of constants and means with quasi-power function parameters”, Math. Scand., 88:1 (2001), 79–95 | MR | Zbl

[9] V. I. Ovchinnikov, “On the description of interpolation orbits in couples of $L_{p}$ spaces when they are not described by the $K$-method”, Interpolation spaces and related topics (Haifa, 1990), Israel Math. Conf. Proc., 5, Bar-Ilan Univ., Ramat Gan, 1992, 187–206 | MR | Zbl

[10] K. I. Oskolkov, “Approximation properties of summable functions on sets of full measure”, Math. USSR-Sb., 32:4 (1977), 489–514 | DOI | MR | Zbl

[11] S. Janson, “Minimal and maximal methods in interpolation”, J. Funct. Anal., 44:1 (1981), 50–73 | DOI | MR | Zbl

[12] Yu. Brudnyi, A. Shteinberg, “Calderón couples of Lipschitz spaces”, J. Funct. Anal., 131:2 (1995), 459–498 | DOI | MR | Zbl

[13] V. I. Ovchinnikov, A. S. Titenkov, “A criterion for contiguity of quasiconcave functions”, Math. Notes, 70:5 (2001), 708–713 | DOI | DOI | MR | Zbl

[14] V. I. Dmitriev, S. G. Krein, V. I. Ovchinnikov, “Osnovy teorii interpolyatsii lineinykh operatorov”, Geometriya lineinykh prostranstv i teoriya operatorov, Izd-vo Yaroslavskogo un-ta, Yaroslavl, 1977, 31–74 | MR | Zbl

[15] V. I. Ovchinnikov, “Interpolation in quasi-Banach Orlicz spaces”, Funct. Anal. Appl., 16:3 (1982), 223–224 | DOI | MR | Zbl

[16] B. Maurey, G. Pisier, “Une théorème d'extrapolation et ses conséquences”, C. R. Acad. Sci. Paris Sér. A, 277:1 (1973), 39–42 | MR | Zbl

[17] G. Ya. Lozanovskii, “A remark on an interpolational theorem of Calderon”, Funct. Anal. Appl., 6:4 (1972), 333–334 | DOI | MR | Zbl

[18] J.-L. Lions, J. Peetre, “Sur une classe d'espaces d'interpolation”, Inst. Hautes Études Sci. Publ. Math., 19:1 (1964), 5–68 | DOI | MR | Zbl

[19] M. Milman, “The computation of the $K$ functional for couples of rearrangement invariant spaces”, Results Math., 5:1-2 (1982), 174–176 | DOI | MR | Zbl

[20] J. Peetre, “Sur le nombre de paramètres dans la définition de certains espaces d'interpolation”, Ric. Mat., 12 (1963), 248–261 | MR | Zbl

[21] G. Sparr, “Interpolation of weighted $L_p$-spaces”, Studia Math., 62:3 (1978), 229–271 | MR | Zbl

[22] G. Bennett, “Inclusion mappings between $l_p$ spaces”, J. Funct. Anal., 13:1 (1973), 20–27 | DOI | MR | Zbl

[23] V. I. Ovchinnikov, “Interpolation orbits in couples of Lebesgue spaces”, Funct. Anal. Appl., 39:1 (2005), 46–56 | DOI | DOI | MR | Zbl

[24] V. I. Dmitriev, “Duality between the interpolation method of constants and that of means”, Soviet Math. Dokl., 15 (1974), 16–19 | MR | Zbl

[25] E. D. Kravishvili, “Metod srednikh s proizvolnym funktsionalnym parametrom”, Trudy matematicheskogo fakulteta. Novaya seriya, 7, VGU, Voronezh, 2002, 58–72

[26] G. Ja. Lozanovskii, “On some Banach lattices. IV”, Sib. Math. J., 14:1 (1973), 97–108 | DOI | MR | Zbl

[27] V. I. Ovchinnikov, “The quasinormed Neumann–Schatten ideals and embedding theorems for the generalized Lions–Peetre spaces of means”, St. Petersburg Math. J., 22:4 (2011), 669–681 | DOI | MR | Zbl

[28] L. Maligranda, “The K-functional for symmetric spaces”, Interpolation spaces and allied topics in analysis (Lund, 1983), Lecture Notes in Math., 1070, Springer, Berlin, 1984, 169–182 | DOI | MR | Zbl

[29] A. A. Sedaev, “Description of interpolation spaces for the pair $(L^p_{\alpha_0},L^p_{\alpha_1})$ and some related problems”, Soviet Math. Dokl., 14 (1973), 538–541 | MR | Zbl

[30] I. C. Gohberg, M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969, xv+378 pp. | MR | MR | Zbl | Zbl

[31] Yu. A. Brudnyi, S. G. Krein, E. M. Semenov, “Interpolation of linear operators”, J. Soviet Math., 42:6 (1988), 2009–2112 | DOI | MR | Zbl

[32] V. I. Dmitriev, “On interpolation of operators in $L_p$ spaces”, Soviet Math. Dokl., 24:2 (1981), 373–376 | MR | Zbl

[33] A. A. Dmitriev, E. M. Semenov, “Optimality of the M. Riesz interpolation theorem in the upper triangle”, Soviet Math. Dokl., 23:3 (1981), 664–666 | MR | Zbl

[34] N. Aronszajn, E. Gagliardo, “Interpolation spaces and interpolation methods”, Ann. Mat. Pura Appl. (4), 68:1 (1965), 51–117 | DOI | MR | Zbl

[35] V. I. Ovchinnikov, “Interpolyatsionnaya teorema dlya ravnomerno ukhudshayuschikh operatorov v simmetrichnykh prostranstvakh”, Vestn. Voronezh. gos. un-ta. Ser. Estestvennye nauki, 3 (1998), 119–126

[36] M. Mastyło, “Interpolation methods of means and orbits”, Studia Math., 171:2 (2005), 153–175 | DOI | MR | Zbl

[37] V. I. Ovchinnikov, “Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters”, J. Funct. Anal., 228:1 (2005), 234–243 | DOI | MR | Zbl

[38] K. Hansson, “Imbedding theorems of Sobolev type in potential theory”, Math. Scand., 45:1 (1979), 77–102 | MR | Zbl

[39] H. Brézis, S. Wainger, “A note on limiting cases of Sobolev embeddings and convolution inequalities”, Comm. Partial Differential Equations, 5:7 (1980), 773–789 | DOI | MR | Zbl

[40] V. I. Ovchinnikov, “Generalized Lions–Peetre interpolation construction and optimal embedding theorems for Sobolev spaces”, Sb. Math., 205:1 (2014), 83–100 | DOI | DOI | MR

[41] A. Cianchi, “Optimal Orlicz–Sobolev embeddings”, Rev. Mat. Iberoam., 20:2 (2004), 427–474 | DOI | MR | Zbl

[42] R. Kerman, L. Pick, “Optimal Sobolev imbeddings”, Forum Math., 18:4 (2006), 535–570 | DOI | MR | Zbl

[43] A. Gogatishvili, V. I. Ovchinnikov, “Interpolation orbits and optimal Sobolev's embeddings”, J. Funct. Anal., 253:1 (2007), 1–17 | DOI | MR | Zbl