Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation
HTML articles powered by AMS MathViewer
- by Zbigniew Slodkowski PDF
- Trans. Amer. Math. Soc. 317 (1990), 255-285 Request permission
Abstract:
This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in ${{\mathbf {C}}^k}$. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the $r$th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.References
- Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99–109. MR 779607, DOI 10.1007/BF01455798
- R. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel, and Y. Sagher, The complex method for interpolation of operators acting on families of Banach spaces, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 123–153. MR 576042
- R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, Complex interpolation for families of Banach spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 269–282. MR 545314
- R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, A theory of complex interpolation for families of Banach spaces, Adv. in Math. 43 (1982), no. 3, 203–229. MR 648799, DOI 10.1016/0001-8708(82)90034-2 R. Coifman and S. Semmes, Interpolation of Banach spaces and nonlinear Dirichlet problems.
- Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
- L. R. Hunt and John J. Murray, $q$-plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J. 25 (1978), no. 3, 299–316. MR 512901
- 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
- Richard Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math. 110 (1984), no. 2, 355–376. MR 726495
- Richard Rochberg, The work of Coifman and Semmes on complex interpolation, several complex variables, and PDEs, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 74–90. MR 942258, DOI 10.1007/BFb0078864
- Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703
- Zbigniew Slodkowski, The Bremermann-Dirichlet problem for $q$-plurisubharmonic functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 2, 303–326. MR 764948 —, Analytic multifunctions, pseudoconvex domains and uniform algebras, Contemp. Math. 32 (1984), 243-258.
- Zbigniew Slodkowski, Local maximum property and $q$-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. 115 (1986), no. 1, 105–130. MR 835588, DOI 10.1016/0022-247X(86)90027-2
- Zbigniew Slodkowski, An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, Trans. Amer. Math. Soc. 294 (1986), no. 1, 367–377. MR 819954, DOI 10.1090/S0002-9947-1986-0819954-1
- Zbigniew Slodkowski, On bounded analytic functions in finitely connected domains, Trans. Amer. Math. Soc. 300 (1987), no. 2, 721–736. MR 876475, DOI 10.1090/S0002-9947-1987-0876475-9
- Zbigniew Slodkowski, Pseudoconvex classes of functions. I. Pseudoconcave and pseudoconvex sets, Pacific J. Math. 134 (1988), no. 2, 343–376. MR 961240
- Zbigniew Slodkowski, Complex interpolation of normed and quasinormed spaces in several dimensions. I, Trans. Amer. Math. Soc. 308 (1988), no. 2, 685–711. MR 951623, DOI 10.1090/S0002-9947-1988-0951623-1
- Zbigniew Slodkowski, Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces, Trans. Amer. Math. Soc. 309 (1988), no. 1, 165–189. MR 957066, DOI 10.1090/S0002-9947-1988-0957066-9 —, Pseudoconvex classes of functions. II, Pacific J. Math. (to appear). —, Complex interpolation of normed and quasinormed spaces in several dimensions. III, Trans. Amer. Math. Soc. (to appear). J. Wermer, letter.
- Richard Rochberg, Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984), no. 2, 745–758. MR 743742, DOI 10.1090/S0002-9947-1984-0743742-6
- Michael Cwikel and Shlomo Reisner, Interpolation of uniformly convex Banach spaces, Proc. Amer. Math. Soc. 84 (1982), no. 4, 555–559. MR 643748, DOI 10.1090/S0002-9939-1982-0643748-5
- Marco Vignati, Complex interpolation and convexity, Proc. Amer. Math. Soc. 99 (1987), no. 4, 705–711. MR 877044, DOI 10.1090/S0002-9939-1987-0877044-2
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 255-285
- MSC: Primary 46M35; Secondary 32A30, 32F05, 46A99, 46B70
- DOI: https://doi.org/10.1090/S0002-9947-1990-0949900-2
- MathSciNet review: 949900