Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation

Author:
Zbigniew Slodkowski

Journal:
Trans. Amer. Math. Soc. **321** (1990), 305-332

MSC:
Primary 46M35; Secondary 32F05, 46B70

DOI:
https://doi.org/10.1090/S0002-9947-1990-0991968-1

MathSciNet review:
991968

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces , where is open and bounded in . The main result asserts that (under a mild assumption on the datum) the norm function belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler proof of this duality relation is also presented in the paper.

**[1]**E. Asplund,*Frechet differentiability of convex functions*, Acta Math.**121**(1968), 31-48. MR**0231199 (37:6754)****[2]**E. Bedford, B. A. Taylor,*The Dirichlet problem for a complex Monge-Ampere equation*, Invent. Math.**37**(1976), 1-44. MR**0445006 (56:3351)****[3]**R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss,*The complex method for interpolation of operators acting on families of Banach spaces*, Lecture Notes in Math., vol. 779, Springer-Verlag, Berlin and New York, 1980, pp. 123-153. MR**576042 (81k:46075)****[4]**-,*A theory of complex interpolation for families of Banach spaces*, Adv. in Math.**33**(1982), 203-229. MR**648799 (83j:46084)****[5]**R. Coifman and S. Semmes,*Interpolation of Banach spaces and nonlinear Dirichlet problems*, Lecture Notes in Math., vol. 1302, Springer-Verlag, Berlin and New York, 1988.**[6]**M. M. Day,*Normed linear spaces*, Springer-Verlag, New York, 1973. MR**0344849 (49:9588)****[7]**W. K. Hayman and P. B. Kennedy,*Subharmonic functions*, Vol. I, Academic Press, London, 1976. MR**0460672 (57:665)****[8]**E. Hewitt and K. Stromberg,*Real and abstract analysis*, Springer-Verlag, New York, 1969. MR**0367121 (51:3363)****[9]**C. O. Kiselman,*The partial Legendre transformation for plurisubharmonic functions*, Invent. Math.**49**(1978), 137-148. MR**511187 (80d:32014)****[10]**V. G. Mazja,*Sobolew spaces*, Springer-Verlag, Berlin, 1985.**[11]**R. Rochberg,*The work of Coifman and Semmes on complex interpolation, several complex variables and PDE's*, U.S.-Swedish Seminar on Function Spaces and Applications, Lund, June 1986 (to appear). MR**942258 (90f:46117)****[12]**-,*Interpolation of Banach spaces and negatively curved vector bundles*, Pacific J. Math.**110**(1984), 335-376. MR**726495 (85m:46077)****[13]**R. Rochberg and G. Weiss,*Some topics in complex interpolation theory*, Topics in Modern Harmonic Analysis, Istituto Nazionale di Alta Matematica, Rome, 1983, pp. 769-818. MR**748883 (86a:46100)****[14a]**Z. Slodkowski,*Analytic multifunctions and their applications*, talk at the Banach Space Conference, Kent, August 1985.**[14b]**-,*Complex interpolation families of normed spaces over several-dimensional parameter space*, Abstracts of the Special Session in Several Complex Variables, 826th Meeting of the AMS, Indianapolis, April 1986.**[14c]**-,*On complex interpolation methods for families of normed spaces over domains in*, talk at the International Conference on Harmonic Measure, Toledo, Ohio, July 1986.**[15]**-,*Local maximum property and**-plurisubharmonic functions in uniform algebras*, J. Math. Anal. Appl.**115**(1986), 105-130. MR**835588 (87j:32050)****[16]**-,*Pseudoconvex classes of functions*. I.*Pseudoconcave and pseudoconvex sets*, Pacific J. Math.**134**(1988), 343-376. MR**961240 (89m:32031)****[17]**-,*Complex interpolation of normed and quasinormed spaces*. I, Trans. Amer. Math. Soc.**308**(1988), 685-711. MR**951623 (89j:32024)****[18]**-,*Pseudoconvex classes of functions*. III.*Characterization of dual pseudoconvex classes on complex homogeneous spaces*, Trans. Amer. Math. Soc.**309**(1988), 165-189. MR**957066 (89m:32032)****[19]**-,*Complex interpolation of normed and quasinormed spaces in several dimensions*. II.*Properties of harmonic interpolation*, Trans. Amer. Math. Soc.**317**(1990), 255-285. MR**949900 (91e:46107)****[20]**A. I. Vol'pert,*The spaces BV and quasilinear equations*, Math. USSR-Sb.**2**(1967), 225-267. MR**0216338 (35:7172)****[21]**S. Kobayashi,*Negative vector bundles and complex Finsler structures*, Nagoya Math. J.**57**(1975), 153-166. MR**0377126 (51:13299)****[22]**R. Coifman and S. Semmes,*Interpolation of Banach spaces, Perron processes and Yang-Mills*.**[23]**S. Semmes,*Interpolation of Banach spaces, differential geometry and differential equations*, preprint. MR**1009123 (91e:46105)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46M35,
32F05,
46B70

Retrieve articles in all journals with MSC: 46M35, 32F05, 46B70

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0991968-1

Keywords:
Plurisubharmonic functions,
Hessian form,
weak derivative,
dual norm,
interpolation family,
harmonic multifunction

Article copyright:
© Copyright 1990
American Mathematical Society