The duals of harmonic Bergman spaces
HTML articles powered by AMS MathViewer
- by Charles V. Coffman and Jonathan Cohen PDF
- Proc. Amer. Math. Soc. 110 (1990), 697-704 Request permission
Abstract:
In this paper we show that for $\Omega$, a starlike Lipschitz domain, the dual of the space of harmonic functions in ${L^p}(\Omega )$ need not be the harmonic functions in ${L^q}(\Omega )$, where $1/p + 1/q = 1$. We show that, as a consequence, the harmonic Bergman projection for $\Omega$ need not extend to a bounded operator on ${L^p}(\Omega )$ for all $1 < p < \infty$. The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.References
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- David Békollé, Projections sur des espaces de fonctions holomorphes dans des domaines plans, Canad. J. Math. 38 (1986), no. 1, 127–157 (French). MR 835039, DOI 10.4153/CJM-1986-007-1
- R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 11–66. MR 604369
- B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135 (English, with French summary). MR 865663 E. Ligowcka, On the duality and interpolation for spaces of polyharmonic functions, Studia Math. T.87.1 (1987), 23-32. P. Grisvard, Elliptic boundary value problems on non-smooth domains, Pittman, 1983.
- J.-L. Lions and E. Magenes, Problemi ai limiti non omogenei. V, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 16 (1962), 1–44 (Italian). MR 146527
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
- Mitsuru Nakai and Leo Sario, Existence of biharmonic Green’s functions, Proc. London Math. Soc. (3) 36 (1978), no. 2, 337–368. MR 481056, DOI 10.1112/plms/s3-36.2.337
- Mitsuru Nakai and Leo Sario, Banach spaces of harmonic functions in $L_{p}$, Bull. Inst. Math. Acad. Sinica 6 (1978), no. 2, 415–417. MR 528660
- Jill Pipher and Gregory Verchota, Area integral estimates for the biharmonic operator in Lipschitz domains, Trans. Amer. Math. Soc. 327 (1991), no. 2, 903–917. MR 1024776, DOI 10.1090/S0002-9947-1991-1024776-7 —, Biharmonic Dirichlet problem on Lipschitz domains in ${{\mathbf {R}}^3}$, preprint. A. A. Solov’ev, ${L^p}$-estimates of integral operators associated with spaces of analytic and harmonic functions, Soviet Math. Dokl. 19 (1978), 764-768.
- Gregory Verchota, The Dirichlet problem for the biharmonic equation in $C^1$ domains, Indiana Univ. Math. J. 36 (1987), no. 4, 867–895. MR 916748, DOI 10.1512/iumj.1987.36.36048
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 697-704
- MSC: Primary 46E15; Secondary 31B05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028042-X
- MathSciNet review: 1028042