Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On linear topological properties of $ H\sp 1$ on spaces of homogeneous type


Author: Paul F. X. Müller
Journal: Trans. Amer. Math. Soc. 317 (1990), 463-484
MSC: Primary 46E15; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9947-1990-0974522-7
MathSciNet review: 974522
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (X,d,\mu )$ be a space of homogeneous type. Let $ B = \{ x \in X:\mu \{ x\} = 0\} $, then $ \mu (B) > 0$ implies that $ {H^1}(X,d,\mu )$ contains a complemented copy of $ {H^1}(\delta )$. This applies to Hardy spaces $ {H^1}(\partial \Omega ,d,\omega )$ associated to weak solutions of uniformly elliptic operators in divergence form. Under smoothness assumptions of the coefficients of the elliptic operators, we obtain that $ {H^1}(\partial \Omega ,d,\omega )$ is isomorphic to $ {H^1}(\delta )$.


References [Enhancements On Off] (What's this?)

  • [C-F-M-S] L. Caffarelli, E. Fabes, S. Mortola and S. Salsea, Boundary behaviour of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), 621-640. MR 620271 (83c:35040)
  • [C-W] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
  • [deG] M. de Guzman, Real variable methods in Fourier analysis, North-Holland, 1981. MR 596037 (83j:42019)
  • [D-J-K] B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with non-smooth coefficients, Ark. Math. 22 (1984), 97-107. MR 735881 (85h:35021)
  • [Dy] E. B. Dynkin, Markov processes, Vol. II, Springer-Verlag, 1965.
  • [F-J-K] E. Fabes, D. Jerison and C. Kenig, Boundary behaviour of solutions of degenerate elliptic equations, Conference on Harmonic Analysis, edited by W. Becker et al., Wadsworth, 1981. MR 730093 (85m:35028)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1977. MR 0473443 (57:13109)
  • [J] P. W. Jones, Constructions with functions of bounded mean oscillation, Thesis, 1978.
  • [J-K] D. Jerison and C. Kenig, Boundary behavior of harmonic in $ NTA$ domains, Adv. in Math. 46 (1982), 80-147. MR 676988 (84d:31005b)
  • [Ma] B. Maurey, Isomorphismes entres espaces $ {H^1}$, Acta Math. 145 (1980), 79-120. MR 586594 (84b:46027)
  • [Mü] P. F. X. Müller, On subsequences of the Haar system and isomorphism between $ {H^1}$ spaces, Studia Math. 85 (1987), 73-90.
  • [M-S-1] A. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), 257-270. MR 546295 (81c:32017a)
  • [M-S-2] -, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271-309. MR 546296 (81c:32017b)
  • [P] K. E. Petersen, Brown motion, Hardy spaces and bounded mean oscillation, Cambridge Univ. Press, 1971. MR 0651556 (58:31383)
  • [Øk] B. Øksendal, Stochastic differential equations, Springer-Verlag, 1985. MR 804391 (87d:60057)
  • [W] T. Wolniewicz, On isomorphisms between Hardy spaces on complex balls, Ark. Math. 27 (1989), 155-168. MR 1004730 (91b:32007)
  • [Wo] P. Wojtaszczyk, Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc $ 1 \leqslant p \leqslant \infty $, Ann. of Math. 118 (1983), 21-34. MR 707159 (84i:32006)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E15, 46B20

Retrieve articles in all journals with MSC: 46E15, 46B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0974522-7
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society