The presentation problem of the conjugate cone of the Hardy space H ^p (0 < p ≤ 1)

Abstract

The Hardy space H ^p is not locally convex if 0 < p < 1, even though its conjugate space (H ^p)* separates the points of H ^p. But then it is locally p-convex, and its conjugate cone (H ^p) _p* is large enough to separate the points of H ^p. In this case, the conjugate cone can be used to replace its conjugate space to set up the duality theory in the p-convex analysis. This paper deals with the representation problem of the conjugate cone (H ^p) _p ^* of H ^p for 0 < p ≤ 1, and obtains the subrepresentation theorem (H ^p) _p ^* ≃ L ^∞(T, C _p ^*).

Keywords

Locally p-convex space / Hardy space / Normed conjugate cone / Shadow cone / Subrepresentation theorem

References

[1] Kalton N J, Peck N T, Roberts J W. An F-space Sampler, 1984, London: Cambridge University Press

[2] Rolewicz S. Metric Linear Spaces, 1985, Warszawa: Polish Scientific Publishers

[3] Duren P L. Theory of H ^p Spaces, 1970, New York: Academic Press

[4] Simmons S. Boundness in linear topological spaces. Trans. Amer. Math. Soc., 1964, 113: 169-180

[5] Jarchow H. Locally Convex Spaces, 1981, Stuttgart: B. G. Teubner

[6] Wang J Y, Ma Y M. The second separation theorem in locally β-convex space and the boundedness theorem in its conjugate cone. J. Math. Res. Exposition, 2002, 22(1): 25-34

[7] Wang J Y. Quasi-translation invariant topological cones and the conjugate cones of locally β-convex spaces. Math. Practice Theory, 2003, 33(1): 89-97

[8] Wang J Y. The Hahn-Banach theorems about β-subseminorms in locally β-convex spaces and their applications. Journal of Changshu Institue of Technology, 2006, 20(4): 19-24

[9] Diestel J, Uhl J J. Vector Measures, 1977, Providence, Rhode Island: Amer. Math. Soc.

[10] Wang J Y. The subrepresentation theorem of the conjugate cone of lp(X) (0 < p < 1). Adv. Math., 2010, 39(6): 709-718

[11] Halmos P R. Measure Theory, 1950, New York: Van Nostrand