Poisson integrals of Pettis integrable functions
by
Francisco J. Freniche
Universidad de Sevilla, Spain.
Coauthors: Juan Carlos García-Vázquez (Universidad de Sevilla, Spain), Luis Rodríguez Piazza (Universidad de Sevilla, Spain)

We prove some results on Poisson integrals of Banach valued Pettis integrable functions on the unit circle T. We show that for every infinite dimensional Banach space X and every 1 <= p < \infty there exists a p-Pettis integrable function F:T --> X satisfying:

(1) lim_{r --> 1^-} ||P_r*F(t)|| = +\infty uniformly in t in T.

(2) The function F does not admit a conjugate function, although it has a conjugate operator.

We also show that for every infinite dimensional Banach space X there exists an analytic Pettis integrable function F:T --> X, such that limsup_{r --> 1^-} ||P_r*F(t)|| = +\infty for almost every t in T.

The first two results have appeared in J. Funct. Anal. vol. 160., pp. 28-41, and the third one solves a question posed in that paper.

(T)

Date received: November 29, 1999