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The Failure of Fatou's Theorem on Poisson Integrals of Pettis Integrable Functions

✍ Scribed by Francisco J Freniche; Juan Carlos Garcı́a-Vázquez; Luis Rodrı́guez-Piazza

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
292 KB
Volume
160
Category
Article
ISSN
0022-1236

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✦ Synopsis

In this paper we prove that for every infinite-dimensional Banach space X and every 1 p<+ there exists a strongly measurable X-valued p-Pettis integrable function on the unit circle T such that the X-valued harmonic function defined as its Poisson integral does not converge radially at any point of T, not even in the weak topology. We also show that this function does not admit a conjugate function. An application to spaces of vector valued harmonic functions is given. In the case that X does not have finite cotype we can construct the function with the additional property of being analytic, in the sense that its Fourier coefficients of negative frequency are null. In the general case we are able to give a countably additive vector measure, analytic in the same sense.

1998 Academic Press P r V +(t)= | T P r (t&s) d+(s).

On the other hand, as for any summability kernel, it is also a well known fact that P r V F converges to F in the norm of L p (T), whenever F # L p (T),

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