The Failure of Fatouʹs Theorem on Poisson Integrals of Pettis Integrable Functions

In this paper we prove that for every infinite-dimensional Banach spaceXand every 1⩽p<∞ there exists a strongly measurableX-valuedp-Pettis integrable function on the unit circle T such that theX-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 thatXdoes 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.