On the global integrability of non-negative harmonic functions

Let H+( D) be the set of all non-negative harmonic functions on a domain D subset of R-d. Let q > 0 and define L-q( D) to be the set of all Borel functions f such that | f |(q) is Lebesgue-integrable on D. Let x(0) is an element of D. N. Suzuki established the following: H+( D) subset of L-q(D) double left right arrow sup {integral(D) h(q) ( x) dx : h is an element of H+(D), h(x(0)) = 1} < infinity. In this paper, we prove results of this kind in a general setting of harmonic spaces covering the elliptic case and the parabolic one as well. The last section deals with some applications of these results.