We give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as $$u = P(\Delta )g(x) + h(x)$$ where$g(x)$is a bounded continuous function,$h(x)$is a bounded real analytic function and$P(d/dt)$is an ultradifferential operator. Also, we show that the boundary value of every heat function with some exponential growth condition determines an ultradistribution in this class. These results generalize the theorem of Matsuzawa [M] for the above class of quasianalytic ultradistributions and partially solve a question of A. Kaneko [Ka]. Our interest lies in the quasianalytic case, although the theorems do not exclude non-quasianalytic classes.

Let$D$be a domain in$R$^{2}whose complement is contained in a pair of rays leaving the origin. That is,$D$contains two sectors whose base angles sum to 2π. We use balayage to give an integral test that determines if the origin splits into exactly two minimal Martin boundary points, one approached through each sector. This test is related to other integral tests due to Benedicks and Chevallier, the former in the special case of a Denjoy domain. We then generalise our test, replacing the pair of rays by an arbitrary number.

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In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic function$u$in a bounded NTA domain$D$in$R$^{$n$}. We achieve this by showing that a weighted Riesz measure of$u$is a Carleson measure, with the Carleson norm bounded by a constant independent of$u$. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions in$D$.