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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$D$.

We consider systems of partial differential equations of the first order in$t$and of order 2$s$in the$x$variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid in$R$^{n}×(0,$T$] for$s$≥2.

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