For an integral domain$R$and a non-zero non-unit$a$ε$R$we consider the number of distinct factorizations of$a$^{$n$}into irreducible elements of$R$for large$n$. Precise results are obtained for Krull domains and certain noetherian domains. In fact, we prove results valid for certain classes of monoids which then apply to the above-mentioned classes of domains.

We show that the$L$^{$p$}norms, 0<$p$<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectively$A$_{∞}with respect to the corresponding harmonic measure or normalized harmonic measure.

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