Abstract
ly concave domains these results do not enable us to obtain precise theorems; the reason for this is that the symbol q) should not change at the moment of reflection of the bicharaeteristic from the boundary. In these domains results have been obtained by completely different methods [8-10]. Finally, in domains of arbitrary type results have been obtained by the energy method but with very essential appeal to results obtained for bicharacteristic ally concave domains [Ii, 12]. Analogous results were obtained independently and practically simultaneously by the author and were announced in [13]; however, the author used an improved energy method which made it possible to remove the condition that ~ be constant on reflection by replacing it by a condition that ~ not decrease. This, in turn, made it possible to obtain the results directly without appealing to results for bicharacteristic aIiy concave domains. It is true that here it is necessary to consider only boundary-value problems satisfying the Shapiro-Lopatins
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