We consider the equation , where is a first order pseudo-differential operator with real symbol . Under a suitable convexity assumption on we find the decay properties for . These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem , . If is a smooth function which satisfies near , and is small in suitably Sobolev norm, we prove global existence theorems provided is greater than a critical exponent.
In this note, we prove that the countable compactness of together with the Countable Axiom of Choice yields the existence of a nonmeasurable subset of . This is done by providing a family of nonmeasurable subsets of whose intersection with every non-negligible Lebesgue measurable set is still not Lebesgue measurable. We develop this note in three sections: the first presents the main result, the second recalls known results concerning non-Lebesgue measurability and its relations with the Axiom...
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