Optimal Gevrey classes for the existence of solution operators for linear partial differential operators in three variables

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For a constant coefficient linear partial differential operator acting on all infinitely differentiable functions or ω-ultradifferentiable functions of Beurling type on euclidean 3-space, the existence of a continuous linear solution operator is investigated. It is shown that there is an optimal weight ω in the sense that a solution operator exists for a weight σ if and only if ω = O ( σ ) , provided that such an operator exists for at least one weight. Furthermore, the optimal class is either a Gevrey class of rational exponent or the class of all infinitely differentiable functions.