Semilinear pseudodifferential equations in spaces of tempered ultradistributions

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of $u$ with sub-exponential growth with respect to $u,$ we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution $u$.

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