Structure of Quasi-analytic Ultradistributions

We study the structure of functions between distributions and hyperfunctions. The structure theorem is known for distributions, non-quasi-analytic ultradistributions and hyperfunctions. In this paper, we try to fill the gap among them. We prove the structure theorem for quasi-analytic ultradistributions. In this paper, we discuss the structure of generalized functions. It is wellknown that any distribution f is locally represented as f = P (D)g ,w hereP (D) is a finite order differential operator with constant coefficients and g is a continuous function, which is the structure theorem for distributions. The structure theorems for non-quasi-analytic ultradistributions ([1, 5]) and hyperfunctions ([3]) are also known. In this paper, we study the structure of functions between them, namely, the structure of quasi-analytic ultradistributions. We prove the structure theorem for non-analytic ultradistributions which includes both nonquasi-analytic and quasi-analytic ones. It is our main theorem to prove that any non-analytic ultradistribution f of the class ∗ is locally represented as f = P (D)g ,w hereP (D) is an ultradifferential operator of the class ∗ and g is an ultradifferentiable function of the class † > ∗. We also claim that this