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For Ξ∈$R$^{$n$},$t$∈$R$and$f$∈$S$($R$^{$n$}) define $$\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$$ . We determine the optimal regularity$s$_{0}such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds where$C$is independent of$f$∈$S$($R$^{$n$}) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].

We give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as $$u = P(\Delta )g(x) + h(x)$$ where$g(x)$is a bounded continuous function,$h(x)$is a bounded real analytic function and$P(d/dt)$is an ultradifferential operator. Also, we show that the boundary value of every heat function with some exponential growth condition determines an ultradistribution in this class. These results generalize the theorem of Matsuzawa [M] for the above class of quasianalytic ultradistributions and partially solve a question of A. Kaneko [Ka]. Our interest lies in the quasianalytic case, although the theorems do not exclude non-quasianalytic classes.

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