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∂u ∂t + it ∂u ∂x = f, f ∈ C∞ 0 (R) , has no solution in the space D′(R2) of distributions, or in the space B(R) of hyperfunctions (see [T] for the historical backgrounds). On the other hand, N. Aronszajn introduced an abstract Frechet space, the Aronszajn space of the traces of the analytic solutions of the heat equations in C × {t ∈ C, Re t > 0}, and M. S. Baouendi [B] simplified the complicated arguments and showed that this equation has a solution in the Aronszajn space. In this talk, we introduce a much simpler space X (R) of real analytic and exponentially decreasing functions and show that the Fourier transformation is an isomorphism on X (R), and also on its strong dual X ′(Rn). As an application, applying the Fourier transformation only we show that the Mizohata operator is solvable in C(Rt;X ′(R)). The complete proofs will be published elsewhere.

The space of exponentially decreasing entire functions and its application to solvability