We consider Fourier ultra-hyperfunctions and characterize them as boundary values of smooth solutions of the heat equation. Namely we show that the convolution of the heat kernel and a Fourier ultra-hyperfunction is a smooth solution of the heat equation with some exponential growth condition and, conversely that such smooth solution can be represented by the convolution of the heat kernel and a Fourier ultra-hyperfunction.
"Fourier Ultra-Hyperfunctions as Boundary Values of Smooth Solutions of the Heat Equation." Tokyo J. Math. 25 (2) 381 - 398, December 2002. https://doi.org/10.3836/tjm/1244208861