傅立叶超函数和扩充傅立叶超函数与爱米特热方程(英文)

Relating Fourier Hyperfunctions and Extended Fourier Hyperfunction to Hermite Heat Equation

证明了傅立叶超函数和扩充傅立叶超函数可用爱米特热方程的解来表示,且用以表示的解有很良好的性质.

It was proved by K. W. Kim, S. Y. Chung and D. Kim that if a C∞- solution u（x,t） of the heat equation in Rn＋1^＋ satisfies │u（x,t）│≤C·exp（ε（1/t＋t│x│）） for any ε〉 0, and some C 〉 0, then its boundary determines a unique Fourier hyperfunction; and conversely, any Fourier hyperfunction is the boundary of such a u（x. t）. Also, S. Y. Chung, D. Kim and K. Kim showed that replacing ＂any ε：〉 0＂ by ＂some ε 〉 0＂, then the above statements are true for extended Fourier hyperfunctions （called Fourier ultra-hyperfunctions also in the literature）. We show that replacing solutions of the heat eauation by solutions U （x,t）of the Hermite heat equation,and exp（ε（1/t＋t＋│x│））by （√1＋e^4t/e^-t）^n e- 2/│x│^2 1＋e^-4te^t/1-e^-4t e^t（1-e^-4t/1＋e^-4t＋1＋e^-4t/e^-2t │x│） then the above results relating Fourier hyperfunctions and extended Fourier hypertunctions to heatequation become the relations with Hermite heat equations. Furthermore we proved that for fixed t,U（x,t）is an element of the space of test functions for extended Fourier hyperfunctions, thus Fourier hyperfunctions and extended Fourier hyperfunctions are limits of such nice functions. This gives also a new proof of the recent result of K. Kim on denseness of test functions in the space of extended Fourier hyperfunctions. Perhaps, the most interesting thing is that if U（x,t） represents a Fourier hyperfunetion or an extended Fourier hyperfunction u, then the Fourier transformation of U（x,t） with respect to x represents the Fourier transformation of u.