局部凸空间上检验函数与广义函数空间的嵌入定理及其应用

设Y是局部凸向量空间,其上装配有GaussianRadon测度γ．A（Y）（或ε（Y）是Y上检验函数空间（或με（Y）是相应的分布函数空间·我们证明了：（或με（Y）,并由此得到μA（Y）（或με（Y））上的Fourier变换公式.其中“*”表示复共轭算子,“”表示连续稠线性嵌入.进一步还得到了A（Y）（或ε（Y））上无穷维伪微分算子A是L2（Y,γ）上连续的充要条件是其共轭算子A’满足A’（L2（Y,γ）L2（Y,γ）．

An Embedding Theorem for Space of Distributions on Locally Convex Space

Let Y be a locally convex vector space with a Gaussian Radon measureγ denote by A(Y) and (y), two spaces of test functions on Y, and denote byμ(Y) and με(y), two spaces of distributions on Y. We prove that A(Y) (or(the symbol "*" means the complex conjugate operation). Using this embedding theorem we give a direct formula forFourier transform of h∈L2(Y, γ) as a distributions on Y. Moreover, by this embeddingtheorem we prove that infinite dimensional pseudodifferential operator A on A(Y) (orε(Y)) is continuous ω.r.t the norm of L2(Y, γ) if and only if for the adjoint operatorA' of A, we have A'(L'(Y, γ)) L2(Y, γ).