Hilbert空间中球的Gauss测度的一个注记

A NOTE ON GAUSSIAN MEASURE OF BALLS IN HILBERT SPACE

设Ｈ是一个可分的Ｈｉｌｂｅｒｔ空间，μ是Ｈ上的一个对称Ｇａｕｓ测度，λ１≥λ２≥…＞０是μ的共变算子的特征值，那么存在一个仅和ｔ有关的正常数ｃｔ，使得对任意的ｒ∈Ｈμ｛ｘ∈Ｈ：‖ｘ‖≤ｔ｝－μ｛ｘ∈Ｈ：‖ｘ＋ｒ‖≤ｔ｝≤ｃｔ（λ１λ２）１２‖ｒ‖２．

Let μ be a symmetric Gaussian measure on a separable Hilbert space H.λ 1≥λ 2≥…>0 be the eigenvalues of its covariance operator.Then there exists a constant c t depending only on t,such that for every r∈H the following estimate is valid: μ{x:x∈H,‖x‖≤t}-μ{x:x∈H,‖x+r‖≤t}≤c t(λ 1λ 2) 12 ‖r‖ 2.