Dimension independent Bernstein–Markov inequalities in Gauss space
Dimension independent Bernstein–Markov inequalities in Gauss space
复制标题
高斯空间中维度无关的伯恩斯坦马尔可夫不等式
DOI：
10.1016/j.jat.2020.105377
复制
发表时间：
2020
影响因子：
0.9
通讯作者：
Ivanisvili, Paata
中科院分区：
文献类型：
--
作者：
Eskenazis, Alexandros;Ivanisvili, Paata
我们在高斯空间中得到以下与维数无关的伯恩斯坦-马尔可夫不等式：对于每一个1≤p<∞，存在一个常数Cp>0，使得对于任意k≥1和Rk上的所有多项式p，我们有‖∇p‖Lp(Rk,dγk)≤Cp(degP)12+1πarctan|p−2|2p−1‖p‖Lp(Rk,dγk)，其中dγk是Rk上的标准高斯测度。我们还证明了在任意函数B∈C2(（0,∞))∩C([0,∞)）与B ',B ' >的温和增长假设下，我们有∫RkB|LP(x)|dγk(x)≤∫RkB10(degP)αB|P(x)|dγk(x)，其中L=Δ−x⋅∇是Ornstein-Uhlenbeck半群的生成子，αB=1+2πarctan12sup∈(0,∞)sB ' (s)B ' (s)+B ' (s)sB ' (s)−2。
We obtain the following dimension independent Bernstein–Markov inequality in Gauss space: for each 1≤p<∞ there exists a constant Cp>0 such that for any k≥1 and all polynomials P on Rk we have ‖∇P‖Lp(Rk,dγk)≤Cp(degP)12+1πarctan|p−2|2p−1‖P‖Lp(Rk,dγk),where dγk is the standard Gaussian measure on Rk. We also show that under some mild growth assumptions on any function B∈C2((0,∞))∩C([0,∞)) with B′,B′′>0 we have ∫RkB|LP(x)|dγk(x)≤∫RkB10(degP)αB|P(x)|dγk(x)where L=Δ−x⋅∇ is the generator of the Ornstein–Uhlenbeck semigroup and αB=1+2πarctan12sups∈(0,∞)sB′′(s)B′(s)+B′(s)sB′′(s)−2.