Dimension independent Bernstein–Markov inequalities in Gauss space

Abstract

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.