The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)
Tom 131 / 1998
Studia Mathematica 131 (1998), 205-214 DOI: 10.4064/sm-131-3-205-214
The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:= d^2/dx^2 - 2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1(dγ)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.