On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

Abstract

Abstract. This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein-Uhlenbeck semigroup e tL . Our approach is to expand the Mehler kernel into Hermite polynomials and applying the powers L N of the Ornstein-Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L. As an application we give an alternative proof of the kernel estimates by [11], making all relevant quantities explicit.

– Off-Diagonal Estimates for the Ornstein–uhlenbeck Semigroup: Some Positive and Negative Results

“…By the same argument, we can prove failure of L p (γ)-L q (γ) off-diagonal estimates for the derivatives (L m e tL ) m∈N of the Ornstein-Uhlenbeck semigroup, with the same conditions on (p, q, t) and the same class of testing sets (E, F). This relies on an identification of the kernel of L m e tL , which has been done by the second author in [12].…”

Littlewood-Paley-Stein Theory and Banach Spaces in the Inverse Gaussian Setting

“…We think that some signs in the expression of [31,Theorem 5.1] are not correct. The new corrected equality is the following…”