L^p-L^q off-diagonal estimates for the Ornstein--Uhlenbeck semigroup: some positive and negative results

2016/07/06 by Amenta, Alex, Teuwen, Jonas
#43A99 #47D06 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1607.01666

Abstract

We investigate L^p(γ)-L^q(γ) off-diagonal estimates for the Ornstein-Uhlenbeck semigroup (e^tL)_{t > 0}. For sufficiently large t (quantified in terms of p and q) these estimates hold in an unrestricted sense, while for sufficiently small t they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.

Related

• Regularization in L₁ for the Ornstein--Uhlenbeck semigroup
• q-Invariant Functions for Some Generalizations of the Ornstein-Uhlenbeck Semigroup
• Maximal operator, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operators
• Littlewood-Paley functions associated with general Ornstein-Uhlenbeck semigroups
• Description of the smoothing effects of semigroups generated by fractional Ornstein-Uhlenbeck operators and subelliptic estimates