Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift

2024/09/30 by Bui, The Anh, Duong, Xuan Thinh, Merz, Konstantin · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2410.00191

Abstract

We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of L^p-Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e., ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup.

Cited by

• Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators

Related

• Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space
• Fractional Kolmogorov operator and desingularizing weights
• Limiting behaviour of intrinsic semi-norms in fractional order Sobolev spaces
• On the limiting behaviour of some nonlocal seminorms: a new phenomenon
• Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential term