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

TL;DR

This paper establishes the equivalence of Sobolev norms for certain non-symmetric operators combining Laplacians with scaling-critical drift, highlighting differences from Hardy operator studies due to the drift's impact on semigroup bounds.

Contribution

It demonstrates the equivalence of Sobolev spaces associated with these operators, extending previous results to include scaling-critical drift terms with new technical restrictions.

Findings

Sobolev norm equivalence for operators with critical drift

Comparison between Sobolev spaces generated by these operators and classical ones

Identification of restrictions related to semigroup gradient bounds

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.