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

The Anh Bui, Xuan Thinh Duong, Konstantin Merz

TL;DR

This work establishes an equivalence between Sobolev norms generated by the scaling-critical Kolmogorov operator Λκ = (-Δ)^{α/2} + (κ/|x|^α) x·∇ and those generated by the pure fractional Laplacian, in a carefully constrained range of p and s dictated by the drift's singularity. The authors develop a robust framework based on heat-kernel bounds, continuous square-function estimates, and novel reversed and generalized Hardy inequalities to relate Λκ to Λ0, despite the non-symmetric gradient perturbation. They identify two critical coupling constants and show the range of admissible Sobolev exponents is restricted by the drift, with additional αs < α−1 restrictions emerging from gradient effects. The results have concrete applications to nonlinear diffusion equations and perturbation theory, enabling analysis that replaces complex operator functions of Λκ with those of the simpler Λ0, and potentially impacting mean-field limits and related quantum/mechanical problems.

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.

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

TL;DR

This work establishes an equivalence between Sobolev norms generated by the scaling-critical Kolmogorov operator Λκ = (-Δ)^{α/2} + (κ/|x|^α) x·∇ and those generated by the pure fractional Laplacian, in a carefully constrained range of p and s dictated by the drift's singularity. The authors develop a robust framework based on heat-kernel bounds, continuous square-function estimates, and novel reversed and generalized Hardy inequalities to relate Λκ to Λ0, despite the non-symmetric gradient perturbation. They identify two critical coupling constants and show the range of admissible Sobolev exponents is restricted by the drift, with additional αs < α−1 restrictions emerging from gradient effects. The results have concrete applications to nonlinear diffusion equations and perturbation theory, enabling analysis that replaces complex operator functions of Λκ with those of the simpler Λ0, and potentially impacting mean-field limits and related quantum/mechanical problems.

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 -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.
Paper Structure (28 sections, 27 theorems, 266 equations)

This paper contains 28 sections, 27 theorems, 266 equations.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Main result
  4. Proof of Theorem \ref{['eqsob']}
  5. Discussion of Theorem \ref{['eqsob']}
  6. Comparison with earlier results
  7. Impact of Theorem \ref{['eqsob']}
  8. Estimates involving the heat kernel of $\Lambda_\kappa$
  9. Spatially averaged estimates
  10. Pointwise estimates
  11. Proof of square function estimates (Theorem \ref{['squarefunctions']})
  12. Bounds for differences of kernels
  13. Integral bounds
  14. An estimate for $|Q_t(x,y)|$
  15. An estimate for $|(Q_t f)(x)|$
  16. ...and 13 more sections

Key Result

Lemma 1.1

Let $\alpha\in(1,2]$ and $M=\alpha$ if $\alpha<2$ and $M=\infty$ if $\alpha=2$. Then, the map $((d+\alpha)/2,d+M)\ni\beta\mapsto\Psi(\beta)$, given by eq:defbeta is strictly monotonously decreasing.

Theorems & Definitions (52)

  • Lemma 1.1
  • Theorem 1.2: Equivalence of Sobolev norms
  • Theorem 1.3
  • Theorem 1.4: Reversed Hardy inequality
  • Theorem 1.5: Generalized Hardy inequality
  • proof : Proof of Theorem \ref{['eqsob']}
  • Remark 1.6
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • ...and 42 more