Logarithmic heat-semigroup growth at the critical weight exponent

Prove that, at the critical weight exponent γ = (2 − k_bc)p − 1 for the Dirichlet or Neumann Laplacian acting on the weighted Sobolev space W^{k,p}_{γ,bc}(ℝ^d_+, w_γ; X), the heat semigroup has the logarithmic growth rate ||T^d_bc(t)|| ≍ (1 + log(1+t))^{1/p′} for all t ≥ 0.

Background

The paper establishes optimal polynomial growth estimates for the Dirichlet and Neumann heat semigroups on weighted Sobolev spaces when the weight exponent lies away from the critical value γ = (2 − k_bc)p − 1. At this critical exponent, the polynomial growth exponent becomes borderline, and the paper does not identify the generator domain or prove the corresponding semigroup estimate.

The authors conjecture that the borderline behavior is logarithmic rather than polynomial. A later discussion in Remark \ref{rem:critical-log} supports this conjecture by deriving logarithmic resolvent estimates and outlining potential upper- and lower-bound arguments, but the result is not proved in the paper.