Off-diagonal decay estimates for the generalized Stokes resolvent

Establish off-diagonal decay estimates for the generalized Stokes resolvent with bounded measurable coefficients that are analogous to the exponential off-diagonal estimates known for second-order elliptic operators, quantifying the influence of data supported in a dyadic annulus on the resolvent solution over a distant ball.

Background

For second-order elliptic operators in divergence form, classical off-diagonal estimates provide exponential decay of the resolvent solution on a ball when the forcing data are supported in a distant dyadic annulus. The paper notes that these estimates are optimal even for the Laplacian.

The pressure term in the generalized Stokes system creates a non-local interaction that obstructs the direct transfer of such elliptic estimates. The paper develops non-local polynomial decay estimates for the generalized Stokes resolvent, but explicitly records that estimates analogous to the classical elliptic off-diagonal bounds remain largely unresolved.

References

So far, analogous estimates for the generalized Stokes resolvent remain largely open; see the discussion in the introduction.

— On the $\mathrm{L}^p$-theory of the generalized Stokes operator with rough coefficients  (2609.10427 - Haardt, 9 Sep 2026) in Section 4, “Decay estimates for the Stokes resolvent,” opening discussion