$\mathrm{L}^p$-extrapolation via non-local decay estimates: theory and applications to the generalized Stokes operator with rough coefficients

Abstract

In this article, we establish $\mathrm{L}^p$-mapping properties for the generalized Stokes operator with bounded measurable coefficients on the full range of exponents that is known to be sharp for elliptic systems with rough coefficients. These include $\mathrm{L}^p$-estimates for the corresponding semigroup, its gradient and the $\mathrm{H}^\infty$-calculus, as well as $\mathrm{L}^q$-$\mathrm{L}^p$ smoothing estimates. As a key ingredient, we develop a non-local framework for $\mathrm{L}^p$-extrapolation that captures the relationship between hypercontractivity, non-local decay estimates and uniform $\L^p$-bounds.