Explicit improvements for Lp$\mathrm{L}^p$‐estimates related to elliptic systems

Abstract

AbstractWe give a simple argument to obtain ‐boundedness for heat semigroups associated to uniformly strongly elliptic systems on by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give explicitly in terms of ellipticity. It is optimal at the endpoint . We also obtain ‐estimates for the gradient of the semigroup, where depends on ellipticity but not on dimension.