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

We give a simple argument to obtain ${\mathrm{L}}^p$-boundedness for heat semigroups associated to uniformly strongly elliptic systems on ${\mathbb{R}}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give $p$ explicitly in terms of ellipticity. It is optimal at the endpoint $p=\mathrm{\infty }$. We also obtain ${\mathrm{L}}^p$-estimates for the gradient of the semigroup, where $p>2$ depends on ellipticity but not on dimension.