Explicit improvements for Lp‐estimates related to elliptic systems

We give a simple argument to obtain Lp‐boundedness for heat semigroups associated to uniformly strongly elliptic systems on Rd 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 = ∞. We also obtain Lp‐estimates for the gradient of the semigroup, where p > 2 depends on ellipticity but not on dimension.