Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)P(D) be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on RnRn and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e−tHe−tH generated by H=P(D)+VH=P(D)+V with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e−t(Δ+V)e−t(Δ+V) as V∈K2(Rn)V∈K2(Rn), the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e−tP(D)e−tP(D) on L1(Rn)L1(Rn). As a consequence of the Gaussian upper bound, the LpLp-spectral independence of H is concluded.