We consider Schrodinger semigroups e.- IH, H =-A+V on Iw ” with V---cIxl- ’ as 1x1--rco, O<c<[(l/2)(n-2)] * with H>O. We determine the exact power law divergence of I~e-‘Hi~p,p and of some IIe-‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x)- 1.x-‘.:Ta 1991 Academic Press, Inc. 1.

Abstract. We study the generation of an analytic semigroup in L p (R d) and the determination of the domain for a class of second order elliptic operators with unbounded coefficients in R d. We also establish the maximal regularity of type L q –L p for the corresponding inhomogeneous parabolic equation. In contrast to the previous literature the coefficients of the second derivatives are not required to be strictly elliptic or bounded. Interior singularities of the lower order terms are also discussed. Regularity properties of elliptic operators 1.

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies [2], from the case of the Dirichlet Laplacian of a region\Omega ` R N to that of the higher order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.