On the L-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

Abstract

We study positive C 0 -semigroups on L p associated with second-order uniformly elliptic divergence-type operators with singular lower-order terms, subject to a wide class of boundary conditions. We obtain an interval ðp min ; p max Þ in the L p -scale where these semigroups can be defined, including the case 2 = 2 ðp min ; p max Þ. We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined. # 2002 Elsevier Science (USA)

Generalized Gaussian estimates and riesz means of Schrödinger groups

“…This occurs, for example, for elliptic operators A that are of higher order or have singular lower order terms [12,21]. However, in many of these cases A still satisfies so-called generalized Gaussian estimates (GGEs); see [11,25].…”

Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem

“…Estimates of a similar nature were also considered in [42], see Propositions 2.6 and 2.8 in this paper. For interesting considerations about L p − L q Gaussian estimates, see also [5].…”

“…Therefore standard heat kernel theory can not be applied to study the semigroup generated by L (c) . However, it was proved in [56], see also [42], that exp(−L (c) ) p→p is finite for all p in the interval ((p * c )…”

“…For example, one can consider the situation where L is an operator with periodic coefficients in divergence form and L 0 is its homogenization, to obtain Gaussian estimates for the difference of the corresponding heat kernels |p t (x, y) − p 0 t (x, y)|. Next, our methods have various applications in the theory of L p to L q Gaussian estimates, developed by Blunck and Kunstmann in [8,9,10,11,12], see also [42] and [5]. Blunck and Kunstmann call such estimates generalized Gaussian estimates.…”

$L^p$-Theory for Elliptic Operators on $\mathbb R^d$ with Singular Coefficients

“…Recent and quite general results in this direction are presented in [22] and [27] using form methods; see also [3] for a different approach based on an approximation procedure. However, the determination of the domain is a much more difficult question which requires more assumptions on the coefficients.…”