Generalized Gaussian estimates and riesz means of Schrödinger groups

Abstract

We show that generalized Gaussian estimates for selfadjoint semigroups (e~M),<=R + on L 2 imply L pboundedness of Riesz means and other regularizations of the Schrodinger group (e" A ),<=R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjostrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.

Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem

“…Note that in virtue of our assumptions the semigroup exp(−z∆), z ∈ C + satisfies condition (4.45) with For the relevance of assumption (4.46), see [14], [9], and Corollary 4.14 above.…”

“…We give below a proof that follows directly from Theorem 4.13. The conclusion of the corollary is instrumental in the theory of Riesz means (see [14,9], and references therein); see also condition (HG α ), p.339 in [32] and its consequences.…”

“…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.…”

“…The so-called generalized Gaussian bounds that such semigroups may satisfy were studied for instance by Davies in [25,Lemmas 23 and 24], and they were extensively discussed by Blunck and Kunstmann (see [8,9,10,11,12]). Estimates of a similar nature were also considered in [42], see Propositions 2.6 and 2.8 in this paper.…”

“…(b) In [8,9,10,11,12], Blunck and Kunstmann develop spectral multiplier theorems for operators which generate semigroups without heat kernel acting on spaces satisfying the doubling condition 4 . As their basic assumption they consider the following form of generalized Gaussian estimates (4.39)…”