Transference principles for semigroups and a theorem of Peller

Abstract

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, culminating in a new proof of classical results by Peller from 1982. The method allows a generalization of his results away from Hilbert spaces to L p -spaces and -involving the concept of γ-boundedness -to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.In the classical transference principle(s) the objects under investigation are derived operators of the formwhere G is a locally compact group and T = (T (s)) s∈G : G → L(X) is a bounded strongly continuous representation of G on a Banach space X. The integral (1.1) has to be understood in the strong sense, i.e.,Since such operators occur in a variety of situations, the applications of transference principles are manifold, and the literature on this topic is vast. We therefore restrict ourselves to mentioning only a few 'landmarks' which we regard as most important for the understanding of the present paper.

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

“…The following corollary covers some ideas which are included in [78, Theorem 5.5.12, etc] using Peller's method, and in [44] 4), and a similar result for bounded holomorphic semigroups on Banach spaces was given by Schwenninger [67]. The results are immediate consequences of Lemma 3.2(2).…”

“…Letting δ ′ → 0+ with δ fixed and using [44], and a smaller class in [78]). We obtain the estimate for all functions in B.…”

“…It is a base for so-called transference principles and their applications in harmonic analysis and operator theory. While classical aspects of the transference techniques can be found in [20], its modern treatment and applications to semigroup theory are contained in [44] (see also [57]).…”

On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

“…2.5]. Using powerful transference principles from [20], Haase and Rozendaal generalized this to arbitrary Banach spaces for f in the analytic multiplier algebra AMp(X) ⊂ H ∞ (C+), p ≥ 1, see in [21]. Note that the latter inclusion is a strict embedding unless p = 2 and X is a Hilbert space (in which case equality holds by Plancherel's theorem).…”

Maximal ergodic inequalities for Banach function spaces

“…In [12] the Transfer Principle of Coifman and Weiss is extended to weighted Orlicz spaces for group actions that are uniformly bounded in a sense determined by the space. Other important contributions regarding the development of the transfer principle include the work of Haase [16], Lin and Wittmann [22], and of Asmar, Berkson and Gillespie ( [2] and [3]). However with each of these the focus of the research is somewhat different to that of the present paper.…”