Functional calculus for generators of symmetric contraction semigroups

Abstract

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\infty$, a H\"ormander-type holomorphic functional calculus on $L^p$ in the sector of angle $\phi^*_p=\arcsin|1-2/p|$. The obtained angle is optimal.Comment: 26 pages, minor corrections and slight changes of notation. Some changes in Sections 4, 6 and

Joint Spectral Multipliers for Mixed Systems of Operators

“…Using [22, Theorem 3. 5 Let now A be a non-negative, self-adjoint operator defined on a space L 2 (Y, μ), where Y is equipped with a metric ζ such that (Y, ζ, μ) is a space of homogeneous type, i.e. μ is a doubling measure.…”

“…5]. Moreover, by a recent result of Carbonaro and Dragičević [5] (see also [8]), every operator for which (CTR) holds satisfies (3.2) with the optimal angle φ j p = φ * p := arcsin |2/ p − 1| and θ j = θ = 3. Put in other words every operator generating a symmetric contraction semigroup has an H ∞ functional calculus on L p in every sector larger than S φ * p .…”

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

“…If interested in the genesis of Bellman functions and the overview of the method, the reader is also referred to [68,81,82]. The method has seen a whole series of applications, yet until recently (see [10,11]) mostly in Euclidean harmonic analysis.…”

“…For the purpose of proving (16) we apply the heat-flow technique developed in [10]. Fix u ∈ L p (µ), v ∈ L q (µ) and consider the functional…”

“…We now briefly describe the technique we utilise for proving equality (2). Our approach is based on [10], which was the first case of Bellman functions being applied for the study of spectral multipliers.…”