Spectral multiplier theorems and averaged R-boundedness

Abstract

Let A be a 0-sectorial operator with a bounded H ∞ (Σ σ )-calculus for some σ ∈ (0, π), e.g. a Laplace type operator on L p (Ω), 1 < p < ∞, where Ω is a manifold or a graph. We show that A has a H α 2 (R + ) Hörmander functional calculus if and only if certain operator families derived from the resolvent (λ−A) −1 , the semigroup e −zA , the wave operators e itA or the imaginary powerswe denote the R-bound of this set. In a Hilbert space X, R[L 2 (J)]-boundedness reduces to the simple estimate J | N(t)x, y | 2 dt 1 2 ≤ C x y for all x, y ∈ H.Assume now that A is a 0-sectorial operator with an H ∞ (Σ σ ) calculus for some σ ∈ (0, π) on a Banach space isomorphic to a subspace of an L p (Ω) space with 1 ≤ p < ∞ (or more generally, let X have Pisier's property (α)). Then our main results, Theorems 6.1 and 6.4 show (among other statements), that the following conditions on the operator function above are essentially equivalent:1. A has an R-bounded H α 2 spectral calculus, i.e.

Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

“…In particular, the argument which shows continuity of t → φ(B − t) in operator norm of X is still valid. We also note that, if we apply ψ ∈ Φ(τ X ), we have (12) φ…”

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

“…, for the imaginary powers in [39], for the resolvents in [38,Proposition 3.9], and for the semigroup, see this paper at Remark 7.2. The case of general p is entirely similar.…”

“…Thus, t → t −β A it x, y belongs to L 2 (R) for all x, y ∈ X if and only if β > m + 1 2 , so that A cannot have a H β 2 calculus for β ≤ m + 1 2 . We refer to [39] for an adequate adaptation of the R-boundedness notion which is equivalent to the R-bounded H β 2 calculus. On the other hand, A it ∼ = t m , so that the assumption in (2) of Theorem 6.1 holds with α = m, and since X is a Hilbert space, also the assumption in (1).…”

“…Our framework also allows us to work out a rather general version of the Paley-Littlewood theory for operators with a H α p calculus, including estimates of the form (for X = L q (U)): 7) and their analogues in Banach spaces, see [38]. For precise characterizations of the exponents of the Hörmander calculus in terms of R-bounds and square functions of the form (1.6), (1.7), also in Banach spaces, see [39,36].…”

Dunkl spectral multipliers with values in UMD lattices

“…A theorem of Hörmander type holds true for many elliptic differential operators A, including sub-laplacians on Lie groups of polynomial growth, Schrödinger operators and elliptic operators on Riemannian manifolds, see [3,21,34,35]. More recently, spectral multipliers have been studied for operators acting on L p (Ω) only for a strict subset of (1, ∞) of exponents [9,18,19,20,48,49], for abstract operators acting on Banach spaces [45], and for operators acting on product sets Ω 1 ×Ω 2 [61,68,69]. A spectral multiplier theorem means then that the linear and multiplicative mapping…”