Spectral multiplier theorems of Hormander type on Hardy and Lebesgue spaces

Abstract

Abstract. Let X be a space of homogeneous type and let L be an injective, non-negative, selfadjoint operator on L 2 (X) such that the semigroup generated by −L fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator, acts as a bounded linear operator on the Hardy space H 1 L (X) associated with L whenever F is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.

Boundedness of Stein’s square functions and Bochner-Riesz means associated to operators on hardy spaces

“…P r o o f. The proof is similar to that of Theorem 3.1 [11] or Theorem 4.6 [17]. We omit the details here.…”

“…P r o o f. The proof is similar to that of [12], Proposition 3.1, or [17], Lemma 2.7, so we omit the details here.…”

“…The proof follows from a slight modification of an argument as in [17], Theorem 4.2. In fact, we can get the desired result by using Proposition 5.2 and Lemma 4.2.…”

Lp-Spectral Multipliers for some Elliptic Systems

“…Via an interpolation argument already used in [21], this also yields the assertion of Theorem 2.3. In fact, this is the idea behind the approach we use in [24]. However, the result in [24] applies to operators satisfying generalized Gaussian estimates of any order, which means that essential technical tools such as the finite propagation speed of the wave equation for A, which one has for second-order operators, cannot be used.…”

“…Quite recently, considerable progress has been made (see [7,15,17,22,24]) concerning operators for which the associated semigroups satisfy generalized Gaussian bounds or Davies-Gaffney estimates (see § 2 for more details). Various generalizations of this result have been given since then, in several directions.…”