L^p-Spectral Multipliers for some Elliptic Systems

Abstract

We show results on L p -spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lamé system. Here, we rely on resolvent estimates recently established by Mitrea and Monniaux.

Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

“…If c ≥ 0, then p = (p * c ) ′ = 1 is included. It is also known (see [36]) that the estimates (GGE p ) hold for some p with 1 ≤ p < 2 (and hence the property (FS) also holds) when L is the second order Maxwell operator with measurable coefficient matrices, or the Stokes operator with Hodge boundary conditions on bounded Lipschitz domains in R 3 , or the time-dependent Lamé system equipped with homogeneous Dirichlet boundary conditions.…”

Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law

“…Further, we have S n u L p (D) 6 ≤ C u L p (D) 6 with a constant C > 0 depending on p, but not on u and n. Note that such an estimate is not available for P n in a general situation. This remarkable uniform L p -boundedness is a consequence of [21], together with generalized Gaussian bounds for the Hodge-Laplacian (see [25], [22]). The deep connection between ∆ H and M is a consequence of the formula −∆ H = curl curl − grad div, which implies ∆ H = M 2 in the range of the Helmholtz projection P H and M 2 = 0 in the range of (I −P H ).…”

“…In the following Lemma, we exploit the fact Proof. The first claim can be found in [25], section 3 or in [22], Lemma 5.4. Consequently, we also have S n P H = P H S n and P n P H = P H P n , since S n and P n are in the functional calculus of ∆ H .…”

“…Proof. In [22], the authors argue on page 239, that both A (1) and A (2) satisfy generalized Gaussian…”

Spectral multiplier theorems of Hormander type on Hardy and Lebesgue spaces

“…It is well-known that p 0 is sharp in the sense that for any r / ∈ [p 0 , p ′ 0 ] there exists an operator L in the given class for which e −tL cannot be extended from [23,Theorem 10]). In another paper ( [41]) we discuss how spectral multiplier theorems of the type presented here apply to the second order Maxwell operator with measurable coefficient matrices and the Stokes operator with Hodge boundary conditions on bounded Lipschitz domains in R 3 as well as the time-dependent Lamé system equipped with homogeneous Dirichlet boundary conditions.…”