Fourier Multiplier Theorems Involving Type and Cotype

Abstract

In this paper we develop the theory of Fourier multiplier operators T m :The case p = q has been studied extensively since the 1980s, but far less is known for p < q. In the scalar setting one can deduce results for p < q from the case p = q. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for p < q other geometric conditions on X and Y , such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for T m without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as 1 p − 1 q remains constant.

Sharp growth rates for semigroups using resolvent bounds

“…if m ∈ M p,p (R n ; L(X)). In the case where X = L p (Ω) for p ∈ [1, ∞) it follows from the proof of [40,Theorem 3.24] or [50, Theorem 2] that m ∈ M p,p (R; L(X)) with the required estimate. Next, assume that p = ∞ and let f := m k=1 1 E k ⊗ x k for m ∈ N, E 1 , .…”

“…To prove Theorem 1.1 we use the connection between stability theory and Fourier multipliers which goes back to e.g. [21,24,30,49] and which was renewed in [39], following the development of a theory of operator-valued (L p , L q ) Fourier multipliers in [38,40]. In particular, Theorem 3.2 gives a Fourier multiplier criterion for a bound as in (1.3) to hold, and Corollary 3.13 gives a characterization of polynomial growth and uniform boundedness of a semigroup in terms of multiplier properties of the resolvent.…”

“…To this end we first prove a multiplier theorem for positive kernels. Part of this result is already contained in[40, Theorem 3.24]. Recall that a subspace X of a Banach lattice Y is a sublattice if x ∨ y, x ∧ y ∈ X for all x, y ∈ X.…”

Stability theory for semigroups using (L,L) Fourier multipliers

“…In doing so we extend the Fourier analytic characterization of exponential stability to this more refined setting. Then, using the theory of operator-valued (L p , L q ) Fourier multipliers which was developed in [55,56] with applications to stability theory in mind, we derive concrete polynomial decay rates from this characterization. These results involve only growth bounds for the resolvent and are new even on Hilbert spaces.…”

“…In this article we argue that for the study of asymptotic behavior it is more natural to consider general p ∈ [1, ∞) and q ∈ [p, ∞]. It was observed in [55,56] that one can derive boundedness of Fourier multipliers from L p (R; X) to L q (R; Y ) for p < q under different geometric assumptions on X and Y than in the case where p = q, and assuming decay of the multiplier at infinity but no smoothness. In fact, the parameters p and q depend on the geometry of X, and the amount of decay which is required at infinity is proportional to 1 p − 1 q .…”