Fourier multiplier theorems on Besov spaces under type and cotype conditions

Abstract

Abstract. In this paper we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents p and q, which depend on the type p and cotype q of the underlying Banach spaces. In a previous paper we considered L p -L q -multiplier theorems. In the current paper we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the L p -L q -setting as well.We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties then the boundedness of the multiplier operator extrapolates to other values of p and q for which

Sharp growth rates for semigroups using resolvent bounds

“…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.…”

Optimal rates of decay for operator semigroups on Hilbert spaces

“…Thus Theorem 3.1 is already optimal in these cases, and our result improves Theorem 3.1 only if the growth of M is sufficiently close to being polynomial. The proof of Theorem 3.2 combines ideas taken from [6, Theorem 4.7] and [15], and is inspired by techniques from operator-valued Fourier multiplier theory; see [31][32][33]. More specifically, the first step of our proof is to decompose each relevant semigroup orbit into what may be viewed as a high-frequency component and a low-frequency component.…”

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

“…To properly define Fourier multipliers for symbols with a singularity at zero, we briefly introduce the class of vector-valued homogeneous distributions. For more on these distributions see [55]. For X a Banach space leṫ S(R; X) := {f ∈ S(R; X) | f (k) (0) = 0 for all k ∈ N 0 }, endowed with the subspace topology, and letṠ ′ (R; X) be the space of continuous linear mappings fromṠ(R; C) to X. ThenṠ(R; X) is dense in L p (R; X) for all p ∈ [1, ∞), and L p (R; X) can be naturally identified with a subspace ofṠ ′ (R; X) for all p ∈ [1, ∞].…”

“…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.…”