A note on operator‐valued Fourier multipliers on Besov spaces

Abstract

Let X be a Banach space. We show that each m : R \ {0} → L(X) satisfying the Mikhlin condition sup x =0 ( m(x) + xm (x) ) < ∞ defines a Fourier multiplier on B In recent years, operator-valued Fourier multipliers on vector-valued function spaces have been extensively studied [1,3,10,16]. They are needed to establish existence and uniqueness as well as regularity for differential equations in Banach spaces. In the L p spaces case, it was Weis who found the formulation of the operator-valued Fourier multiplier theorem: let X be a Banach space and let m : R \ {0} → L(X) be a C 1 -function, assume that X is a UMD space (see [5,7]), 1 < p < ∞ and that both sets {m(x) : x = 0} and {xm (x) :where we denote by L(X) the set of all bounded linear operators on X. It is known that in Weis' result the R-boundedness may be replaced by the norm-boundedness only when X is isomorphic to a Hilbert space [4].In the vector-valued Besov spaces case, things are different. It was Amann [1] (see also Weis [17]) who discovered that the assumption of R-boundedness is not needed and no hypothesis on the geometry of the underlying Banach space is required in the Besov spaces case: let m : R → L(X) be a C 2 -function, assume thatthen for 1 ≤ p, q ≤ ∞, s ∈ R, m defines a Fourier multiplier on B s p,q (R; X) (see also [10] for further development, and [2] for the corresponding result in the periodic Besov spaces case).In this paper, we are interested in the following questions: what happens when we consider a larger class of multiplier functions m in the Besov spaces case? In view of known Fourier multiplier results in the scalar L p spaces case, one may consider the following two natural classes: the first one is the set of all

Hölder–Besov boundedness for periodic pseudo-differential operators

“…In the general case of Compact Lie groups we refer the reader to the works of Alexopoulos, Anker, Coifman, Ruzhansky, Turunen and Wirth [2,3,18,29,30,31,32,33]. The general case of operator-valued Fourier multipliers on the torus has been investigated by Arendt, Bu, Barraza, Denk, Hernández, and Nau in [4,5,6,9,10,11]. L p and Hölder estimates of periodic pseudo-differential operators can be found in [12,13,14,19] and [26].…”

“…In the special case p = q = ∞, Λ r (T) = B r ∞,∞ (T) is nothing else but the familiar space of all Hölder continuous functions of order 0 < r < 1. There are several possibilities concerning the conditions to impose on a symbol σ in the attempt to establish a periodic Fourier multiplier theorem of boundedness on Besov spaces and Lebesgue spaces for its corresponding operator (1.1) (see [5,6,9,10,11]). In this paper we investigate the action of periodic Fourier multipliers and periodic pseudo-differential operators from Hölder spaces into Besov spaces.…”

Operator-valued Fourier multipliers on toroidal Besov spaces

“…There, in Theorem 4.2, the autors proved that each sequence M : Z → L (E) satisfaying the variational Marcinkiewicz condition is a Fourier multiplier on B s p,q (T, E) if and only if 1 < p < ∞ and E is a UMD-space. The corresponding result of this theorem for Besov spaces on the real line has been established by Bu and Kim in [BK05b]. The variational Marcinkiewicz condition, giving in [AB04] is equivalent to the bounded variation condition (5.4) in case n = 1.…”

“…In contrast to extensive theory on E−valued distributions in general and Fourier multiplier theorems on L p (R n , E) and B s p,q (R n , E) (and its aplications to partial differential equations) in particular, the contribution in literature to E−valued periodic distributions is rather sparse. The classical Fourier multiplier theorems of Marcinkiewicz and Mikhlin are extended to vector-valued functions and operator-valued multipliers on Z n , which satisfy certain R-boundedness condition, in [Zim89], [BK05a], [BK05b], [ŠW07] and [Na12], for example. More specifically, they established Fourier multiplier theorems on L p (T n , E) if 1 < p < ∞, E is a UMD-space and, instead uniform boundedness, a R-boundedness condition similar to condition (5.4) in this work holds.…”

The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory

“…The analytical and spectral properties for periodic operators have been treated in the work of Ruzhansky and Turunen [50, 51], Delgado [26], Molahajloo and Wong [42–44], and in the works of the authors [15–17, 20, 23, 25, 38, 39]. For the nuclearity of multilinear periodic operators in ‐spaces we refer the reader to the works of the authors [21, 22] and to Delgado and Wong [31] for the linear case. If , Amann [2], Arendt and Bu [4, 5], Barraza, Gonzalez and Hernández [6], Barraza, Denk, Hernández and Nau [7], Rabinovich [46], Bu and Kim [11–13] and Bu [14] investigated the mapping properties of the vector‐valued pseudo‐differential operators on ‐spaces and Besov spaces. The applications to PDE's studied by Denk and Nau [32], Keyantuo, Lizama, and Poblete [36] and references therein.…”