Operator-Valued Fourier Multiplier Theorems on Triebel Spaces

Hölder–Besov boundedness for periodic pseudo-differential operators

“…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.…”

“…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].…”

Regularity of Difference Equations on Banach Spaces

“…However, one requires more on the smoothness of the multipliers (see Theorem 2.5.33). We recall the definition of periodic Triebel-Lizorkin spaces in the vector-valued case [28]. We use the same notations S.R/, D.OE0; 2 /, D 0 .OE0; 2 I X/,ˆ.R/ as in the preceding section…”

“…The following result gives a sufficient condition for an operator-valued sequence to be a F s p;q -multiplier (see [28]). Theorem 2.5.33.…”

“…34. In [28] by using Theorem 2.5.33 the authors gave a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions. Remark 2.5.…”