Operator-Valued Triebel–Lizorkin Spaces

Abstract

This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on R d . As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel-Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

Mapping properties of operator-valued pseudo-differential operators

“…See [60] for the proof of this interpolation. When α = 0 and 1 ≤ p < ∞, it is proved in [60] that F 0,c…”

“…Note that the mixture Triebel-Lizorkin space F α p (R d , M) coincides with L p (N ) when α = 0 and 1 < p < ∞. On the other hand, the arguments of [15] are based on a careful analysis of the L 2 and BMO cases, while our proof in the case p = 1 (the main case) relies entirely on the atomic decomposition of F α,c 1 (R d , M) obtained in [60].…”

“…Compared to the standard proof of the boundedness on Hardy spaces of a usual Calderón-Zygmund operator with a commutative or noncommutative convolution kernel, the present proof is much subtler and more technical. We need a careful analysis of a pseudo-differential operator on smooth (sub)atoms given in [60]. Now we state the main results of this paper.…”

“…T c σ f (s) = We will mainly work on the column operators T c σ and establish their mapping properties on local Hardy spaces h c p (R d , M) (introduced in [59]) and inhomogeneous Triebel-Lizorkin spaces F α,c p (R d , M) (introduced in [60]). The main part of the proof concerns the case p = 1 for both kinds of spaces.…”

“…In the next section, we introduce some elementary notation and knowledge on noncommutative L p -spaces, and the definitions of local Hardy spaces in [59] and inhomogeneous Triebel-Lizorkin spaces in [60]. Then we present the smooth atomic decompositions of these spaces obtained in [60]. In section 2, we give the concrete definitions and some easily deduced useful facts on operator-valued pseudo-differential operators.…”