An operator-valued T1 theory for symmetric CZOs

Abstract

We provide a natural BMO-criterion for the L 2 -boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the L 2 -boundedness of the commutators [R j , b] whenever b belongs to the Bourgain's vector-valued BMO space, where R j is the j-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.

An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces

“…This indicates a continuing demand for checkable criteria for the boundedness of at least special classes of non-convolution operators, as long as the full analogue of the scalar-valued T (1) and T (b) theorems seems out of reach. In a recent work [14], Hong, Liu and Mei achieve such a result, in the very style of a T (1) theorem, for operator-valued singular integrals with a certain symmetry assumption, satisfied in particular by all even operators. Under natural assumptions, their result gives the L p (R d ; X)-boundedness of these operators when p = 2 and X = H is a Hilbert space, and a weaker substitute result (replacing either the domain or the target with a suitable non-commutative Hardy space) when p ∈ (1, ∞) \ {2} and X is non-commutative L p -space.…”

“…We stress that all other assumptions of Theorem 1.3 are essentially the same as in any other operator-valued T (1) theorem in the literature (like [13,15,19]), and the key novelty is the condition (1.4) inspired by [14]. This improves on all previous results on the level of general UMD spaces by means of replacing their more complicated BMO-type spaces by the plain BMO(R d ; L (X)), which is just the classical BMO space with absolute values replaced the norm in L (X); it achieves this at the cost of requiring the additional symmetry imposed by the equality in (1.4).…”

“…In this paper, we obtain an extension of the Hong-Liu-Mei [14] result to all Banach spaces X with the unconditionality property of martingale differences (UMD). Our result is a pure L p estimate for all p ∈ (1, ∞), without the need of substitute Hardy spaces, and it is obtained in the maximal generality of Banach spaces (namely, UMD spaces) in which such results could be hoped for.…”

“…For our purposes, we use the operator-valued dyadic representation theorem from [13]. (Hong, Liu and Mei [14] adapt the approach of [16] instead; this would also have been a relevant alternative here. )s The rest of this paper is structured as follows.…”

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

“…The theory has developed rapidly in further directions over the last decade. Other interesting results and applications can be found in [3,5,19,20,25,31,50,51] and the references therein.…”