Singular integral operators and norm ideals satisfying a quantitative variant of Kuroda's condition

Abstract

A norm ideal C is said to satisfy condition (QK) if there exist constants 0 < t < 1 and 0 < B < ∞ , such that ∥ X [ k ] ∥ C ⩽ Bk t ∥ X ∥ C for every finite-rank operator X and every k ∈ N , where X [ k ] denotes the direct sum of k copies of X. Let μ be a regular Borel measure whose support is contained in a unit cube Q in R n and let K j be the singular integral operator on L 2 ( R n , μ ) with the kernel function ( x j - y j ) / | x - y | 2 , 1 ⩽ j ⩽ n . Let { Q w : w ∈ W } be the usual dyadic decomposition of Q, i.e., { Q w : | w | = ℓ } is the dyadic partition of Q by cubes of the size 2 - ℓ × ⋯ × 2 - ℓ . We show that if C satisfies (QK) and if ∥ ∑ w ∈ W 2 | w | μ ( Q w ) ξ w ⊗ ξ w ∥ C ′ < ∞ , where C ′ is the dual of C ( 0 ) and { ξ w : w ∈ W } is any orthonormal set, then K 1 , … , K n ∈ C ′ . This is a very general obstruction result for the problem of simultaneous diagonalization of commuting tuples of self-adjoint operators modulo C .