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

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 Rⁿ and let K_j be the singular integral operator on L² (Rⁿ, 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∈W2 ^w μ (Qw) ξ_w ⊗ ξ_w∥_C′ < ∞, where C′ is the dual of C₍₀₎ and {ξ_w: w ∈ W} is any orthonormal set, then K₁,..., 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.