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

摘要

A norm ideal C is said to satisfy condition (QK) if there exist constants 0 < t < 1 and 0 < B < infinity, such that parallel to X([k])parallel to(C) <= Bk(t)parallel to X parallel to(C) for every finite-rank operator X and every k epsilon N, where X-[k] denotes the direct sum of k copies of X. Let mu be a regular Borel measure whose support is contained in a unit cube Q in R" and let K-j be the singular integral operator on L-2(R-n, mu) with the kernel function (x(j)-y(j))/vertical bar x-y vertical bar(2), 1 <= j <= n. Let {Q(w) : w epsilon W} be the usual dyadic decomposition of Q, i.e., {Q(w) : vertical bar w vertical bar= l} is the dyadic partition of Q by cubes of the size 2(-l) x (...) x2(-l). We show that if C satisfies (QK) and if parallel to Sigma(w epsilon W)(2 vertical bar w vertical bar)mu(Q(w))xi(w)circle times xi(w)parallel to(C') < infinity, where C' is the dual of C-(0) and {xi(w) : w epsilon W} is any orthonormal set, then K-1,...,K-n epsilon C'. This is a very general obstruction result for the problem of simultaneous diagonalization of commuting tuples of self-adjoint operators modulo C. (c) 2005 Elsevier Inc. All rights reserved.