A last theorem of Kalton and finiteness of Connes' integral

We connect finiteness of the noncommutative integral in Alain Connes' noncommutative geometry with the study of tensor multipliers from classical Banach space theory. For the Lorentz function space Lambda(1)(R-d) = {f is an element of L-0(R-d) : integral(infinity)(0)mu(s, f)(1 + log+(s(-1)))ds < infinity} where mu(s, f), s > 0, denotes the decreasing rearrangement of f, and log(+) denotes the positive part of log on (0, infinity), we prove using tensor multipliers the formula phi((1-Delta R-d)M--d/4(f)(1-Delta R-d)(-d/4)) = VolS(d-)(1)/d(2 pi)(d) integral R(d)f(x)dx, f is an element of Lambda(1)(R-d). Here -Delta R-d is the selfadjoint extension of minus the Laplacian on R-d, M-f denotes the operation of pointwise multiplication, the operator (1-Delta R-d)M--d/4(f)(1-Delta R-d)(-d/4) has a bounded extension which is a compact operator from the Hilbert space L-2(R-d) to itself, and phi is any continuous normalised trace on the ideal of compact operators on L-2(R-d) with series of singular values at most logarithmically diverge. The formula fails given only f is an element of L1(R-d), and previously had been shown by different methods for the smaller set of functions f is an element of L-2(R-d) that have compact support. We prove a similar formula for the Laplace-Beltrami operator on a compact Riemannian manifold without boundary. We discuss how the integral formula incorporates a last theorem of Nigel Kalton. We also extend to the case p = 2 a classical result of Cwikel on weak estimates p > 2 of operators of the form M-fg(-i del), f is an element of L-p(R-d), g is an element of L-p,L-infinity(R-d) where del is the gradient operator. (C) 2020 Elsevier Inc. All rights reserved.