A last theorem of Kalton and finiteness of Connes' integral

Abstract

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 spaceΛ1(Rd)={f∈L0(Rd):∫0∞μ(s,f)(1+log+⁡(s−1))ds<∞} where μ(s,f), s>0, denotes the decreasing rearrangement of f, and log+ denotes the positive part of log on (0,∞), we prove using tensor multipliers the formulaφ((1−ΔRd)−d/4Mf(1−ΔRd)−d/4)=VolSd−1d(2π)d∫Rdf(x)dx,f∈Λ1(Rd). Here −ΔRd is the selfadjoint extension of minus the Laplacian on Rd, Mf denotes the operation of pointwise multiplication, the operator (1−ΔRd)−d/4Mf(1−ΔRd)−d/4 has a bounded extension which is a compact operator from the Hilbert space L2(Rd) to itself, and φ is any continuous normalised trace on the ideal of compact operators on L2(Rd) with series of singular values at most logarithmically diverge. The formula fails given only f∈L1(Rd), and previously had been shown by different methods for the smaller set of functions f∈L2(Rd) 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 Mfg(−i∇), f∈Lp(Rd), g∈Lp,∞(Rd) where ∇ is the gradient operator.