Let X C Rn and let K be a trace class operator on L2(X) with corresponding kernel K(x,y) € L2(X x X). An integral formula for tr K, proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of K into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be

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... sufficient for positive operators. The proofs make use of the boundedness of the Hardy-Littlewood maximal function on L2(R").