In several papers a result called Segal’s lemma appears. This result is Lemma 1.4 in I. E. Segal’s paper Construction of nonlinear local quantum processes I. in Annals of Math. 92 (1970) 462–481. Here I describe a slightly modified and more detailed formulation and proof of this result. I present a slighly different version of Segal’s proof. I finish this note with the original formulation and proof of Segal’s result.

Segal’s lemma. Let (M_{i}, μ_{i}) (i = 1, …, n) be separable measure spaces, and let T_{i} be an integral operator on L_{p}(M_{i}, μ_{i}) with positive kernel K_{i}, i.e. let

(T_{i}f)(x_{i}) = ∫_{M_{i}} K_{i}(x_{i}, y_{i})f(y_{i})μ_{i}( dy_{i}), x_{i} ∈ M_{i}, y_{i} ∈ M_{i}, i = 1, …, n,

where f(·) is a function in L_{p}(M_{i}, μ_{i}). Let T_{i} be a contraction from L_{p}(M_{i}, μ_{i}) to L_{q}(M_{i}, μ_{i}), for certain given 1 ≤ p < ∞ and 1 ≤ q < ∞ for each i. Then the algebraic tensor product T_{1} × ⋯ × T_{n} is a contraction from L_{p}(M_{1} × ⋯ × M_{n}, μ_{1} × ⋯ × μ_{n}) to L_{q}(M_{1} × ⋯ × M_{n}, μ_{1} × ⋯ × μ_{n}) i.e.

(∫_{M_{1}×···×M_{n}} |(T_{1} × ⋯ × T_{n}f)(x_{1}, …, x_{n}))|^{q} μ_{1}( dx_{1}) … μ_{n}( dx_{n}))^{1/q}

≤ (∫_{M_{1}×···×M_{n}} |f(y_{1}, …, y_{n})|^{p} μ_{1}( dy_{1}) … μ_{n}( dy_{n}))^{1/p}

for all f(y_{1}, …, y_{n}) ∈ L_{p}(M_{1} × ⋯ × M_{n}, μ_{1} × ⋯ × μ_{n}), where

(T_{1} × ⋯ × T_{n}f)(x_{1}, …, x_{n})
= ∫_{M_{1}×···×M_{n}} K_{1}(x_{1}, y_{1}) ⋯ K_{n}(x_{n}, y_{n}) f(y_{1}, …, y_{n})μ_{1}( dy_{1}) … μ_{n}( dy_{n}).

Proof. It is enough to prove the estimate of the Lemma for n = 2, because then the lemma follows for general n by simple induction. It suffices to prove this inequality only for functions f with the additional property f(y_{1}, y_{2}) ≥ 0 for all (y_{1}, y_{2}) ∈ M_{1} × M_{2}. Because of this additional condition we can omit the absolute value in the subsequent calculations, since we are working with non-negative functions.

Let us fix a function f(y_{1}, y_{2}) ∈ L_{p}(M_{1} × M_{2}, μ_{1} × μ_{2}) with y_{1} ∈ M_{1} and y_{2} ∈ M_{2} such that f(y_{1}, y_{2}) ≥ 0 for all (y_{1}, y_{2}) ∈ M_{1} × M_{2}. Then the function f(y_{1}, y_{2}) with a fixed point y_{1} ∈ M_{1} is a function in L_{p}(M_{2}, μ_{2}) for almost all y_{1} ∈ M_{1}. Hence we can define for all y_{1} ∈ M_{1} the L_{p}(M_{2}, μ_{2}) valued measurable function F (y_{1}) on M_{1} by the formula (F (y_{1}))(y_{2}) = f(y_{1}, y_{2}) for all y_{1} ∈ M_{1} and y_{2} ∈ M_{2}, and F (y_{1}) has the norm

∥F (y_{1})∥_{p} = (∫_{M_{2}} f(y_{1}, y_{2})^{p}μ_{2}( dy_{2}))^{1/p}. The function ∥F (y_{1})∥_{p}, y_{1} ∈ M_{1}, is in the Banach space L_{p}(M_{1}, μ_{1}), since ∫ ∥F (y_{1})∥_{p}^{p}μ_{1}( dy_{1}) = ∫_{M_{1}×M_{2}} f(y_{1}, y_{2})^{p}μ_{1}( dy_{1})μ_{2}( dy_{2}) < ∞. We also define the L_{q}(M_{2}, μ_{2}) valued function G(x_{1}) for all x_{1} ∈ M_{1} by the formula (G(x_{1}))(y_{2}) = ∫_{M_{1}} K_{1}(x_{1}, y_{1})f(y_{1}, y_{2})μ_{1}( dy_{1}). Then

(G(x_{1}))(y_{2}) = ∫_{M_{1}}K(x_{1}, y_{1})(F (y_{1}))(y_{2})μ_{1}( dy_{1}) = (∫_{M_{1}}K_{1}(x_{1}, y_{1})F (y_{1})μ_{1}( dy_{1})) (y_{2}), i.e. G(x_{1}) = ∫_{M_{1}} K_{1}(x_{1}, y_{1})F (y_{1})μ_{1}( dy_{1}) for all x_{1} ∈ M_{1}.

An upper bound will be given on ∥G(x_{1})∥_{p} = (∫_{M_{2}}[(G(x_{1}))(y_{2})]^{p}μ_{2}( dy_{2}))^{1/p} for a fixed point x_{1} ∈ M_{1}. In this estimate the following result will be applied. If (M, μ) is a separable measure space, X is a Banach space, H(y_{1}), y_{1} ∈ M, is a measurable non-negative function on M, U(y_{1}), y_{1} ∈ M, is an X valued measurable function on M, then ∥∫_{M} H(y_{1})U(y_{1})μ( dy_{1})∥ ≤ ∫_{M} H(y_{1})∥(U(y_{1}))∥μ( dy_{1}).

With the choice (M, μ) = (M_{1}, μ_{1}), X = L_{p}(M_{2}, μ_{2}), H(y_{1}) = K_{1}(x_{1}, y_{1}), U(y_{1}) = F (y_{1}) for all y_{1} ∈ M_{1} this result yields that

∥G(x_{1})∥_{p} = ∥∫_{M_{1}} K_{1}(x_{1}, y_{1})F (y_{1}) μ_{1}(dy_{1})∥_{p}
≤ ∫_{M_{1}} K_{1}(x_{1}, y_{1})∥F (y_{1})∥_{p}μ_{1}( dy_{1}) = (T_{1}(∥F ∥_{p})))(x_{1}).

This inequality together with the contraction property of T_{1} yields the following estimate on the L_{q}(M_{1}, μ_{1})-norm of the function ∥G(x_{1})∥_{p} with arguments x_{1} ∈ M_{1} by the L_{p}(M_{1}, μ_{1})-norm of the function ∥F (y_{1})∥_{p} with arguments y_{1} ∈ M_{1}.

∥∥G(·)∥_{p}∥_{q} ≤ ∥T_{1}(∥F (·)∥_{p})∥_{q} ≤ ∥∥F (·)∥_{p}∥_{p} .

Let us also observe that since ∥F (y_{1})∥_{p} = (∫_{M_{2}} |f(y_{1}, y_{2})|^{p}μ_{2}( dy_{2}))^{1/p} for almost all y_{1} ∈ M_{1}

∥∥F (·)∥_{p}∥_{p} = (∫_{M_{1}} (∫_{M_{2}} |f(y_{1}, y_{2})|^{p}μ_{2}( dy_{2})) μ_{1}( dy_{1}))^{1/p}

= (∫_{M_{1}×M_{2}} |f(y_{1}, y_{2})|^{p}μ_{1}( dy_{1})μ_{2}( dy_{2}))^{1/p} = ∥f(y_{1}, y_{2})∥_{p}.

Define the function u(x_{1}, y_{2}) = ∫_{M_{1}} K_{1}(x_{1}, y_{1})f(y_{1}, y_{2})μ_{1}( dy_{1}) which equals

(G(x_{1}))(y_{2}) for all x_{1} ∈ M_{1} and y_{2} ∈ M_{2}, and let us estimate the number

S(u(·)) = (∫_{M_{1}×M_{2}} |(∫_{M_{2}} K_{2}(x_{2}, y_{2})u(x_{1}, y_{2})μ_{2}( dy_{2}))|^{q} μ_{1}( dx_{1})μ_{2}( dx_{2}))^{1/q} .

The application of the contraction property of T_{2} for f(x_{2}) = u(x_{1}, x_{2}) with a fixed x_{1} ∈ M_{1} yields the estimate

∫_{M_{2}} |(∫_{M_{2}} K_{2}(x_{2}, y_{2})u(x_{1}, y_{2})μ_{2}( dy_{2}))|^{q} μ_{2}( dx_{2}) ≤ (∫_{M_{2}} |u(x_{1}, x_{2})|^{p}μ_{2}( dx_{2}))^{q/p}

= (∫_{M_{2}} |(G(x_{1}))(x_{2})|^{p}μ_{2}( dx_{2}))^{q/p} = ∥G(x_{1})∥_{p}^{q}.

Hence

S(u(·)) ≤ (∫_{M_{1}} ∥G(x_{1})∥_{p}^{q}μ_{1}( dx_{1}))^{1/q} = ∥ ∥G(·)∥_{p}∥_{q} ≤ ∥ ∥F (·)∥_{p}∥_{p} = ∥f(y_{1}, y_{2}∥_{p}.

On the other hand,

S(u(·)) = (∫_{M_{1}×M_{2}} |(∫_{M_{1}×M_{2}} K_{2}(x_{2}, y_{2})K_{1}(x_{1}, y_{1})f(y_{1}, y_{2})μ_{1}( dy_{1})μ_{2}( dy_{2}))|^{q}

μ_{1}( dx_{1})μ_{2}( dx_{2}))^{1/q} = ∥(T_{1} × T_{2})f(y_{1}, y_{2})∥_{q},

and the statement of the lemma holds.

Here I write down Segal’s result in its original form. I present both its formulation and its proof.

LEMMA 1.4. Let M_{i} (i = 1, …, n) be separable measure spaces, and let T_{i} be an integral operator on L_{p}(M_{i})) with positive kernel K_{i}, which is a contraction from L_{p}(M_{i}) to L_{q}(M_{i}), for certain given p and q for each i. Then the algebraic tensor product T_{1} × ⋯ × T_{n} is a contraction from L_{p}(M_{1} × ⋯ × M_{n}) to L_{q}(M_{1} × … M_{n}).

PROOF. It suffices by associativity to treat the case n = 2. Now if B is any separable Banach space, and if L_{p}(M_{1}, B) denotes the space of all strongly measurable B-valued functions F on M_{1} which are p-th power integrable, with the norm ∥F ∥ = (∫ ∥F (x)∥^{p} dx)^{1/p}, then the operator T′_{1}: F → G, where G(x) = ∫ K_{1}(x, y)F (y) dy, exists and is a contraction from L_{p}(M_{1}, B) to L_{q}(M_{1}, B). For the mapping y → K_{1}(x, y)F (y) is easily seen to be strongly measurable from M_{1} to B, for each x; and

∥G(x)∥ ≤ ∫ K_{1}(x, y)∥F (y)∥ dy,

i.e., ∥G(·)∥ ≤ T_{1}(∥F (·)∥), so that ∥∥G(·)∥_{B}∥_{q} ≤ ∥T_{1}(∥F (·)∥)_{B}∥_{q} ≤ ∥∥F (·)∥_{B}∥_{p}. This shows the absolute integrability of the integral defining G(x) almost everywhere, and gives the estimate ∥T′_{1}∥ ≤ 1.

In addition, the operation T′′_{2} from L_{q}(M_{1}, B) to L_{q}(M_{2}, B′), where B′ is another separable Banach space and T_{2} is a contraction from B to B′, defined by the equation (T′′_{2}F)(x) = T_{2}F (x), F ∈ L_{q}(M_{1}, B), is easily seen to be a contraction. Now taking B as L_{p}(M_{2}) and B′ as L_{q}(M_{2}), and making the natural identifications of L_{p}(M_{1}, B) with L_{p}(M_{1} × M_{2}) and of L_{q}(M_{1}, B′) with L_{q}(M_{1} × M_{2}) which are justified by the Fubini theorem, it follows that the contraction T′′_{2}T′_{1} extends the algebraic tensor product T_{1} × T_{2}; the latter is therefore a contraction, as stated.