Celebratio Mathematica

Y. Fourès and I. E. Segal: “Causality and analyticity,” Trans. Amer. Math. Soc. 78 (1955), pp. 385–405. MR 69401 Zbl 0064.36805 article

We treat an abstract version of a linear mechanical system, demonstrating a connection between causality in the system and analyticity of its gain function. This function is in general merely Lebesgue measurable; it is suitably analytic if and only if the system is causal in the sense that the future cannot influence the past. This result is for the “continuous” case in which the output is globally a continuous function of the input. Systems described by partial differential equations with constant coefficients will not in general be of this type, and we are led thereby to an examination of the influence of causality in the “discontinuous” case.

Y. Choquet-Bruhat and I. E. Segal: “Solution globale des équations de Yang–Mills sur l’univers d’Einstein,” C. R. Acad. Sci. Paris Sér. I Math. 294 : 6 (1982), pp. 225–230. MR 654042 Zbl 0501.58023 article

Y. Choquet-Bruhat, S. M. Paneitz, and I. E. Segal: “The Yang–Mills equations on the universal cosmos,” J. Funct. Anal. 53 : 2 (1983), pp. 112–150. MR 722506 Zbl 0535.58022 article

Global existence and regularity of solutions for the Yang–Mills equations on the universal cosmos \( \tilde{\mathbf{M}} \), which has the form \( R^1 \times S^3 \) for each of an 8-parameter continuum of factorizations of \( \tilde{\mathbf{M}} \) as \( \textit{time}\times\textit{space} \) are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the \( \textit{field}\oplus\textit{potential} \) in \[ L_{2,r}(S^3)\oplus L_{2,r+1}(S^3) ,\] where \( r \) is an integer \( > \) 1 and \( L_{2,r} \) denotes the class of sections whose first \( r \) derivatives are square-integrable; if \( r= 1 \), the problem is soluble locally in time. When \( r \) is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in \( C^{\infty}(\tilde{\mathbf{M}}) \). These results extend earlier work and approaches. Solutions of the equations on Minkowski space-time \( M \), extend canonically (modulo gauge transformations) to solutions on \( \tilde{\mathbf{M}} \) provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on \( \mathbf{M}_0 \), follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is \( O(|x_0|^{-5}) \), where \( x_0 \) is the Minkowski time coordinate; analysis solely in \( \mathbf{M}_0 \), earlier yielded the estimate \( O(|x_o|^{-2}) \) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang–Mills equuions in \( \mathbf{M}_0 \), is finite, in fact absolutely convergent.

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