Fritz John pp 364-365 | Cite as

# Commentary on [25]

## Abstract

The use of Fourier transforms to study partial differential equations with constant coefficients, in particular in constructing fundamental solutions, is classical, going back to Cauchy. For elliptic operators *P*(*x, D*) with variable coefficients, many authors had constructed parametrices, i. e., kernels *K*(*x, y*) depending on two points, and singular only at *x* = *y*, such that *P*(*x, D x*) *K*(*x,y*) - *δ*(*x* - *y*) = *S*(*x, y*) is a *C* ^{∞} function. In this paper John constructed fundamental solutions *K*(*x, y*), i. e., with *S*(*x, y*) ≡ 0, for general elliptic operators of arbitrary order with analytic coefficients. This is taken up again in Chapter 3 of his book, Plane Waves and Spherical Means Applied to Partial Differential Equations [37].

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