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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Nirenberg, L. (1985). Commentary on [25]. In: Moser, J. (eds) Fritz John. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-5406-5_24
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DOI: https://doi.org/10.1007/978-1-4612-5406-5_24
Publisher Name: Birkhäuser, Boston, MA
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