Abstract
Let P(D) be a hypoelliptic pdo with constant coefficients and let E(M) (and(WM,∞ b′) be the weighted spaces of C∞-functions (resp. of distributions) defined by Palamodov (resp. Gelfand/Shilov), where M is a radially symmetric weight function, eventually satisfying some mild technical conditions. Then P(D) has no continuous linear right inverse in E(M). If the index of hypoellipticity w.r.t. some x is greater than 1 and is minimal in a certain sense, then P(D) has no right inverse in (WM,∞)b′.
If P(D) is elliptic however, then a continuous linear right inverse for P(D) exists in (WM,∞)b′.
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Langenbruch, M. Partial differential equations without solution operators in weighted spaces of (generalized) functions. Manuscripta Math 56, 353–374 (1986). https://doi.org/10.1007/BF01180774
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DOI: https://doi.org/10.1007/BF01180774